Emergent Braided Matter of Quantum Geometry
We review and present a few new results of the program of emergent matter as braid excitations of quantum geometry that is represented by braided ribbon networks. These networks are a generalisation of the spin networks proposed by Penrose and those in models of background independent quantum gravit...
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Bilson-Thompson, S. Hackett, J. Kauffman, L. Wan, Y. 2019-02-18T11:29:11Z 2019-02-18T11:29:11Z 2012 Emergent Braided Matter of Quantum Geometry / S. Bilson-Thompson, J. Hackett, L. Kauffman, Y. Wan // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 106 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 83C45; 83C27; 81T99; 81V25; 20F36; 18D35; 20K45; 81P68 DOI: http://dx.doi.org/10.3842/SIGMA.2012.014 https://nasplib.isofts.kiev.ua/handle/123456789/148393 We review and present a few new results of the program of emergent matter as braid excitations of quantum geometry that is represented by braided ribbon networks. These networks are a generalisation of the spin networks proposed by Penrose and those in models of background independent quantum gravity theories, such as Loop Quantum Gravity and Spin Foam models. This program has been developed in two parallel but complimentary schemes, namely the trivalent and tetravalent schemes. The former studies the braids on trivalent braided ribbon networks, while the latter investigates the braids on tetravalent braided ribbon networks. Both schemes have been fruitful. The trivalent scheme has been quite successful at establishing a correspondence between braids and Standard Model particles, whereas the tetravalent scheme has naturally substantiated a rich, dynamical theory of interactions and propagation of braids, which is ruled by topological conservation laws. Some recent advances in the program indicate that the two schemes may converge to yield a fundamental theory of matter in quantum spacetime. This paper is a contribution to the Special Issue “Loop Quantum Gravity and Cosmology”. The full collection is available at http://www.emis.de/journals/SIGMA/LQGC.html. SBT is grateful to the Ramsay family for their support through the Ramsay Postdoctoral Fellowship. JH is grateful to his Thesis Advisor Lee Smolin for his discussion and critical comments. YW is in debt to his Supervisor Mikio Nakahara for his constant support and generosity. YW is also supported by “Open Research Center” Project for Private Universities: matching fund subsidy from MEXT, Japan. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Emergent Braided Matter of Quantum Geometry Article published earlier |
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Emergent Braided Matter of Quantum Geometry Bilson-Thompson, S. Hackett, J. Kauffman, L. Wan, Y. |
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Emergent Braided Matter of Quantum Geometry |
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emergent braided matter of quantum geometry |
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Bilson-Thompson, S. Hackett, J. Kauffman, L. Wan, Y. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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We review and present a few new results of the program of emergent matter as braid excitations of quantum geometry that is represented by braided ribbon networks. These networks are a generalisation of the spin networks proposed by Penrose and those in models of background independent quantum gravity theories, such as Loop Quantum Gravity and Spin Foam models. This program has been developed in two parallel but complimentary schemes, namely the trivalent and tetravalent schemes. The former studies the braids on trivalent braided ribbon networks, while the latter investigates the braids on tetravalent braided ribbon networks. Both schemes have been fruitful. The trivalent scheme has been quite successful at establishing a correspondence between braids and Standard Model particles, whereas the tetravalent scheme has naturally substantiated a rich, dynamical theory of interactions and propagation of braids, which is ruled by topological conservation laws. Some recent advances in the program indicate that the two schemes may converge to yield a fundamental theory of matter in quantum spacetime.
|
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1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/148393 |
| citation_txt |
Emergent Braided Matter of Quantum Geometry / S. Bilson-Thompson, J. Hackett, L. Kauffman, Y. Wan // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 106 назв. — англ. |
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2025-11-25T21:22:36Z |
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2025-11-25T21:22:36Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 014, 43 pages
Emergent Braided Matter of Quantum Geometry‹
Sundance BILSON-THOMPSON :1, Jonathan HACKETT :2, Louis KAUFFMAN :3
and Yidun WAN :4
:1 School of Chemistry and Physics, University of Adelaide, SA 5005, Australia
E-mail: sundance.bilson-thompson@adelaide.edu.au
:2 Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Waterloo, ON N2L 2Y5, Canada
E-mail: jonathanhackett@gmail.com
:3 Department of Mathematics, University of Illinois at Chicago,
851 South Morgan Street, Chicago, Illinois 60607-7045, USA
E-mail: kauffman@uic.edu
:4 Open Research Centre for Quantum Computing, Kinki University,
Kowakae 3-4-1, Higashi-osaka 577-0852, Japan
E-mail: ywan@alice.math.kindai.ac.jp
Received August 31, 2011, in final form March 12, 2012; Published online March 24, 2012
http://dx.doi.org/10.3842/SIGMA.2012.014
Abstract. We review and present a few new results of the program of emergent matter
as braid excitations of quantum geometry that is represented by braided ribbon networks.
These networks are a generalisation of the spin networks proposed by Penrose and those in
models of background independent quantum gravity theories, such as Loop Quantum Gravity
and Spin Foam models. This program has been developed in two parallel but complimen-
tary schemes, namely the trivalent and tetravalent schemes. The former studies the braids
on trivalent braided ribbon networks, while the latter investigates the braids on tetravalent
braided ribbon networks. Both schemes have been fruitful. The trivalent scheme has been
quite successful at establishing a correspondence between braids and Standard Model par-
ticles, whereas the tetravalent scheme has naturally substantiated a rich, dynamical theory
of interactions and propagation of braids, which is ruled by topological conservation laws.
Some recent advances in the program indicate that the two schemes may converge to yield
a fundamental theory of matter in quantum spacetime.
Key words: quantum gravity; loop quantum gravity; spin network; braided ribbon net-
work; emergent matter; braid; standard model; particle physics; unification; braided tensor
category; topological quantum computation
2010 Mathematics Subject Classification: 83C45; 83C27; 81T99; 81V25; 20F36; 18D35;
20K45; 81P68
1 Introduction
1.1 An invitation to emergent matter of quantum geometry
What is spacetime? What is matter? Physicists and philosophers have pondered these questions
for centuries. In fact, an ultimate goal of modern physics is to find a unified answer for both
questions. Recently, in order to answer these questions, a novel approach towards emergent1
‹This paper is a contribution to the Special Issue “Loop Quantum Gravity and Cosmology”. The full collection
is available at http://www.emis.de/journals/SIGMA/LQGC.html
1Here we mean coexisting quantum geometry and matter because our program indicates that a background
independent quantum gravity theory may have built-in matter as topological excitations of the quantum geometry
described by the theory.
mailto:sundance.bilson-thompson@adelaide.edu.au
mailto:jonathanhackett@gmail.com
mailto:kauffman@uic.edu
mailto:ywan@alice.math.kindai.ac.jp
http://dx.doi.org/10.3842/SIGMA.2012.014
http://www.emis.de/journals/SIGMA/LQGC.html
2 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
matter as topological excitations of quantum geometry has been put forward and extensively
developed [12, 13, 14, 15, 38, 39, 40, 41, 45, 46, 67, 93, 101, 103]. Provided with the results of
two recent papers [38, 39] along this course, an article that offers a precise review and outlook
of this research line seems timely.
A brief historical account is as follows. In 2005, Bilson-Thompson proposed a topological mat-
ter model, the helon model [12], which is based on the preon models of Harari and Shupe [44, 88]
and is more elementary than the Standard Model (SM) of particles by interpreting the elemen-
tary particles as braids of three ribbons. At the time it was proposed, the helon model took the
form of a combinatoric game rather than a rigorous theory. In this model, the integral twists of
ribbons of braids are interpreted as the quantized electric charges of particles. The permutations
of twists on certain braids naturally account for the color charges of quarks and gluons. This
model incorporates a simple scheme of the color interaction and the electro-weak interaction
with lepton and baryon number manifestly conserved. It may also be able to account for the
three generations of elementary fermions.
In 2006, Bilson-Thompson, Markopoulou and Smolin [15] coded the helon model in certain
background independent Quantum Gravity models such as Loop Quantum Gravity (LQG) and
Spin Foam (SF) models, by identifying helons with emergent topological excitations of embedded
trivalent spin networks that label the states in LQG. Hereafter this will be called the trivalent
scheme. Developments of the trivalent scheme [13, 14, 40, 67] allowed the helon model to be
used as a dictionary between the 3-strand braids of embedded trivalent spin networks and the
SM particles. The trivalent scheme led to a new perspective; instead of treating the helon model
as yet another model of elementary particles, one can encode it in LQG and SF models to make
a theory of both spacetime and matter. The dynamics governing particle interactions would
then be a consequence of the dynamics of the discrete building blocks of quantum spacetime.
In this setting, matter is emergent from quantum spacetime, and the corresponding low energy
effective theories may give rise to general relativity coupled with quantum fields.
Unfortunately, results on the stability of braided states [15] in the trivalent scheme strongly
suggested that the dynamics of spacetime would allow particle propagation, but not interac-
tions. In effect, braids in the trivalent scheme are “too stable”. To address this issue, and
because of the geometrical correspondence between framed 4-valent spin networks and 3-space,
a 4-valent scheme was developed [41, 45, 46, 93, 101, 102, 103]. In the 4-valent scheme, the
topological structures that can potentially be identified with particles are also 3-strand braids,
each of which is formed by the three common edges of two adjacent 4-valent nodes of embedded,
framed 4-valent spin networks. The 4-valent scheme gives rise to forms of braid propagation and
interaction that are analogous to the dynamics of particles. Nevertheless, the lack of sufficient
super-selection rules over an enormous zoo of 3-strand braids in the 4-valent scheme withholds
a Rosetta Stone that maps the braids to the SM particles. On the other hand, the 4-valent
braids may be more elementary, high-energy entities whose low energy limit produces the SM
particles [101, 102].
Very recently, two papers by Hackett [38, 39] and work by Bilson-Thompson reported here
in Section 4.3, provide a framework that may encode both the trivalent and 4-valent schemes.
This would allow the economical reproduction of SM particle states that occurs in the trivalent
scheme, and the propogation and interactions that occur in the 4-valent scheme to be combined
into a single theory.
As a historical remark, the idea that matter is topological defects of spacetime is an old dream
that dates back to 1867 when Lord Kelvin proposed that atoms were knots in the ether [98].
Kelvin’s idea failed largely due to the limited knowledge of atomic and subatomic structure at
the time. Nevertheless, this dream has persisted in physicists thereafter. Various proposals of
topological matter have arisen as physicists deepen and broaden their recognition of nature. An
example is the topological Geon model due to Wheeler and others [18, 27, 30, 78, 105] but the
Emergent Braided Matter of Quantum Geometry 3
geons therein were unstable and classical. To make stable geons [30], Finkelstein invented the
notion of topological conservation laws that also led to advances in condensed matter physics,
e.g., topologically conserved excitations in the sine-Gordon theory. Finkelstein’s idea had not
been compatible with quantum gravity until the recent work by Markopoulou et al. [56, 64, 68]
that motivated our work. Analogously, certain condensed matter systems have quasi-particles
as collective modes, e.g., phonons and rotons in superfluid He4. A recent example in condensed
matter physics is a unification scheme due to Wen et al. [37, 58], where gauge theories and
linearised gravity appear to be low energy effective descriptions of a new phase, the string-net
condensate of lattice spin systems.
In the rest of the Introduction, we briefly introduce concepts and ideas that underpin our
approach to emergent matter. We leave main discussions on the trivalent and 4-valent schemes
to other sections.
1.2 Noiseless subsystems
To appreciate the ideas of emergent matter of embedded, framed spin networks, one needs to
understand two notions, namely noiseless subsystems and spin networks. Let us address the
former first. Noiseless subsystems, put forward in quantum information and computation for
quantum error correction [47, 51, 55, 106], are subsets of states of a quantum system that
are preserved under the evolution algebra of the system, and hence protected from any error.
Markopoulou et al. adopted the idea of noiseless subsystems to solve the problem of the low
energy limit in background independent quantum gravity theories [28, 56, 66].
Background independence brings in difficulties that make taking the low energy limit of
a background independent theory of quantum gravity [64] a big open issue, although attempts
have been made in various approaches of quantum gravity. In LQG, e.g., [5, 9, 87, 95] use the
method of coherent states, which are, according to quantum physics, the quantum states closest
to classical ones. For another example, in SF models, [1, 2, 3, 21, 77] utilize n-point correlations
functions, which is reasonable because all semiclassical observables are either correlation func-
tions or their derivatives. In view of this, Markopoulou and Kribs [56, 64, 66] proposed a new
way to resolve this issue by looking for conserved quantities in background independent theories
of gravity. Since the aforementioned noiseless subsystems are conserved states under the evolu-
tion algebra of a quantum system, there should be conserved quantities associated with them.
This is why one can adapt the method of noiseless subsystems to find conserved quantities of
quantum geometry in a large class of background independent theories.
The first application of noiseless subsystems in LQG [28] gives a possible explanation of
black hole entropy and how symmetries can emerge from a diffeomorphism invariant formulation
of quantum gravity. The noiseless substructures of braided ribbon networks, which appear to
encode the particle states in models of quantum gravity such as LQG, are braids whose associated
conserved quantities are called reduced link invariants [13, 15, 38, 39, 40].
1.3 Spin networks
Penrose invented spin networks as a fundamental discrete description of spacetime [75, 76]; later,
Rovelli and Smolin found a more generalized version of spin networks to label the states in LQG
Hilbert space [85]. Although the context of spin networks in this article is mainly LQG and its
path integral formulation, SF models, it will be clear that our results do not really depend on
these models but find their natural home in a generalisation of Penrose’s version. Spin networks
also arise in lattice gauge theories [19, 20, 34, 52] and topological field theories [23, 31, 70, 99],
which are not discussed here.
4 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
1.3.1 Penrose’s spin networks
Penrose noticed the fundamental incompatibility between General Relativity and Quantum
Physics, the problem of the concept of continuum, and the divergences in quantum field theories.
He thought that resolving this incompatibility demands a discrete notion of spacetime at the
Planck scale, where the classical notion of spacetime is no longer valid. Consequently, the con-
cept of time and space gives way to a more fundamental notion, the microscopic causal relation
between quantum events2. Knowing that spin (or angular momentum) is intrinsic and charac-
teristic to both quantum systems and classical spacetime, Penrose used combinatoric graphs,
consisting of lines intersecting at vertices, to represent the fundamental states of spacetime.
Each line in a graph is labeled by a spin, an integer or half integer. Hence, such a graph is
called a spin network. Later on, [69] showed that the classical 3-dimensional angles of space can
be recovered from trivalent spin networks. Note that these spin networks are unembedded and,
opposed to those in LQG, are a direct construction of fundamental quantum spacetime, rather
than obtained from quantizing spacetime or General Relativity by any means.
1.3.2 Spin networks from LQG
LQG is a non-perturbative, canonical quantization of General Relativity3. The background
independence of General Relativity actually does not leave any room for perturbative quantiza-
tion [4, 6, 83, 89]. LQG assumes a pd` 1q-dimensional differential manifold M with a foliation
M “ Σ ˆ R, without metric but merely a differential structure with a Lie algebra4 valued
connection 1-form field. We take d “ 3.
Our approach does not directly depend on any specific techniques and results of LQG but is
only inspired by them, so we do not review the technical settings and quantization procedure in
LQG.
In LQG, the states are spin networks, which are graphs embedded in Σ (Fig. 1). An edge e is
a flux line, labelled by an irreducible representation je of a Lie group (usually SUp2q or SOp3q).
A vertex is labelled by an intertwiner that is the invariant tensor of the labels on the edges
meeting at the vertex.
Figure 1. A portion of a generic spin network. Some labels are omitted.
LQG has produced many physical results, which are extensively reviewed in [82, 90, 92]. The
result most relevant to this review is the evidence that space at the Planck scale is discrete be-
cause spin network states are eigenstates of operators corresponding to geometric measurements
such as area and volume. For example the area operator  acting on a 2-dimensional surface S
has the spectrum
ÂrSs |Γ〉9l2p
ÿ
iPtΓYSu
a
jipji ` 1q |Γ〉 ,
2The underlying philosophy is relationalism, as opposed to reductionism, reviewed in [82, 83, 89].
3Other non-perturbative approaches to quantum gravity also exist; however, here we focus on LQG.
4Usually sup2q or sop3q.
Emergent Braided Matter of Quantum Geometry 5
where lp is the Planck length, and Γ is a spin network that has no vertices but only edges
in S.5 Likewise, the intertwiners on the nodes of a spin network in a region determine the
3-volume of the region. This fundamental discreteness resolves the singularity problem and also
eliminates ultraviolet divergences, as it provides a natural cutoff at the Planck scale to the
physical spectrum of the theory.
In general, a vertex of a spin network can have any valence greater than two, the number
of edges meeting at the vertex, as seen in Fig. 1. We may consider a basis of spin networks
with definite valences, i.e., spin networks respectively with three, four, and higher valences,
such that a generic LQG state is a linear combination of these basis states. one may also think
that trivalent spin networks may be sufficient to provide a complete basis that spans all spin
networks. This is plausible and is suggested by Rovelli [83, 85] for the case of SUp2q and SOp3q.
The trivalent spin networks represent a basis of LQG state space. That is, associated with a spin
network Γ is a state |Γ〉, and for two such states |Γ〉 and |Γ1〉, 〈Γ|Γ1〉 “ δΓΓ1 . Spin network labels
on edges and nodes are representations of the group elements, labeling the graphs in the classical
configuration space, and the corresponding intertwiners.
Trivalent spin networks have difficulty in representing 3-space because their nodes have zero
3-volume. But each 4-valent node yields a 3-volume6 [25, 61]. [65] also suggests one may carry
this correspondence to any higher valence. In this review, we shall study both trivalent and
4-valent spin networks.
Trivalent spin networks acquire dynamics by evolving under the action of the Hamiltonian
constraint operator of LQG, which also helps to realize the 4D diffeomorphism invariance of
the theory. The well-accepted form of the Hamiltonian constraint acts only on vertices (Fig. 2)
Thiemann [94, 96, 97]; hence, it behaves as a local move that evolves a spin network state to
another. In fact, LQG has a path integral formulation, SF models, casting the evolution of spin
networks in a systematic, covariant way [83].
Figure 2. Sketch of of the Hamiltonian constraint acting on a vertex. Irrelevant details are ignored.
In many studies of SF models [10, 33], spin networks and their histories are unembedded,
combinatoric graphs. The key difference between embedded and unembedded spin networks is
that the edges in the former can knot, braid, and link. The role of these knots, braids, and links
has been a big open issue. Nevertheless, in this article we will show the correspondence between
some of these topological structures of embedded spin networks and matter. [42] offers another
perspective.
According to SF models [7, 8, 81, 83], trivalent spin networks evolve under two more moves,
shown in Fig. 3(a) and (b), including the one in Fig. 2.
Figure 3. (a): expansion and contraction move. (b): exchange move.
Trivalent spin networks are the dual skeletons of triangulations of 2D surfaces (Fig. 4), in
which a node (an edge) is dual to a 2-simplex, i.e., a triangle (a side of the triangle). This is
5If Γ has edges on S, a degeneracy arises and calls for regularization methods to obtain the correct spectrum [83].
6This correspondence is at the Planck level. Whether it holds in a continuum limit is still open.
6 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
Figure 4. Duality between a 2-simplex and a trivalent vertex.
consistent with 2D because the vertices of trivalent spin networks have zero 3-volume. Hence,
the evolution moves of trivalent spin networks are dual to the Pachner moves [74] that relate tri-
angulations of the same surface (Fig. 5). This topological interpretation indicates that summing
over histories of the evolution of certain fundamental building blocks produces a quantum space-
time. That is, one can build pn ` 1qD spacetime from the evolution of n-valent spin networks.
This picture is partly implemented in SF models and fully implemented in another formulation
of quantum gravity, Group Field Theories (GFT) [71, 72, 73].
A subtlety exists, however. A spin network with structureless edges and nodes contains less
information than an exact dual of a 2D simplicial triangulation, in which two triangles can be
glued along a side in two opposite ways. To remedy this, trivalent vertices and their edges
should be framed to disks and ribbons respectively (Fig. 5(a)). We refer to these (embedded)
framed spin networks as (braided) ribbon networks7. We also name the evolution moves
on (braided) ribbon networks the adapted Pachner moves (Fig. 58). The criteria for a legal
(a) (b) (c)
Figure 5. Pachner moves. (a) 1 Ñ 3. (b) 2 Ñ 2.
2D Pachner move is that the triangles before and after the move bound a 3D tetrahedron
(Fig. 5). Interestingly, Major and Smolin [16, 62, 63, 91] suggested that when LQG contains
a non-zero cosmological constant, the corresponding spin networks become framed: edges and
vertices become ribbons and disks in the trivalent case, and become tubes and spheres in the
4-valent case. We shall loosely refer to the (embedded) framed 4-valent spin networks as 4-valent
(braided) ribbon networks. Likewise, 4-valent ribbon networks evolve under a set of 4-valent
adapted dual Pachner moves (Section 3.4). Moreover, labels of framed spin networks turn out
to be representations of the quantum group, e.g., SUqp2q. In GFTs, the spin networks arise as
framed but unembedded.
Our third ansatz requires the embedded 4-valent spin networks to evolve under adapted dual
Pachner moves. These moves are adapted and are thus different from those in SF models. In
many studies of spin foam models, the spin networks and spin foam histories are taken to be
abstract, or unembedded. Our results do not directly apply to these models, as the topological
structures our results concern arise from the embedding of the graphs in a topological three
manifold. But neither do such models give dynamics for states of loop quantum gravity, which
are embedded. A question therefore is how to build spin-foam like dynamics for embedded spin
networks. The adapted dual Pachner moves to be defined in Section 3.4 for the 4-valent braided
ribbon networks will give an answer.
7For embedded spin networks, the duality is in general only local, i.e., restricted to a single node. This
restriction is unnecessary in 2D because a braided ribbon network is dual to a topological manifold globally [24].
8This figure is adopted from [67] with the author’s permission.
Emergent Braided Matter of Quantum Geometry 7
1.4 Three Ansatzes
All of the above encourages a unification scheme in which matter emerges as topological excita-
tions of the braided ribbon networks. We thus posit that braided ribbon networks are the most
fundamental entities of nature9, beyond LQG and SF Models and actually in accordance with
Penrose’s original proposal of spin networks, with, however, a great deal of generalization. More
precisely, the unification scheme and its results we have obtained are based on three ansatzes:
1. Spacetime is pre-geometric and discrete at the fundamental scale.
2. The discrete space is a superposition of basis states represented by braided ribbon net-
works.
3. The braided ribbon networks evolve under a set of local moves.
These ansatzes are independent of the spacetime dimension. Here, we consider only p3 ` 1q-
dimensional spacetime, consistent with the observable universe10. The braided ribbon networks
are in general graced with spin network labels, which are otherwise removed in this article
because our results obtained so far do not depend on them. The set of local moves include only
the adapted dual Pachner moves; however, it may extend to incorporate other moves in future.
We hope that classical spacetime would exist as certain limit of the pre-geometric history of the
evolution of these graphs.
2 The trivalent scheme
The trivalent scheme is the most natural case in which to embed the helon model [12], which
provides a mapping between braided network states and the fermions and bosons of the SM.
Here we will briefly discuss the helon model in terms of abstract braided structures, and how
those structures may be mapped to particle states and quantities such as hypercharge, baryon
number and lepton number. We will then discuss how these braids may be characterised by
appropriately chosen topological invariants. At the end of this section we will discuss how such
braided structures may be embedded in framed spin networks. The possibility of a unified
treatment of trivalent and tetravalent networks is discussed in Section 4.3.
2.1 The helon model
In the helon model, the subcomponents of SM particles occurring in certain preon models are
replaced by a framed braid on three strands. The strands are joined to two surfaces of non-zero
size (which we may think of as one disk at each end), and we will suppose there is a way of
distinguishing these end surfaces so that the braids have a “top” and “bottom” (as shown in
the left of Fig. 6). For brevity, let us refer to these end surfaces as “caps”. Such a braid (all
three strands and the two caps) constitutes a two-dimensional surface and we will immediately
restrict our attention to orientable surfaces. The three strands between the caps can in general be
distinguished by their relative crossings, and it becomes meaningful to speak of the first, second,
and third strand. An entire braid on three strands will represent a single type of fermion or
boson.
9Since all the information due to embedding can be characterized purely combinatorially, to be pointed out
later, embedded spin networks (or combinatorial spin networks with embedding data) are more general than the
unembedded ones [49, 50].
10Why our spacetime is (3+1)-dimensional is a big open issue in physics and philosophy. The earliest reasonable
argument, due to Ehrenfest in 1917 [29], was that atoms are unstable unless in 3D. Recently, anthropic arguments
also arose. Nevertheless, all these arguments sound a posteriori, and a theory that naturally gives rise to our
p3` 1q spacetime is still missing.
8 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
The individual strands in a braid can carry twists, as mentioned above, and we identify
right-handed and left-handed twists as positive and negative electric charges. This is the basis
of the name ‘helon’ (evoking the image of a helix) for a single strand. The requirement that we
consider only orientable surfaces restricts us to twists that are multiples of 2π, so let us interpret
a twist through ˘2π on any helon (that is, any strand) as an electric charge of ˘e{3. Twists
through ˘4π, ˘6π, and so on will not be considered, as we shall see below that extra twists can
be regarded as equivalent to crossings. Besides the helons carrying twist through ´2π and `2π,
there is a third type to consider, carrying no twist. We will denote the three types of helons
as H´, H`, and H0 respectively.
Adapting a scheme originally devised by Harari [44] and Shupe [88], we construct braids
composed of three H`s (corresponding in electric charge to positrons), three H´s (corresponding
to electrons), a single H` and two H0s (corresponding to anti-down quarks), a single H´ and
two H0s (corresponding to down quarks), a single H0 and two H`s (corresponding to up quarks),
a single H0 and two H´s (corresponding to anti-up quarks), and three H0s (corresponding to
neutrinos). This scheme reproduces the fermions of the first generation of the SM, with no extra
particles. Braids consisting of a mix of H` and H´ helons are not allowed when constructing
fermions (but are in fact used to construct the gluons). We identify the permutations of braids
containing two helons of one type, and one of another (e.g. H`H`H0) with the three colour
charges of QCD, and write the helons in ordered triplets for convenience (this is, of course,
simply notation). The quarks are then as follows (subscripts denote colour):
H`H`H0 (uB) H`H0H` (uG) H0H`H` (uR)
H0H0H` (dB) H0H`H0 (dG) H`H0H0 (dR)
H´H´H0 (uB) H´H0H´ (uG) H0H´H´ (uR)
H0H0H´ (dB) H0H´H0 (dG) H´H0H0 (dR)
while the leptons are
H`H`H` pe`q H0H0H0 (νe) H´H´H´ (e´).
Note that in this scheme we have identified neutrinos, but not anti-neutrinos. This has
occurred because while the H´ may be regarded as the anti-partner to the H`, there is no anti-
partner to the H0 helon. This apparent problem will be turned to our advantage in Section 2.3.
2.2 Topological invariants of trivalent braids
The braids on three strands may be characterised by an invariant called the pure twist number,
first described in [14]. This is a triple of real numbers which count the twist remaining on each
strand when the braid is deformed such that all crossings are removed. This is possible because
any braid on n strands can be written as a product of the generators σ1, . . . , σn´1, and their
inverses, where σi crosses the ith strand in front of the pi` 1qth strand, and σ´1
i crosses the ith
strand behind the pi` 1qth strand. The sequence of σ factors defining a braid is called its braid
word. Clearly, in the case of braids on three strands we are only concerned with σ1, σ2 and
their inverses. The generators induce permutations of the strand ordering. The generator σ1
induces the permutation P1,2 (that is, it swaps the 1st and 2nd strands), while the generator σ2
induces the permutation P2,3. Notice also that the same permutation is induced by a generator
or its inverse, σ´1
i . Therefore the generators contain more information than the permutations –
in particular the direction of the crossing is specified by the generators (as shown in Fig. 6).
It is convenient to define a standardised form for framed braids in which all the twist is
isotoped to the top of the braid. Then we can write rr, s, tsB where B is an ordinary braid word
and rr, s, ts is a triple of multiples of half-integers which catalogue the twists in the ribbons.
We shall call this triple of numbers the twist-word. Thus a framed braid on three strands is
Emergent Braided Matter of Quantum Geometry 9
Figure 6. A braid, with the bottom cap denoted by a rectangular block (left) and the generators of the
braid group on three strands (right).
Figure 7. Swapping crossings for twist.
Figure 8. Converting a braid generator into twists.
completely specified by the twist word and the braid word. Since these braids are taken to
exist on networks embedded in a manifold of three dimensions, we also allow the braids to be
deformed such that the top end of the braid is flipped over (effectively feeding it through the
strands). This allows us to undo crossings and hence simplify the braid structure, as illustrated
in Fig. 7 where we show how a disk with three untwisted ribbons emerging from it (but with two
ribbons crossed), can be converted into a disk with uncrossed ribbons and oppositely-directed
half-twists in the upper and lower ribbons by flipping the disk over (in the illustration, a negative
half-twist in the lower ribbon and positive half-twists in the upper ribbons). In Fig. 8, we show
the same process performed on a disk whose (crossed) upper ribbons have been bent downwards
to lie besides and to the left of the (initially) lower ribbon. This configuration is nothing other
than a framed braid on three strands corresponding to the generator σ1 (with cap at the bottom
omitted). Keeping the ends of the ribbons fixed as before and flipping over the cap so as to
remove the crossings now results in three unbraided (i.e. trivially braided) strands, with a positive
half-twist on the leftmost strand, a positive half-twist on the middle strand, and a negative half-
twist on the rightmost strand. Hence the associated twist-word is r12 ,
1
2 ,´
1
2 s. This illustrates
that in the case of braids on three strands, each of the crossing generators can be isotoped to
uncrossed strands bearing half-integer twists. By variously bending the top two ribbons down
to the right of the bottom ribbon, and/or taking mirror images, and flipping the cap at the top
of the braid appropriately we can determine that the generators may be exchanged for twists
according to the pattern:
σ1 Ñ
„
1
2
,
1
2
,´
1
2
, σ´1
1 Ñ
„
´
1
2
,´
1
2
,
1
2
,
σ2 Ñ
„
´
1
2
,
1
2
,
1
2
, σ´1
2 Ñ
„
1
2
,´
1
2
,´
1
2
. (2.1)
10 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
All braids on three strands can be built up as products of these generators. It should therefore
be clear to the reader that we may entirely eliminate the crossings from a braid on three strands.
When we do so we uncross the strands one generator at a time (hence permuting them by the
permutation Pσi associated with the crossing σi being eliminated) and introduce the twists
indicated in equation (2.1). In general, this means that we iterate the process
ra1, a2, a3srb1, b2, b3sσiσj . . . σm Ñ ra1 ` b1, a2 ` b2, a3 ` b3sσiσj ¨ ¨ ¨σm
Ñ Pσipra1 ` b1, a2 ` b2, a3 ` b3sqrx, y, zsσj ¨ ¨ ¨σm, (2.2)
where rx, y, zs is the twist-word associated to σi (as listed in equation (2.1), when i is specified),
and Pσipra, b, csq is the permutation of ra, b, cs arising from σi. The process given in equation (2.2)
is iterated until the braid word becomes the identity.
We shall refer to the form of a braid in which all the crossings have been exchanged for twists
as the pure twist form. The list of three numbers which characterise the twists on the strands in
the pure twist form will be referred to as the pure twist-word. The pure twist-word is of interest
because it is a topological invariant (since it is obtained when a braid is reduced to a particular
simple form i.e. all crossings removed). The iteration of equation (2.2) therefore describes an
algorithm for calculating the pure twist-word of any three-strand braid. This algorithm was
first described in [14], where further details and examples can be found.
One criticism that has been levelled at this research program is that braids appear somewhat
ad-hoc and unnatural, however it follows from the discussion above that the use of braids in the
helon model is a convenience, and that each such braid can be related to a topological invariant
which is independent of the way the structure is drawn. When we speak of a particular braid
corresponding to a type of fermion, we are therefore simply referring to an equivalence class of
topological structures and using one distinctive member of that equivalence class to refer to the
entire class. We will see shortly how such general topological structures may be embedded in
trivalent spin networks.
It should be noted in the interests of clarity, however, that it is too early to assign specific
choices of braiding structure (and hence specific equivalence classes) to any SM particles. Al-
though the electric and colour charges are determined by the twist assignments listed above
as permutations of H`s, H´s, and H0s, the task of matching topological characteristics to
properties such as spin, generation, hypercharge, etc. is still a work-in-progress. This is dis-
cussed below. Furthermore, a way of modelling transitions between generations using braids
(and hence describing Cabbibo-mixing and neutrino oscillations) has not yet been found. In
light of this, it seems premature to assign definite topological structures to the gauge bosons,
and to the fermions of each generation – although such an assignment is a definite goal of this
research program. The braids shown in Figs. 6 and 9 represent our best attempts at the present
time of matching particle properties to the braid multiplication. Their role in the wider models
discussed here is yet to be fully determined.
2.3 Quantum numbers of particle states
Given the discussion above, it is clear that the node at the “top” of a braid defines an internal
direction which is unaffected by rotating the diagram of a braid. The pure twist form is found
by working through each crossing in turn, starting with the crossing closest to the node at the
“top”. Therefore rotating a diagram of a braid on a page does not change which equivalence class
that braid belongs to (and hence we may always perform such a rotation without compromising
the validity or usefulness of our model). Let us then pick a certain braid with untwisted strands
(i.e. composed entirely of H0s) to act as a basic structure for one generation of fermions. Unlike
rotation, taking the mirror image of such a braid will produce a member of a different topological
equivalence class, in general. Such a braid and its mirror image can be regarded as left-handed
Emergent Braided Matter of Quantum Geometry 11
Figure 9. The fermions formed by adding zero, one, two or three charges to a neutral braid. Charged
fermions come in two handedness states each, while ν and ν come in only one each. (3) denotes that
there are three possible permutations, identified as the quark colours. The bands at top and bottom
represent the nodes and connections to the external network (“the rest of the Universe”).
and right-handed fermions. By adding one, two, or three twisted strands (H`s or H´s, but not
both at the same time), we construct left-handed and right-handed fermions with overall positive
and negative charge. This is illustrated in Fig. 9. Notice that for any given non-zero magnitude
of charge, there are four states (left-handed and right-handed particle, and left-handed and right-
handed antiparticle, where positive and negative electric charge distinguish between particles
and antiparticles), but for the case of zero charge there exist only two states (left-handed and
right-handed). We identify these with the neutral left-handed fermion (neutrino) and right-
handed antifermion (anti-neutrino).
The helon model allows us to describe electric charge and colour charge in terms of the
topological structure of braids, but we can similarly describe hypercharge and isospin. We
begin by assigning a scalar quantity β to braids, such that β “ `1 for the braids on the top
row of Fig. 9, and β “ ´1 to the braids on the bottom row. This effectively distinguishes
between fermions with a net positive and net negative charge (and establishes a definition of
the equivalent quantity for the neutrinos). Of course β provides only a very crude distinction.
To rectify this shortcoming we will define a new quantity, given by one-third the number of
“more positive” helons, minus one-third the number of “less positive” helons. We shall denote
this quantity by the symbol Ω. To clarify, H` helons are considered “more positive” than H0
helons, which are “more positive” than H´ helons. If NpH`q is the number of H` helons,
NpH0q the number of H0s and NpH´q the number of H´s within a triplet, and remembering
that H` and H´ helons never occur within the same braided triplet, we may write
Ω “ β
ˆ
1
3
NpH`q `
1
3
NpH´q ´
1
3
NpH0q
˙
.
Hence we have Ω “ `1 for the e`, Ω “ `1{3 for the u, Ω “ ´1{3 for the d, and Ω “ ´1 for
the anti-neutrino. For the electron, anti-up, down, and neutrino the signs are reversed. With
this definition, noting that NpH0q “ 3 ´ pNpH`q ` NpH´qq and the total electric charge of
a fermion is given by
Q “ β
ˆ
1
3
NpH`q `
1
3
NpH´q
˙
,
it is easy to show that
Q “
1
2
pβ ` Ωq . (2.3)
For the quarks and anti-quarks Ω reproduces the SM values of strong hypercharge, while for the
leptons β reproduces the SM values of weak hypercharge. We also observe that for quarks β{2
12 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
reproduces the values of the third component of strong isospin, while for leptons Ω{2 reproduces
the values of the third component of weak isospin (in short, the roles of β and Ω as isospin
and hypercharge are reversed for quarks and leptons). With these correspondences the Gell-
Mann–Nishijima relation Q “ I3` Y {2 for quarks may trivially be derived from equation (2.3).
This construction does not distinguish between left-handed and right-handed fermions, and so
it does not match all values of weak isospin and hypercharge in the SM. The reader should
remember that the values of isospin and hypercharge in the SM are assigned in an entirely ad
hoc manner, to reflect the observed asymmetry of the weak interaction, while in the helon model
these values are constructed. It is possible that with further work another set of topologically-
defined quantities may be found which will enable the helon model to describe the asymmetry
of the weak interaction.
2.4 Interactions and embedding in trivalent networks
Although the helon model does not provide a dynamical framework, it is possible to represent
the electroweak interactions in a simple manner, by forming the braid product of several braids.
The product of two braid diagrams is accomplished by adjoining the strands of the second braid
to the corresponding strands of the first braid, as in Fig. 10. The product of two braids, A1
and A2, is therefore a single braid, the braid word of which is the concatenation of the generators
in the braid words of A1 and A2. It therefore follows that if A2 is the top-to-bottom mirror
image of A1 the product of A1 and A2 will contain no crossings. The reader can easily check
that this is true for any braid in the top row of Fig. 9, and the corresponding braid in the
bottom row. Such a process suggests particle-antiparticle annihilation. More generally, when
the product of two braids is formed, and then decomposed into several braids, the twists on the
strands may be shuffled so that the outgoing braids are different from the incoming braids. In
this way an interaction such as u ` e´ Ñ d ` νe may be modelled, with the structure of the
intermediate braid product suggesting the structure of a boson. A more detailed discussion of
how particle interactions can be modelled using braid products is given in [12].
Figure 10. The product of two braids.
Having established the possibility of a mapping between braids and SM particles, we now
turn to the matter of embedding the helon model in trivalent spin networks. In this scheme,
a braid occurs as a single node from which three strands (helons) emerge. That is, the cap at
the top of a braid is identified as a trivalent node. The strands braid around each other, and
then join to the rest of the network at three other nodes (see Fig. 11). It was shown in [15]
that such embedded braids can be characterised by the linking of the edges of the braid. If we
take a diagram of a topological substructure in a trivalent network, and trace along the left and
right edges of each strand – discarding any unlinked closed loops – we obtain a diagram of a link
corresponding to that structure. This construction is illustrated in Fig. 12. The link obtained
may in general be simplified by applying a series of Reidemeister moves to the diagram, to
obtain a “reduced link”. The reduced link is an invariant of a braid, so that it does not change
no matter how much the network of nodes and strands is deformed within the manifold in which
Emergent Braided Matter of Quantum Geometry 13
Figure 11. A framed braid on three strands (right), equivalent to a braided substructure in a network
(left).
it is embedded. Likewise, the standard trivalent evolution moves (1-3 expansion move and 2-2
exchange move) do not change the reduced link. The definition of the reduced link provides
another way of seeing that, as claimed above, rotating a diagram of a braid on a page does not
change its equivalence class, since a rotated reduced link is equivalent to an unrotated reduced
link, under the Reidemeister moves.
Figure 12. Finding the reduced link of a topological structure within a ribbon network.
It is here that the difficulty regarding interactions occurs. In order for any process analogous
to the braid product to take place, two braids must be able to combine and then split apart.
If the outgoing particle states are to be different from the incoming particle states, then the
associated reduced links should also be different. But as noted above, it was shown that this is
not possible under the 1-3 and 2-2 evolution moves. While this ensures that individual leptons
and quarks will not decay (since the associated reduced link cannot be changed), it also appears
to prohibit any non-trivial particle interactions. It is possible that this issue could be addressed
by postulating a new evolution move. However in Section 4.3 we wish to explore an alternative
approach, in which we construct a correspondence between braided networks in the trivalent
and tetravalent cases.
3 The 4-valent scheme
Having seen the results and limitations of the trivalent scheme, we shall move on to the 4-valent
scheme, which, although begun as an extension to the trivalent one, turned out to be a rich,
fully dynamical theory of braids of 4-valent braided ribbon networks. The 4-valent scheme has
been studied and cast in two parallel but complementary formalisms, namely the graphic and
the algebraic formalisms. While the former offers a more intuitive picture, the latter provides
a more convenient playground for theorem-proving and investigating the properties of braids
and their dynamics. Consequently, in this section, we shall adopt the graphic formalism for
illustrative purposes only but the algebraic formalism extensively.
14 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
3.1 4-valent braided ribbon networks
4-valent braided ribbon networks are framed 4-valent spin networks embedded in R3. The local
duality between a node of a 4-valent braided ribbon network and a tetrahedron allows us to
represent a node by a 2-sphere with four circular punctures and its edges by tubes that are
welded at the punctures. This is depicted in Fig. 13. We assume that each node is rigid and
non-degenerate, such that it can only be translated and rotated, its punctures where its edges
are attached are fixed, and no more than two edges of a node are co-planar11. Because a tubular
edge is dual to a triangular face of a tetrahedron, a tube implicitly carries three racing stripes
that record the twist of a tube, which dictates how two tetrahedra are glued on a common face.
Section 4 gives a further discussion of the racing stripes.
(a) (b)
A
B
C
D
A
B
C
D
(c)
A
C
B
DRotation axis
(d)
(e)
(f)
B
1 in unit of /3p
B
1 in unit of /3p
=
B+1
=
Figure 13. (a) is a tetrahedron and its dual node; a tube has three racing stripes. In a diagram, a node
can have two states, (b) ‘ and (c) a. (f) shows an unambiguous label of the two right-handed π{3 twists
in (d) and (e).
In a projection, we simplify the tube-sphere notation in Fig. 13(a) to a circle-line notation
((b) or (c)), in which solid lines piercing through the circle represent tubes that are above in
the 3D notation, while a dashed line represents two lines as the tubes that are under. There is
no information loss in doing so because one can always arrange a node in the either the state
in Fig. 13(b) or (c) by isotopy before projecting it. Owing to the local duality between a node
and a tetrahedron and the symmetry on the latter, if we grab an edge of a node, the other three
edges are still on an equal footing, inducing a rotation symmetry w.r.t. the edge being grabbed
(Fig. 13(b) & (c)), which will be explained shortly. Therefore, in a projection one can assign
states to a node w.r.t. its rotation axis. If the rotation axis is an edge in the back (front), the
node is in state ‘ (a) and is called a ‘-node (a-node), as in Fig. 13(b) ((c)).
Nonetheless, if the three edges other than the rotation axis of a node must be distinguished,
e.g., to be seen shortly in the case of the nodes of a braid, a degeneracy of each state of the
node arises. Consider Fig. 13(b), the node is in ‘ w.r.t. edge B; rotating the node about B
produces permutations of the other edges and puts them in different configurations. Ignoring
the twists and crossings of the edges that are created by rotations, which will be our next topic,
it takes a full rotation for the node to roll back to its original configuration. In configurations
pA, D, Cq12, pD, C, Aq, and pC, A, Dq, the node is in ‘ w.r.t. B, whereas in pA, C, Dq,
pC, D, Aq, and pD, A, Cq, the node flips to a. That is, each state has a 3-fold degeneracy; or
in other words, each state is a triplet. The six sub-states in total record the full configuration
of the node w.r.t. the rotation axis.
11The reader may wonder why our discussion of braided ribbon graphs relies so much on the classical structure
of 3D space. As the model is not assumed to take place at some large scale where a 3D tangent space has emerged,
this is an important epistemological point and can be answered in the following way: One should regard these
networks as holding combinatorial conditions that happen to be expressed in the language of three-dimensional
space. It would make the statement of the model overly complex to state such combinatorial conditions abstractly
entirely in terms of graphs and orderings, but this can be done in principle. We realize that this situation is not
entirely satisfactory.
12This notation records the order of the three edges from top to bottom w.r.t. the rotation axis (see Fig. 13(b)).
Emergent Braided Matter of Quantum Geometry 15
If we denote a full rotation by 2π, then the amount of rotation keeping a node within a state
triplet is 2π{3; however, a π{3 rotation causes to jump back and forth between two state triplets.
Note again that these types of rotations are not the ones with a rigid metric but rather discrete
and purely topological. Details of these rotations will be studied in Section 3.3.1.
Naturally, an edge can be twisted discretely. The discussion above of rotations shows that
the smallest distinguishable twist is π{3; higher distinguishable twists in the projection must be
integral multiples of π{3. Fig. 13(d)-(f) shows how the handedness and hence the sign of a twist
is unambiguously defined. This unambiguity appears to be more lucid with the help of racing
stripes, as in Fig. 14, which purports that one can define the handedness of twists on a tube
without referring to either end of the tube. Bearing this in mind, we will stick to our simpler
notation without racing stripes throughout Section 3.
a
b
c
a’
b’
c’
a
b
c a’
b’
c’
This end fixed
2p/3
This end fixed
2p/3
Thumb
Thumb
a
b
c a’
b’
c’
Figure 14. Unambiguity of the handedness of twists. A tube is represented by two circles as ends
connected by three lines as racing stripes. Twists are depicted by the crossings of the racing stripes.
3.2 Braids
The graphic notation enables us to find an interesting type of topological excitations of 4-valent
braided ribbon networks, namely 3-strand braids or 4-valent braids, defined in Fig. 15(a).
A 3-strand braid is made of two end-nodes that share three strands, which are generically
T
l
T
a
T
b
T
c
S
l
S
r
T
r
X
(a)
e
l
e
r
T
a’
T
b’
T
c’
u u
-1
d d
-1
(b)
+2e e’
b
(c)
Figure 15. (a) A generic 3-strand braid. (b) The four generators of X. (c) An example.
braided and twisted. Each end-node has an external edge attached elsewhere in the net-
work. This definition is unambiguous because a braid can always be arranged horizontally as
in Fig. 15(a). We disallow the strands of a braid to tangle with any other edges in the network,
including the braid’s external edges.
A 3-strand braid is characterized by an 8-tuple tTl, Sl, Ta, Tb, Tc, X, Sr, Tru that consists of
a pair of end-node states pSl, Srq, a pair of external twists pTl, Trq, a crossing sequence X,
and a triple of internal twists pTa, Tb, Tcq. pSl, Srq is valued in t`,´u. S’s inverse is ´S or S̄.
X codes the braiding (from left to right) of the three strands; it must be an element in B3,
the group of ordinary braids of three strands, and is thus generated by the four crossings in
Fig. 15(b). All twists are valued in Z in units of π{3.
An X of order n “ |X|, the number of crossings, reads X “ x1x2 ¨ ¨ ¨xi ¨ ¨ ¨xn, where xi P
tu, u´1, d, d´1u is the i-th crossing. We also assign `1 or ´1, the crossing number, to each
generator according to its handedness, i.e., u “ d “ 1 and u´1 “ d´1 “ ´1. Thus, an xi is an
abstract crossing in a multiplication but `1 or ´1 in a summation. Suppose X 1 is a segment
16 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
of X, then |X| ě |X 1| and three cases exist: Assume X 1 “ x1x2 ¨ ¨ ¨xn, 1) if X “ X 1xn`1 ¨ ¨ ¨ , we
write X 1 ď X, 2) if X “ xi1xi2 ¨ ¨ ¨ximX
1, we write X 1 ű X, and 3) otherwise, we write X 1 ă X.
Note that ď‰ű unless X “ X 1. X clearly induces a permutation σX of the three strands of
a braid, which takes value in the permutation group S3 “ t1, p1 2q, p1 3q, p2 3q, p1 2 3q, p1 3 2qu.
A braid can take either of these two algebraic forms,
Sl
Tl
rpTa, Tb, TcqXs
Sr
Tr
, (3.1)
Sl
Tl
rXpTa1 , Tb1 , Tc1qs
Sr
Tr
, (3.2)
where the triple pTa, Tb, Tcq on the left of the X and the triple pTa1 , Tb1 , Tc1q on the right are
related by pTa, Tb, TcqσX “ pTa1 , Tb1 , Tc1q, indicating that σX is left-acting on the triple of internal
twists. Equivalently, we have pTa, Tb, Tcq “ σ´1
X pTa1 , Tb1 , Tc1q, where σ´1
X is the inverse of σX and
is right-acting. For instance, the braid in Fig. 15(c) can be written as `
0rp0, 2, 0qu´1ds
´
0 or
`
0ru´1dp2, 0, 0qs
´
0 , with induced permutations σu´1d “ p3 1 2q and σ´1
u´1d
“ p2 3 1q. Besides,
pTa, Tb, Tcq “ pTd, Te, Tf q means Ta “ Td, Tb “ Te, and Tc “ Tf , and we also have pTa, Tb, Tcq ˘
pTd, Te, Tf q “ pTa ˘ Td, Tb ˘ Te, Tc ˘ Tf q.
We shall see that different types of braids pattern the corresponding 8-tuples differently. For
a trivial braid, its X is trivial, and σX “ 1; hence, the generic notation uniquely boils down to
Sl
Tl
rTa, Tb, Tcs
Sr
Tr
.
Four-valent braids are noiseless topological excitations of braided ribbon networks [15, 64,
68]. We emphasize that 4-valent braids are 3D structures that are better studied in their 2D
projections, which are called braid diagrams hereafter. In fact, we are dealing with equivalence
classes of braids because each braid is equivalent to an infinite number of braids under a set of
isotopy moves (to be introduced soon), whose 2D projections relate equivalent braid diagrams.
Equivalence classes of braids are in one-to-one correspondence with those of braid diagrams [93,
102, 103]; hence, we need not distinguish a braid from its diagrams. That is, by a braid we
mean all its isotopic braids, and we study this braid by its equivalence class of braid diagrams.
In what follows, we may use braids and braid diagrams interchangeably.
The generators of the X of a 4-valent braid obey the well-known braid relations of B3, which
are
udu´1 “ d´1ud, u´1du “ dud´1, udu “ dud. (3.3)
We assume in any X, these relations have been applied, such that, e.g., udu´1d´1 should have
been written as udu´1d´1 “ d´1udd´1 “ d´1u by the first relation above. This assumption
ensures that each 4-valent braid we study has the least number of crossings among all the braids
related to it by equation (3.3). As such, we consider braid diagrams with the same number of
crossings and equivalent crossing patterns as the same!
3.3 Equivalence moves
As aforementioned, equivalent braids and hence equivalent braided ribbon networks are related
by a set of local equivalence moves that act on the nodes and edges of a network without altering
the diffeomorphism class of the embedding of the network. One type of these moves are the well-
known Reidermeister moves [79], which are translations of nodes and continuous deformation
of ribbons [102, 103]. We shall focus on the other type, discrete rotations of nodes, which are
peculiar to braided ribbon networks.
3.3.1 π{3-rotations: generators of rotations
With respect to any of its four edges, a node admits discrete, purely topological rotation sym-
metries that are not those with a rigid metric and do not affect the diffeomorphism class of the
Emergent Braided Matter of Quantum Geometry 17
(a)(b)
-1 +1
+1
+1 -1
(c)
z z+1 -1
-1
a c
a
c
b
z
c
b a
b
p/3
z
p/3
z
Figure 16. (b) & (c) are results of (a) by rotating the ‘-node in (a) w.r.t. edge z in two directions
respectively. Red dots are assumed connected elsewhere and fixed.
embedding of the node. Section 3.1 points out that each end-node state is a triplet preserved by
a 2π{3 rotation but mapped to the other state by a π{3 rotation. Because two consecutive π{3
rotations comprise a 2π{3 rotation, π{3 rotations are generators of all possible discrete rotations
of a node w.r.t. an edge of the node. We now show how π{3 rotations affect a subgraph, in
which a node is rotated. In either a ‘ or a a state, a node can be rotated in two opposite
directions. Fig. 16 shows the case where a node is in a ‘-state w.r.t. its rotation axis before the
π{3 rotation is done. The effect of a π{3 rotation on a a-node is the mirror image of Fig. 16.
A π{3 rotation of a node always flips the state of the node, creates a crossing of two edges of
the node, and generates a ˘1 twist on the rotation axis and a ¯1 twist on each of the other
three edges.
Since our key topological structures are braids, we may ask how the rotations act on a braid.
In this case, e.g. in Fig. 17, we only allow an external edge of a braid to be a rotation axis [103].
In Fig. 17 the left braid is equivalent to the right one with one less crossing. This observation
motivates a classification of braids as if they are isolated structures. We name a few important
definitions, whose details are referred to [93, 103]. A braid is reducible if it is equivalent to
a braid with fewer crossings and otherwise irreducible. A braid is left-, right-, or two-way-
reducible if it can be reduced by rotations on either its left, right, or both end-nodes. A braid
equivalent to a trivial braid, i.e., a braid without crossings, is completely reducible. The
algebraic form of a rotation on a braid as a whole can be found in [46].
1 2
+1
z
p/3
z
1 2 z
2
p/3
z2
-1
-1
-1
Figure 17. Two braids are equivalent under a π{3-rotation of node 2.
As equivalence moves, the rotations and translations should have associated invariants. Al-
though such an invariant exist on arbitrary sub-graphs of braided ribbon networks, in which
equivalence moves act [93, 103], we are more interested in restricting the sub-graphs to be 4-
valent braids only. In this case, a braid bears two invariants of discrete rotations, namely its
effective twist and effective state [46],
Θ “ Tl ` Tr `
c
ÿ
i“a
Ti ´ 2
|X|
ÿ
i“1
xi, χ “ p´q|X|SlSr. (3.4)
Moreover, we shall see that both quantities in equation (3.4) are also conserved quantities of
braid interactions.
3.3.2 Braid representations
As an equivalence class, a braid should be studied in terms of certain convenient representatives
of the class. Each braid bears a unique representation, in which the braid has twist-free
18 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
external edges13. Here is why. Were there two braids with twist-free external edges in one
equivalence class, they must be related by rotations, which is contradictory to the condition
that any rotation creates twists on an external edge. This representation is convenient for the
study of braid dynamics in general.
All irreducible braids in a class must have the same number of crossings; otherwise, the
longer ones should either be reducible or belong to another class. Therefore, irreducible braids
in a class must have the smallest number of crossings. We henceforth call an irreducible braid
an extremum of a class. A braid is in an extremal representation if it is represented by its
extremum. The extremal representation is not unique because a braid has an infinite number
of extrema [46, 102]. An extremal representation is called a trivial representation if the
associated extremum has no crossings.
3.4 Dynamics: evolution moves
To define the models our results apply to, we have to choose a set of dynamical moves. In SF
and other models of spin networks, dual Pachner moves are a common choice [7, 81, 83], as seen
in Section 1.3.2 for the trivalent scheme. We now discuss the dual Pachner moves on 4-valent
braided ribbon networks.
Let us fix a non-singular topological manifold M and choose a triangulation of it in terms
of tetrahedra embedded in M whose union is homeomorphic to M. Any such simplicial tri-
angulation of M has a natural dual that is a framed 4-valent graph embedded in M. The
framing determines how the tetrahedra are glued on their faces. Thus, a Pachner move on the
triangulation should result in a local move in the framed graph, i.e., the dual Pachner move.
Nevertheless, not every embedding of a framed four valent graph in M is dual to a trian-
gulation of M. Examples of obstructions to finding the dual include the case of braids (e.g.,
Fig. 15). This is an embedding of a graph that could not have arisen from taking the dual of
a regular simplicial triangulation of M. We note that these obstructions are local, in the sense
that a sub-graph of the embedded graph could be cut out and replaced by another sub-graph
that would allow the duality to a triangulation of M.
The question then is how to define the dual Pachner moves on sub-graphs that are not dual
to any part of a triangulation ofM. The answer is that we do not. We thus have the basic rule:
Basic rule14: The evolution moves on 4-valent braided ribbon networks are the dual Pachner
moves that are allowed only on subgraphs which are dual to a 3-ball.
The Pachner moves and the dual evolution moves that obey the basic rule on the 4-valent
braided ribbon networks, namely the 2 Ñ 3 (3 Ñ 2) and 1 Ñ 4 (4 Ñ 1) moves, are respectively
explained by Figs. 18 and 19 and their captions. The result of a dual Pachner move is unique
up to equivalence moves, i.e., the discrete rotations and adapted Reidemeister moves. The basic
rule we posit, which dictates the legitimacy of a dual Pachner move, actually boils down to the
following conditions.
Condition 3.1. A legal dual Pachner move falls into the following three cases.
1. Two nodes allow a 2 Ñ 3 move iff they share only one edge and can be put by equivalence
moves in the same state with the common edge twist-free pFig. 18pbq or its mirror imageq.
2. Three nodes allow a 3 Ñ 2 move iff they and their edges can be set by equivalence moves
as either Fig. 18pcq or its mirror image, where there is a contractible loop made of three
twist-free edges.
13This uniqueness is defined modulo the ordinary braid relations, as explained in Section 3.2.
14This rule and other possible rules are discussed in detail in [67].
Emergent Braided Matter of Quantum Geometry 19
3 2a
2 3a
yx
1
2
a
c
b
a’
b’
c’
a a’
c c’
b b’
3
3 2a
2 3a
(b)
(c) (d)
(a)
Figure 18. The 2 Ø 3 Pachner moves between (a) two and (b) three tetrahedra. The dual 2 Ø 3 move
between (c) two nodes in the ‘-state and (d). (c) and (d) are dual to (a) and (b). The red and dashed
green lines in (b) outline the three tetrahedra. Green edges in (d) are generated by the dual 2 Ñ 3 move.
If the two nodes in (c) are in the a-state, the result of a 2 Ñ 3 move is the left-right mirror image of (d).
(a) 4 1a
1 4a
a
b
c
d
c
b
a
d
Dual x 1 2
3
4
(b) (c)
Figure 19. (a) The 1 Ø 4 Pachner moves between one and four tetraheda. The dual 1 Ø 4 moves are
between (b) and (c). Red edges in (c) are the generated by the dual 1 Ñ 4 move.
3. A 1 Ñ 4 move is always doable; however, a 4 Ñ 1 on four nodes is doable iff the four
nodes together with their edges can be arranged in the form of either Fig. 19pcq or its
mirror image, in which there is a contractible loop and all common edges are twist-free.
In view of Condition 3.1, a 2 Ñ 3 move is illegal on the two end-nodes of the braid in
Fig. 15(c), so is a 3 Ñ 2 move on any three nodes that contain this pair, which makes this
braid stable under single evolution moves. This generalises to the statement that any nontrivial
braid with internal twists is stable under single evolution moves. The stable braids are thus
considered noiseless subsystems [93] and local excitations with conserved quantities [41, 46].
Section 3.7 has more on the stability and locality of 4-valent braids.
The dual Pachner moves in SF models contain only the permutation relations of edges, which
are sufficient only for triangulations. In contrast, our dual Pachner moves are adapted to the
embedded case by recording the spatial relation (under or above) of the edges and nodes in
a projection of the 3D graph; they are thus called the adapted dual Pachner moves and are
able to endow embedded 4-valent spin networks with a spin-foam like dynamics. This answers
the question raised at the end of Section 1.3.2.
3.5 Dynamics: propagation, direct and exchange interaction of braids
In order that the stable braids, as topological excitations of the 4-valent braided ribbon networks,
can be candidates for particles or pre-particles, they must be dynamical. Indeed, the evolution
moves endow the stable braids with a rich dynamics: they can propagate and interact. We
briefly address braid propagation first.
Since a braid can be considered an insertion in an edge, it makes sense to speak of them
propagating to the left or to the right along that edge. To help visualize this in the diagrams
we will always arrange a braid so that the edge of the graph it interrupts runs horizontally on
the page. There are two types of propagation of braids, namely induced propagation and active
propagation.
Under the evolution moves, especially the 1 Ø 4 moves, the ambient network of a braid may
expand on one side of a braid but contract on the other side, such that the braid effectively moves
20 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
Braid B
1 x
e
1
e
2
3
4
a
c
b
a
c
b
g
w
e’e
g’
g’’
v
y
a’
c’
b’
c
ro
s
s
in
g
s
c
ro
s
s
in
g
s
o
f
B
’a’
c’
b’
a’
c’
b’
1
e
2
3
4
a
c
b
g
g’
g’’
a’
c’
b’
1
e
2
3
4
a
c
b
g
g’
g’’
a’
c’
b’
1
e
2
3
4
a
c
b
g
g’
g’’
a’
c’
b’
1
e
2
3
4
a
c
b
g g’
g’’
a’
c’
b’
+1 -1
-1-1
+1 -1
-1
-2+2
-2
-2
-2
-2
-2
-2
1
e
2
3
4
a
c
b
g
g’
g’’
a’
c’
b’
-2
-2
3
p/3
c
3
p/3
c’ This rotation cancels
the twists on g and g’
Translate node 3 along c’ to the right
b
c
a
w
e’
w
e’
w
e’
w
e’
w
e’
w
e’
w
e’
2 3 move on nodes x & ya
(a)
(b)
(c)
(d)
(f)
(e)
4 1 move on nodes1, 2, 3, and 4a
(g)
(h)
-1
Translate node 2, 3, & 4 to the left.
Twists on c cancel
Braid B’ c
ro
s
s
in
g
s
o
f
B
’
c
ro
s
s
in
g
s
o
f
B
’
c
ro
s
s
in
g
s
o
f
B
’
c
ro
s
s
in
g
s
o
f
B
’
c
ro
s
s
in
g
s
o
f
B
’
c
ro
s
s
in
g
s
o
f
B
’
Figure 20. An example of direct right-interaction.
towards the side of contraction. We call this phenomenon induced propagation because the
braid is not directly involved in the evolution and remains the same. Any braid can propagate
in this induced way.
Opposed to induced propagation is active propagation, which occurs only to specific net-
work configurations and braids; it is called active because the braid’s structure undergoes inter-
mediate changes (and probably permanent change of its internal twists) during the propagation.
Braids that can propagate in this way are called actively propagating and otherwise station-
ary or non-actively propagating. Nevertheless, active braid propagation needs some special
care that may modify the overall settings of the 4-valent scheme, we thus refer to [93, 102] for
details.
Two braids may propagate and meet each other in a situation, such that they can interact.
Two interaction types exist: direct interaction and exchange interaction. We shall illustrate
these striking behaviours of the braids by figures and present some key results of the dynamics
in the algebraic notation.
We first elucidate the direct interaction. In a direct interaction of two adjacent braids,
one can merge with the other, through a sequence of evolution moves. We shall describe this
by a complete example and then by the generalised definition. Fig. 20 depicts all the steps
that the two braids in (a) take to become the single braid in (h) to complete an interaction. To
supplement this figure, a few remarks are in order. In this example, the equivalence moves – the
rotations and translations – are nondynamical but are there only to set the nodes and edges in
a proper configuration that manifests the legitimacy of an evolution move, e.g. the final 4 Ñ 1
move (compare it to Fig. 19).
Emergent Braided Matter of Quantum Geometry 21
X
B
1 x
e
1
e
2
3
4
(a) a
c
b
a
c
b
g
2 3
of x & y
a
w
e’e
g’
g’’
v
translate nodes 2, 3, and 4 with edges g, g’, g’’
to the left, passing all the crossings; then rear-
range nodes 1, 2, 3, & 4 by equivalence moves
in a proper configuration ready for a 4 1 mo-a
ve; finally, do the 4 1 move.a
(b)
(c)
y
a’
c’
b’
a’
c’
b’
a’
c’
b’
X
B
X
B
X
B
’
X
B
’
X
B
’
B B’
Ta
Tb
Tc
Ta
Tb
Tc
T’a
T’b
T’c
a b2
a b
2
e
tangled
translation
3
3
e
(d)
Figure 21. (a)-(c) Definition of the direct right-interaction. (d) An obstructed translation.
Braid B in Fig. 20(a) is an example of what we call actively interacting braids because
all the moves are done basically on B’s nodes and edges; B1, except its left end-node, plays no
role. Moreover Fig. 20 is an example of direct right-interaction, in the sense that the actively
interacting braid merges with the braid on its right. In fact, a direct interaction always involves
at least one actively interacting braid.
For simplicity braid B1 is assumed twist-free. One may notice that the twist of ´2 on strand
c of braid B appears again in the braid in Fig. 20(h). This is not a coincidence but an instance
of certain conservation laws that braid interactions follow.
Bearing this example in mind, Fig. 21 serves as a transparent graphical definition of the
direct right-interaction of two braids. Note that translating the nodes 2, 3, and 4 together
with their common edge g, g1, and g2 to the left, passing through all of B’s crossings XB in
Fig. 21 is certainly not always possible because XB may obstruct the translation by creating
a tangle like that in Fig. 21(d). This translation is guaranteed viable only when braid B is at
least completely right-reducible (see Fig. 20). Note also that nodes 1, 2, 3, and 4 with their
edges cannot always be arranged by equivalence moves to meet Condition 3.1(3), such that the
final 4 Ñ 1 move can be done to complete the interaction. If all steps in Fig. 21 are possible,
braid B must be actively interacting.
It is a theorem [93, 41] that if a braid is actively interacting, this constrains the braid to be
equivalent to a trivial braid with both end-nodes in the same state (e.g. Fig. 22). Recall that an
actively interacting braid is studied most conveniently in its trivial representation, which should
read S
Tl
rTa, Tb, Tcs
S
Tr
.
Braid B
1 x
e
b
c
a
-1
-1
+1
+1Braid B
1 x
e
a
c
b
-2
b
c
a
(a) (b)
Figure 22. The braid B in Fig. 20 is equivalent to a trivial braid.
The discussion above is all about direct right-interaction. Direct left-interaction is defined
analogously and can be visualized as the left-right mirror image of right-interaction. Thus, in
what follows, we assume direct left-interaction is understood. We denote a direct interaction of
two braids B1 and B2 by B1 `d B2 “ B; whether it is a left- or right-interaction is manifest
contextually.
A formal division of braids in now natural. We temporarily denote the set of stable braids
by BS
0 . Nevertheless, for reasons to become transparent we enlarge BS
0 by adding two more
braids:
B˘0 “
˘
0 r0, 0, 0s
˘
0 , (3.5)
22 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
which are completely trivial15. B˘0 are actually unstable because they are dual to a 3-ball. If
we tolerate their instability they are obviously actively interacting. As such, let us still call the
enlarged set the set of stable braids but denote it by BS . BS admits a disjoint union of three
subsets:
BS “ Bb \Bf \Bs,
where Bb, Bf , and Bs are the sets respectively of actively interacting braids (including B˘0 ), all
actively propagating braids that do not actively interact, and stationary braids. The meaning of
the superscripts will be made clear later. Although actively interacting braids are also actively
propagating [93, 101], they are excluded from Bf . It is a theorem that @B P Bb, the effective
state of B, χB ” 1 [46].
We now dwell on the algebra of direct interactions. For simplicity, let us consider B,B1 P Bb,
B “ S
Tl
rTa, Tb, Tcs
S
Tr
on the left of B1 “ S1
T 1l
rT 1a, T
1
b, T
1
cs
S1
T 1r
. If B and B1 satisfy the interaction
Condition 3.1(1) trivially, i.e., if Tr ` T 1l “ 0 and S “ S1, then the direction interaction of B
and B1 is simply
B `d B
1 Tr`T
1
l“0
“ S
Tl
rpTa, Tb, Tcq ` pT
1
a, T
1
b, T
1
cqs
S
T 1r
“ S
Tl
rTa ` T
1
a, Tb ` T
1
b, Tc ` T
1
cs
S
T 1r
. (3.6)
If, however, B and B1 satisfy the interaction condition nontrivially, i.e. if either B’s right end-
node or B1’s left end-node must be rotated to make them meet Condition 3.1(1), some algebra
is needed. We would refer to [41, 46] for the details; rather, we present the Lemma 3.1 as the
general result. Note that we put an actively interacting braid in its trivial representation and
an arbitrary braid in its unique representation.
Lemma 3.1. Given a braid B “ S
Tl
rTa, Tb, Tcs
S
Tr
P Bb on the left of a non-actively interacting
braid, B1 “ Sl
0rpT
1
a, T
1
b, T
1
cqXs
Sr
0 , with the interaction condition satisfied by p´qTrS “ Sl, the direct
interaction of B and B1 results in a braid
B2 “
p´qTlS
0 rppP
S
´Tl
pTa, Tb, Tcqq`pP
p´qTrS
´Tl´Tr
pT 1a, T
1
b, T
1
cqq`pTl ` Tr, ¨, ¨qqXlpp´q
TrS,´Tl´TrqX
s
Sr
0 ,
where pTl ` Tr, ¨, ¨q is the short for pTl ` Tr, Tl ` Tr, Tl ` Trq.
The PSl
m in the B2 in Lemma 3.1 is a permutation on the triple, determined by Sl and m and
valued in the group S3. Fig. 16 and its mirror images readily show P``1 “ p1 2q, P`´1 “ p2 3q,
P´`1 “ p3 2q, and P´´1 “ p1 2q. More general equalities can be derived recursively and found
in [41]. The functions XlpSl,mq and XrpSr, nq return crossing sequences generated by the
rotations needed to set B’s right end-node and B1’s left end-node ready for a 2 Ñ 3 move. We
quote its definition in the following equation but refer their detailed properties to [46]:
Xlp`,mq “
#
pudq´m{2 if m is even,
dpudqp´1´mq{2 if m is odd,
Xlp´,mq “
#
pduq´m{2 if m is even,
upduqp´1´mq{2 if m is odd,
Xrp`, nq “
#
pudq´n{2 if n is even,
pudqp1´nq{2d´1 if n is odd,
15Note again that at this point spin network labels are not in play. Including spin network labels has two
immediate consequences. Firstly, like any other braid in BS , B˘0 are not just two braids but an infinite set of
braids coloured by different sets of spin network labels. Secondly, B˘0 are only trivial topologically but neither
algebraically nor physically.
Emergent Braided Matter of Quantum Geometry 23
Xrp´, nq “
#
pduq´n{2 if n is even,
pduqp1´nq{2u´1 if n is odd,
(3.7)
where n,m P Z. In equation (3.7), a positive exponent of a crossing sequence means, for example,
pudq2 “ udud, while a negative one means, for instance, pudq´2 “ d´1u´1d´1u´1. Lemma 3.1
is independent of the trivial representation chosen for B [46]. Equation (3.6) is merely a special
case of Lemma 3.1. We are now ready for a main result.
Theorem 3.1. Given B “ S
Tl
rTa, Tb, Tcs
S
Tr
P Bb, and any braid, B1 “ Sl
0rpT
1
a, T
1
b, T
1
cqXs
Sr
0 P BS,
such that B ` B1 “ B2 P BS, the effective twist number Θ is an additive conserved quantity,
while the effective state χ is a multiplicative conserved quantity, namely
ΘB2 “ ΘB `ΘB1 , χB2 “ χBχB1 .
This theorem, proved in [46], demonstrates that the two invariants of equivalence moves
of braids, Θ and χ, are also conserved charges of direct interactions. Conservation laws play
a pivotal role in revealing the underlying structure of a physical theory. By invariants and
conserved quantities we are able to determine how the content of our theory may relate to
particle physics and if extra inputs are required. In Section 3.6, we try to identify our conserved
quantities with particle quantum numbers.
Since χB ” 1 @B P Bb, Theorem 3.1 shows that if B directly interacts with a braid with
χ “ ´1, the result must be a braid with χ “ ´1 too and is thus not in Bb. Moreover,
Bb `d BSzBb Ď BSzBb.
Because fermions usually do not directly interact with each other but can interact with
(gauge) bosons, the fact that a direct interaction always involves at least one actively inter-
acting braid implies an analogy between actively (non-actively) interacting braids and bosons
(fermions). The evidence of this analogy will become stronger soon, after we study exchange
interaction. This analogy manifests the meaning of the superscript “b”, as for bosons, of Bb,
the set of actively interacting braids. As to the set of non-actively interacting braids, we di-
vided it into Bf and Bs, and we are more inclined to consider the former analogous to the
set of fermions because the braids in Bf are chiral [93], analogous to the chirality of the SM
elementary fermions, which manifest the superscript “f” in Bf .
The result of a direct interaction of two braids is unique! Nonetheless, this uniqueness may
cease to hold if the braided ribbon networks are graced with spin network labels, such that the
result of an evolution move becomes a superposition of outcomes with the same topology but
different sets of labels.
It is natural to ask if a braid can decay, like a particle. The answer is Yes. A braid decay
is a reversed process of a direct interaction. Hence, through decay a braid always radiates
an actively interacting braid, to its left or its right. We thus differentiate a left-decay from
a right-decay, which are symbolized respectively by B
ø
Ñ B1`B2, indicating that B1 P Bb, and
B
ù
Ñ B1 `B2 because B2 P Bb.
Fig. 23 defines right-decay. Fig. 23(a) shows a right-reducible braid, B“ SlrpTa, Tb, TcqXX 1s
Sr ,
with a chosen reducible crossing segment X 1, X 1 ű Xmaxred ű X, where Xmaxred is the maximal
reducible segment of X. This indicates that, in contrast to direct interaction, a braid may decay
in multiple ways, each corresponding to a choice X 1 that must be specified in a decay process.
As the reverse of direct left-interaction, Fig. 23 should be easily understood. Its algebraic form is
B “ SlrpTa, Tb, TcqXX 1s
Sr ù
Ñ SlrppTa, Tb, Tcq ´ σ
´1
X pT
1
a, T
1
b, T
1
cqqXs
S1 ` S1rpT 1a, T
1
b, T
1
cqX 1s
Sr
“ B1 `B2,
where S1 “ p´q|X
1|Sr, B
2 P Bb, and the “`” denotes the adjacency of B1 and B2. Left-decay
is defined similarly. The relation between decay and direct interaction ensures that effective
twist Θ and effective state χ are also additively and multiplicatively conserved in braid decay.
24 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
g
g’
g’’
X
Translate the three nodes on the dashed loop to the left of X’
3 2 movea
1 4 ona S
r
(b)(a)
Sl
B
X’ X X’
X X’S’
-S’ X X’S’ S’
(c)
(d)
B’ B’’
Ta
Tb
Tc
T’aT”a
Sr
Sl Sr
SrSrSl
SrSl
-S’
-Sr
-Sr
Ta
Tb
Tc
Ta
Tb
Tc
T’b
T’c
T’a
T’b
T’c
-T’a
-T’b
-T’c T”b
T”c
Figure 23. Definition of braid decay: B
ù
Ñ B1`B2, B,B1 P BS , B2 P Bb. The dashed lines emphasize
the dependence of their relative positions on the state Sr.
Not only a reducible braid but also an irreducible – in fact any – braid can radiate. What an
irreducible braid emits is either of B˘0 in equation (3.5). In fact, because a 1 Ñ 4 can always take
place on either end-node of any braid, a subsequent 3 Ñ 2 on three of the four nodes generated
by the 1 Ñ 4 move results in either a B`0 or a B´0 , depending on the state of the end-node. This
is why B˘0 are included in BS although they are unstable. It is therefore plausible that B˘0 are
analogous to gravitons.
We now proceed to the exchange interaction of braids. Unlike a direct interaction, an ex-
change interaction can be defined on the whole BS as a map, `e : BS ˆ BS Ñ BS ˆ BS .
Exchange interactions get this name because each such process always involves an exchange of
a virtual actively interacting braid. It is useful to keep track of the direction of the flow of the
actively interacting braid during an exchange interaction. Therefore, we differentiate a left and
a right exchange interaction, respectively denoted by
Ð
`e and
Ñ
`e. The arrow indicates the “flow”
of the virtual actively interacting braid.
(a)
g
2 3 move of the two nodes in statea S
g’
g’’
translate the three new nodes - two in S and one in S
- with edges g, g’, g’’ to the left, passing crossing se-
quence X ; then rearrange the them by equivalence
moves to a proper configuration ready for a 3 2.
-
1B
a
S1l S S X2X
1A
(b)
S
-S
(d)
S’ S’
B1 B2
B’1 B’2
(c) -S’
S’
3 2 on the two nodes in and that in .a -S’ S’
S’ S
B
Ta
(e)
Tb
Tc
T’1a
T’1b
T’1c
T’2a
T’2b
T’2c
T2a
T2b
T2c
Ta
Tb
Tc
-Ta
-Tb
-Tc
T1a
T1b
T1c
T1a
T1b
T1c
T1a
T1b
T1c
T2a
T2b
T2c
T2a
T2b
T2c
X2
X2
X2
X
1B
X
1A
X
1B
X
1A
X
1B
X
1A
X
1B
X1B
S1l
-S
S
2r
S
2r
S
2r
S
2r
S1l
S1l
-S’
Figure 24. (a)-(d): Definition of right exchange interaction: B1
Ñ
`eB2 Ñ B1
1`B
1
2, B1, B2, B
1
1, B
1
2 P B
S .
(e) The virtual actively interacting braid exchanged from B1 to B2.
Emergent Braided Matter of Quantum Geometry 25
The graphical definition of a right exchange interaction is illustrated in Fig. 24. The left
exchange interaction is defined likewise. We now make a few remarks.
Firstly, we begin with the B1 “
S1lrpT1a, T1b, T1cqX1AX1B
sS and B2 “
SrpT2a, T2b, T2cqX2s
S2r ,
which are two stable braids, in Fig. 24(a). The zero external twists are omitted as B1 and B2 are
in their unique representations. B1’s right end-node and B2’s left end-node are set in the same
state, S, to satisfy the interaction condition. We also assume that B1 has a reducible crossing
segment, say X1B.
Secondly, if X1B is not trivial, i.e. if B1 is right-reducible, translating the three nodes on the
loop in (b) and rearranging them by equivalence moves into a proper configuration ready for
a 3 Ñ 2 move generates a pair of triples of internal twists, ´pTa, Tb, Tcq and pTa, Tb, Tcq, and
causes S1 “ p´q|X1B |S, as in (c). Taking the permutations induced by X1A and by X1B into
account, in (d) we have the relations:
pT 11a, T
1
1b, T
1
1cq “ pT1a, T1b, T1cq ´ σ
´1
X1A
pTa, Tb, Tcq, (3.8)
pT 12a, T
1
2b, T
1
2cq “ pTa, Tb, Tcq ` σ
´1
X1B
pT2a, T2b, T2cq, (3.9)
which produce the two adjacent braids, B11 and B12, related to B1 and B2.
Thirdly, according to direct interaction, the only possible triple pTa, Tb, Tcq in Fig. 24(c)
is exactly the same as the triple of internal twists of an actively interacting braid – B “
S1rpTa, Tb, TcqX1B
sS – in Fig. 24(e). For an actively interacting braid of the form in Fig. 24(e),
S1 “ p´q|X1B |S; hence, B’s left and right end-nodes are respectively in the same states as that
of the left end-node of B12 in Fig. 24(d) and that of the right end-node of B1 in Fig. 24(a). Thus,
the form of braid B12 in Fig. 24(d) must be precisely the result of the direct interaction of B
and B2, which by Lemma 3.1 is
B `d B2 “
S1rppTa, Tb, Tcq ` σ
´1
X1B
pT2a, T2b, T2cqqX2s
S2r “ B12,
which validates the relation in equation (3.9).
Therefore, the process of the right exchange interaction defined in Fig. 24 is as if B1 and B2
interact with each other via exchanging a virtual actively interacting braid B, then become B11
and B12. Or one may say that an exchange interaction is mediated by an actively interacting
braid. This reinforces the analogy between actively interacting braids and bosons16. Note that
in an exchange interaction, there does not exist an intermediate state in which only the virtual
actively interacting braid is present because our definition of a braid requires the presence of
its two end-nodes. The following theorem summarizes the above as another main result. (The
case of left exchange interaction is similar.)
Theorem 3.2. Given two adjacent braids, B1, B2 P BS. B1 “
S1lrpT1a, T1b, T1cqX1AX1B
sS is
on the left and has a reducible crossing segment X1B, and B2 “
SrpT2a, T2b, T2cqX2s
S2r , there
exists a braid B P Bb, B “ S1rpTa, Tb, TcqX1B
sS with S1 “ p´q|X1B |S, such that it mediates the
exchange interaction of B1 and B2 to create B11, B
1
2 P B
S, i.e.,
B1
Ñ
`e B2 Ñ B11 `B
1
2 “
S1lrppT1a, T1b, T1cq ´ σ
´1
X1A
pTa, Tb, TcqqX1A
sS
1
` S1rppTa, Tb, Tcq ` σ
´1
X1B
pT2a, T2b, T2cqqX2s
S2r .
The fourth remark is, if X1B “ I, however, pTa, Tb, Tcq “ p0, 0, 0q, and hence pT 11a, T
1
1b, T
1
1cq “
pT1a, T1b, T1cq, pT
1
2a, T
1
2b, T
1
2cq “ pTa, Tb, Tcq, and S1 “ S. That is, B11 “ B1 and B12 “ B2. Thus,
the virtual actively interacting braid exchanged in the interaction is either B`0 or B´0 , which
were aforementioned to be analogous to gravitons.
16More generally, this should imply the analogy between actively interacting braids and particles that mediate
interactions, which should potentially include super partners of gauge bosons.
26 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
Here is the final remark. For the same reason that a braid can decay in multiple ways,
two braids can also have exchange interactions in more than one way, as opposed to direct
interaction. This non-uniqueness of exchange interaction has an analogy in particle physics.
For instance, quarks have both electric and color charges, both photons and gluons can mediate
forces on quarks. The relation between actively interacting braids and bosons is not yet an actual
identification, however. In fact, if each actively interacting braid corresponded to a boson, there
would be too many “bosons”. The underlining physics of how two braids can have exchange
interactions in different ways deserves further study.
It should be emphasized that each individual exchange interaction is a process that yields
a unique result17. An expression like B1
Ñ
`e B2 is only formal. Only when the exact forms of B1
and B2 with their reducible segments are explicitly given does B1
Ñ
`e B2 acquire a precise and
unique meaning. In computing an exchange interaction, we have to specify our choice of the
reducible crossing segment of the braid that gives out the virtual actively interacting braid. For
any such choice Theorem 3.2 holds.
The following Theorem shows that the additive and multiplicative conserved quantities of
direct interaction in Theorem 3.1 are also conserved in the same manner under exchange inter-
action.
Theorem 3.3. Given two neighbouring stable braids, B1, B2 P BS, such that an exchange
interaction pleft or right or bothq on them is doable, i.e., B1`eB2 Ñ B11`B
1
2, B11, B
1
2 P B
S, the
effective twist Θ is an additive conserved quantity, while the effective state χ is a multiplicative
conserved quantity, namely
ΘB11
`ΘB12
`e
“ ΘB1 `ΘB2 , χB11χB12
`e
“ χB1χB2 ,
independent of the virtual actively interacting braid being exchanged during the exchange inter-
action.
This Theorem is proved in [101]. Consequently, exchanges of actively interacting braids give
rise to interactions between braids that are charged under the topological conservation rules.
The conservation of Θ is analogous to the charge conservation in particle physics.
3.6 Dynamics: CPT and braid Feynman diagrams
In this section we discuss the charge conjugation, parity, and time reversal symmetries of stable
braids, and the braid Feynman diagrams. We shall present some key results with a few remarks.
3.6.1 C, P, and T
Though not separately, as a theorem the combined action CPT is a symmetry in any Lorentz
invariant, local field theory. Being a physical model of QFT, the SM respects CPT-symmetry
too. This motivates the search for the possible discrete, non-equivalent, transformations of 4-
valent braids and the testing of their correspondence with C, P, and T transformations. Whether
our braids would eventually be mapped to or more fundamental than the SM particles, they
should possess quantum numbers that are transformed by C, P, and T. Conversely, investigating
the action of discrete transformations on our braid excitations can help us to construct quantum
numbers of a braid.
A large number of possible discrete transformations of the 4-valent braids exist, however, for
example the permutation group S3 on the triple of internal twists, several copies of Z2 that flip
17With spin network labels the result is not unique any more because two topologically equal braids can
be decorated by different sets of labels, and an interaction should result in a superposition of braids labelled
differently.
Emergent Braided Matter of Quantum Geometry 27
a twist, an end-node state, and a crossing respectively, etc. The challenge then is to filter out
the unwanted discrete transformations. Surprisingly, the braid dynamics introduced above turns
out to select exactly seven legal discrete transformations of the braids, as C, P, T, and their
products add up to seven in total (eight including identity) [45]. Now we briefly show how this
“super-selection” works and direct the reader to [45, 102] for details. Since QFT does not bear
a transformation that magically changes a particle to something else, the dynamics of the braids
of embedded 4-valent spin networks, namely propagation and interaction, naturally constrains
what discrete transformations are allowed on braids, which is summarized as a guideline in the
following condition.
Condition 3.2. A legal discrete transformation D on an arbitrary braid B must be an auto-
morphism on Bb, Bf , and Bs separately.
We expect the discrete transformations to be representation independent; therefore, the study
of their effects should be made on braids in their generic forms (Fig. 15(a) and equation (3.1)).
Table 1 displays the result, in which explicit identification of the legal discrete transformations
with C, P, and T is made.
Table 1. The group of discrete transformations on a generic braid diagram. In column-3, a ´ (`) means
the propagation direction of the braid is flipped (unaffected). For comparison, column-4 is the action of
the group on a one-particle state, with 3-momentum p, 3rd component σ of spin J , and charge n. The
primed and unprimed twists are related by pTa1 , Tb1 , Tc1q “ pTa, Tb, TcqσX , as in Fig. 15(a).
Discrete Action on Prop- Action on
Transformation B “ Sl
Tl
rpTa, Tb, TcqXs
Sr
Tr
Direction |p, σ, ny
1
Sl
Tl
rpTa, Tb, TcqXs
Sr
Tr
` |p, σ, ny
C S̄l
´Tl
r´pTa, Tb, TcqIXpXqs
S̄r
´Tr
` 9 |p, σ, ncy
P S̄r
Tr
rpTa1 , Tb1 , Tc1qRpXqs
S̄l
Tl
´ 9 |´p, σ, ny
T Sr
Tr
rpTc1 , Tb1 , Ta1qScRpXqs
Sl
Tl
´ 9p´qJ´σ |´p,´σ, ny
CP Sr
´Tr
r´pTa1 , Tb1 , Tc1qX´1s
Sl
´Tl
´ 9 |´p, σ, ncy
CT S̄r
´Tr
r´pTc1 , Tb1 , Ta1qIXScRpXqs
S̄l
´Tl
´ 9p´qJ´σ |´p,´σ, ncy
PT S̄l
Tl
rpTc, Tb, TaqScpXqs
S̄r
Tr
` 9p´qJ´σ |p,´σ, ny
CPT Sl
´Tl
r´pTc, Tb, TaqIXScpXqs
Sr
´Tr
` 9p´qJ´σ |p,´σ, ncy
In column 2 of Table 1, R, IX , and Sc are discrete operations on the crossing sequence X of
a braid, respectively defined by R : X “ x1x2 ¨ ¨ ¨xn ÞÑ xnxn´1 ¨ ¨ ¨x1, IX : X “ x1x2 ¨ ¨ ¨xn ÞÑ
x´1
1 x´1
2 ¨ ¨ ¨x´1
n , and Sc: @xi P X, xi ÞÑ d if xi “ u and xi ÞÑ u if xi “ d. Hence, X´1 “ IXRpXq.
These operations are commutative and are elaborated in [45, 102].
In Table 1, we chose to denote C, P, and T transformations in the Hilbert space by calligraphic
letters C, P, and T because braids are topological excitations of embedded spin networks that
are the states in the Hilbert space describing the fundamental spacetime. One can easily check
that the eight discrete transformations including identity in the first column of Table 1 indeed
form a group, which is actually the largest group of legal discrete transformations of 3-strand
braids [45, 102]. This is the first reason why they can be identified with C, P, T, and their
products.
28 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
We emphasize again that these discrete transformations of braids are not equivalence moves;
they take a braid to inequivalent ones, as seen in Table 1. Staring at the 3rd column of the
table, one readily finds that some characterizing quantities of a braid, e.g., the effective twist
and effective state, are invariant under some transformations but not under others. Table 1 is
obtained by utilizing these topological characterizing quantities, without involving spin network
labels. We do not count in the phase and sign factors in the 4th column of Table 1 either. All the
transformations are restricted to local braid states, rather than a full evolution picture. Given
these, surprisingly, the map between the legal discrete transformations of braids and those on
single particle states appears to be unique.
According to column-4 in Table 1, the transformations P, T , CP, and CT reverse the three
momentum of a one-particle state. But then, what is the momentum of a braid? Fortunately,
we need not to explicitly define the 3-momentum of a braid for the moment to tell the discrete
transformations that can flip the momentum because it should always agree with the braid’s
local propagation direction, however it is defined [45]. Therefore, the discrete transformations
reversing the 3-momentum of a braid are exactly those flipping the propagation direction of the
braid. This also helps to pin down Table 1.
The effective twist Θ and effective state χ are representation-independent invariants and
dynamically conserved quantities of a braid, while charges, e.g. electric and color charges, are
quantum numbers of a particle. Hence, only Θ, χ, and functions of them can be candidates
for certain charges of a braid. As we know, the electric charge of a particle is an integral
multiple of 1{3, an additively conserved quantity, and a result of Up1q gauge symmetry. The
effective twist of a braid has three similar traits. We have seen the first two and now talk about
the third property. The framing that inflates a spin network edge to a tube is in fact a Up1q
framing. That is, a tube is essentially an isomorphism from Up1q to Up1q, which is characterized
by its twists. A twist-free tube is an identity map, whereas a twisted tube represents a non-
trivial isomorphism. These suggest we interpret Θ or an appropriate function of it as the
“electric charge” of a braid, which may in turn explain the origin and quantization of electric
charge.
Here is the final remark on Table 1. On a single particle state, a CPT has one more effect
than a C because it also turns σ, the z-component spin, to ´σ. We notice that in Table 1, the
last transformation does more to the braid than the second transformation: It swaps the first
and the third elements in the triple of internal twists of the braid. Although we do not know
yet what features of a braid correspond to σ, we can now consider the last transformation in
Table 1 as CPT . [45, 102] argue that spin network labels should play the role that determines
the “spin” of a braid state.
C, P, and T group stable braids into CPT-multiplets. The braids in a multiplet are not
equivalent but may share some traits. It is instructive to find how a CPT-multiplet of braids is
characterized. Theorem 3.4 shows that only CPT-multiplets of actively interacting braids have
a topological character.
Theorem 3.4. All actively interacting braids in a CPT-multiplet have the same number of
crossings if each of them is in its unique representation. This number uniquely characterizes the
CPT-multiplet.
The proof of the theorem can be found in [45, 102]. Theorem 3.4 does not apply to non-
actively interacting braids. In fact, we can always find two non-actively interacting braids
with m crossings (m ą 1), in their unique representations, which are not related to each other
by any discrete transformation. For example, the 2-crossing braids SlrpTa, Tb, Tcqud´1sSr and
S1lrpT 1a, T
1
b, T
1
cquus
S1r can never belong to the same CPT-multiplet, regardless of their internal
twists and end-node states. Nevertheless, it is still true that all the non-actively interacting
braids in a CPT-multiplet have the same number of crossings if they are in the same type
Emergent Braided Matter of Quantum Geometry 29
of representation [45]. This is simply because the discrete transformations do not alter the
representation type and the number of crossings of a braid.
Having seen the effects of C, P, and T on single braid excitations, we now discuss the action of
these discrete transformations on braid interactions. Braid interactions turn out to be invariant
under CPT, and more precisely, under C, P, and T separately [45, 101]. By this we mean,
say, for a direct interaction under a C, CpBq `d CpB1q “ CpB `d B
1q, while under a P, it
means PpB1q `d PpBq “ PpB `d B
1q. Note that the P-transformation of a direct interaction
swaps its direction. A subtlety arises in the case of T, however. An interaction involves the
time evolution of a spin network. To apply our T-transformation to an interaction, one should
reverse all the dynamical moves. Hence, a T-transformation turns a direct interaction into
a decay. That is, to show the invariance of B `d B
1 ÝÑ B2 under time reversal, it suffices to
show that T pB2q ù
Ñ T pB1q ` T pBq. Analogously, the invariance of an exchange interaction,
B1
Ñ
`e B2 Ñ B11 ` B12, under time reversal reads T pB12q
Ð
`e T pB11q Ñ T pB2q ` T pB1q. The case
of braid decay follows similarly [101].
This observation of the absence of CP-violation in our theory does not comply with the SM
of particles, which seems to infer an issue that the interactions of braids are deterministic, in
the sense that an interaction of two braids produces a definite new braid. Nevertheless, this
may not be a problem at all because we have only worked with definite vertices of interactions.
In terms of vertices we have a definite result for an interaction as to the case in QFT; this is
similar to what has been done in SF models or GFTs. Besides, one can certainly argue that
if our braids are more fundamental entities, the CP-violation in particle physics need not hold
at this level. Putting this CP-violation problem aside, however, a fully quantum mechanical
picture should be probabilistic18.
If the absence of CP-violation was truly an issue, we would consider braids with the same
topological structure but a different sets of spin network labels as physically different. One
may adapt some SF methods to assign amplitudes to the adapted dual Pachner moves of the
braided ribbon networks. An evolution move may then yield outcomes with the same topo-
logical configuration but different spin network labels; each outcome has a certain probability
amplitude. As a result, an interaction of two braids may give rise to superposed braids, each of
which has a certain probability to be observed, with the same topological content but different
set of spin network labels. With this, CP-violating interactions may arise.
Note that the current study of discrete transformations of braids would not be impacted by
just adding spin network labels in a straightforward way in to our scheme. One reason is that
the discrete transformations of the braids do not change the spin network label of each existing
edge of the network. One may try to construct discrete transformations that change the spin
network labels on braids, but one does not have a priori a reason to make a special choice among
the many arbitrary ways of doing this.
3.6.2 Asymmetry of braid interaction
Both direct and exchange interactions are asymmetric. A brief description is as follows. The
statement that direct interaction is asymmetric means given an actively interacting braid B
and an arbitrary braid B1, in general either of the direct right interaction B `d B
1 or the
left interaction B1 `d B cannot occur because of the violation of the corresponding interaction
condition. Even if both interactions are feasible, B `d B
1 and B1 `d B are two inequivalent
braids in general, which reads B `d B
1 fl B1 `d B. Two exceptions exist. In the cases where B
and B1 meet certain constraints, B `d B
1 and B1 `d B can simply be equal [46].
On the other hand, interestingly, B `d B
1 and B1 `d B can be related by discrete transfor-
mations. Because P-transformation swaps the two braids undergoing a direct interaction, i.e.,
18One should note that a few theoretical physicists may not agree on this.
30 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
PpB `d B
1q “ PpB1q `d PpBq, we immediately have
B1 `d B “ PpB `d B
1q, if B “ PpBq, B1 “ PpB1q,
where the P can be replaced by CP by the same token. Note that, however, time reversal cannot
relate B `d B
1 and B1 `d B because it turns a direct interaction into a decay.
As aforementioned, the set of actively interacting braids is closed under direct interaction.
This and the asymmetry of direct interaction then give rise to the following theorem [46, 93].
Theorem 3.5. The set of actively interacting braids Bb is an algebra under direct interaction,
namely Bb `d Bb “ Bb. This algebra is associative and non-commutative.
It follows that braid decay is also asymmetric but we shall skip this and move on to the
asymmetry of the exchange interaction, which is subtler.
The asymmetry of the exchange interaction is three-fold. Firstly, for B1, B2 P B
S , in general
B1
Ñ
`eB2 fl B2
Ñ
`eB1 (B1
Ð
`eB2 fl B2
Ð
`eB1), which is called the asymmetry of the first kind.
Secondly, in general B1
Ñ
`e B2 fl B1
Ð
`e B2, which is termed the asymmetry of the second
kind. The asymmetry of the third kind states that generically B1
Ñ
`e B2 fl B2
Ð
`e B1. As in
the case of direct interaction, when B1 and B2 satisfy certain constraints, symmetric exchange
interactions arise; however, a subtlety should be noted. Since two braids may have different
exchange interactions, B1
Ñ
`eB2 and B2
Ñ
`eB1 cannot be equal for all possible interactions of B1
and B2 may interact. The right question to ask is, taking the right exchange interaction as an
example: For any B1 and B2, does there exist an instance of B1
Ñ
`e B2 and one of B2
Ñ
`e B1
among all possible ways of these two interactions, such that B1
Ñ
`eB2 “ B2
Ñ
`eB1? [101] answers
this question for the first two kinds of asymmetry. This asymmetry of the third kind is new and
not studied in [101] but it would not be hard by following the derivations in [101].
Like direct interactions, asymmetric exchange interactions may be related by discrete trans-
formations, but only for the asymmetry of the third kind. Since in the asymmetry of the third
kind, the positions of the braids and the interaction direction are both swapped, according to
Table 1, we immediately see that only P and CP are able to do this because they are the only
ones that can flip a braid horizontally without changing the vertical order of the braid’s internal
twists. Therefore, we obtain equation (3.10), in which P can be replaced by CP,
B2
Ð
`e B1 “ PpB1
Ñ
`e B2q, if B1 “ PpB1q, B2 “ PpB2q. (3.10)
3.6.3 Braid Feynman diagrams
An effective theory of the dynamics of 4-valent braids based on Feynman diagrams is possible,
which are called braid Feynman diagrams. Unlike the usual QFT Feynman diagrams having
no internal structure, each braid Feynman diagram is an effective description of the whole
dynamical process of a braid interaction and has internal structures that record the evolution
of the braid and its surroundings.
We use and for respectively outgoing and ingoing braids in Bf ,
and for respectively outgoing and ingoing non-actively propagating braids19 in Bs.
Outgoing and ingoing braids in Bb are better represented by and respectively.
In accordance with left and right-decay, we will henceforth denote left and right direct
interactions by
Ð
`d and
Ñ
`d respectively. Note that if the two interacting braids are both
in Bb, the direction of the direct interaction is irrelevant because the result is independent
of which of the two braids plays the active role in the interaction. Since Bb
Ñ
`d Bb Ď Bb and
19These braids can still propagate in an induced way.
Emergent Braided Matter of Quantum Geometry 31
Bb
Ñ
`d pB
f \Bsq Ď Bf \Bs, the only possible single vertices of right direct interaction and
of right decay are respectively listed in Fig. 25(a) and (b), whose left-right mirror images are
vertices of direct left-interaction and left-decay. The arrows over the wavy lines in Fig. 25 are
important for they differentiate a direct interaction from a decay.
B
1
B’
1
B
B
1
B’
1
B
B
1
B’
1
B
B
2
B’
2
B
B
2
B’
2
B
B
2
B’
2
B
B
2
B’
2
B
B
1
B’
1
B
(a)
(b)
Figure 25. (a) All vertices of direct right-interaction. (b) All vertices of right-decay. Time flows up.
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2(a) (b) (c) (d)
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2(e) (f) (g) (h)
B1
B’1
B B2
B’2(i)
B1
B’1
B B2
B’2(k)
B1
B’1
B B2
B’2(j)
B1
B’1
B B2
B’2(l)
B1
B’1
B B2
B’2(m)
B1
B’1
B B2
B’2(n)
B1
B’1
B B2
B’2(o)
B1
B’1
B
(p)
B2
B’2
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2 (r)(q)
B1
B’1
B B2
B’2(s)
B1
B’1
B B2
B’2(t)
B1
B’1
B B2
B’2
B1
B’1
B B2
B’2 (v)(u)
B1
B’1
B B2
B’2(w)
B1
B’1
B
B2
B’2(x)
Figure 26. Possible right exchange interaction 2-vertices. Time flows up.
Fig. 26 depicts all possible basic 2-vertex diagrams for right exchange interaction, whose
left-right mirror images are certainly the basic diagrams for left exchange interaction. These
diagrams manifest the invariance of the exchange interaction under C, P, T and their products.
The diagram of an exchange interaction manifests whether it allows symmetric instances.
Taking the asymmetry of the first kind as an example, if a diagram looks formally the same as
its left-right mirror image with the arrow over the virtual braid not mirrored, the corresponding
exchange interaction allows symmetric instances. It follows that Fig. 26(a), (b), (e), (f), (p),
and (u)-(x) are such diagrams.
The diagrams in Figs. 25 and 26 crystallize the analogy between braids in Bb and bosons,
as the topological conservation laws permit them to be singly created and destroyed and as ex-
changes of these excitations give rise to interactions between braids charged under the topological
conservation rules.
Multi-vertex and loop braid Feynman diagrams can be constructed out of these basic vertices.
As a result, there should exist an effective field theory based on these diagrams, in which the
32 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
probability amplitudes of each diagram can be computed. For this one should figure out the
terms evaluating external lines, vertices, and propagators of braids. In a more complete sense,
an action of the effective fields representing braids that can generate these braid Feynman
diagrams should be devised. Each effective field is a function of the representations of the
group elements in the characterizing 8-tuple (and spin network labels if necessary) of a braid;
these representations label a line in the corresponding braid Feynman diagram. An easier task
is to assign a reasonable probability amplitude of each braid Feynman diagram. In either case,
the very first challenge is to find an appropriate mathematical language to study the 4-valent
scheme analytically. In the next section we will briefly mention three possible formalisms.
3.7 Discussions and outlook
The 4-valent scheme resolves some issues and limitations persisting in the 3-valent approach but
also raises new issues. In the first place, we obtain a p3` 1q-dimensional evolution of quantum
states of spacetime, which has intrinsic dynamics of the braid excitations of these states. Because
of the framing and embedding of spin networks, strands of a braid excitation admit twists only
in units of 1/3. The twists of a braid are directly related to its electric charge, which naturally,
rather than by hand, gives rise to charge quantization and fractional charges such as quark
charges. The 4-valent theory also contains another natural selection: braid dynamics magically
picks out a group of exactly eight discrete transformations, including the identity, which can be
identified with analogues of C, P, T, and their products.
Some issues the 4-valent scheme raises have been discussed more or less previously. In what
follows, we shall analyse some issues that bear on the interpretation of these results. In the last
section, however, we shall introduce ideas, future work, and work in progress, which may remove
these issues.
Stability and locality
We argued that the stable braids are local excitations of 4-valent braided ribbon networks
because they are noiseless subsystems of the networks. Nevertheless, the comparison with topo-
logical field theories that do not bear any local degrees of freedom seems to set the locality
of 4-valent braids in doubt. In fact, locality in background independent theories of quantum
gravity is delicate, as it is correlated with two other important issues, namely the problem of
the concept of spacetime and that of a low energy limit. Moreover, the locality and stability of
a braid are also entangled.
A background independent quantum gravity theory usually lacks a metric that directly mea-
sures spatial locality. But the graph metric of a spin network may define the network locality.
In this sense, a stable braid is local because it is confined between two nodes. Unfortunately,
the issue in the stability of a braid may damage its locality because Condition 3.1 only protects
a stable braid from being undone but does not prevent the braid’s end-nodes from being ex-
panded by 1 Ñ 4 moves. In the latter scenario, the three strands of a braid may turn out to
be attached to nodes far from each other on the network, causing the braid to be nonlocal. We
might strengthen the stability condition by further forbidding the action of a 1 Ñ 4 move on
either end-node of a stable braid but the price is the loss of braid decay.
Moreover, the locality discussed above is micro-locality, opposed to which is macro-locality
that is defined in the low energy end of the theory. Markopoulou and Smolin proposed these two
notions of locality and found that they do not match in general [68], which is also exemplified
in [100]. The quantum spacetime in our context is pre-geometric, as it is a sum over quantum
histories of superposed pre-geometric spin networks; it is conjectured that continuous spacetime
emerges as a certain limit of this quantum spacetime. Hence, requiring the micro-locality defined
on each spin network to match the macro-locality in continuous spacetime makes no sense.
Emergent Braided Matter of Quantum Geometry 33
Macro-locality is more relevant to the known physics; however, it is obtained from micro-
locality. This leads back to the problem of low energy limit, to resolve which Markopoulou et
al. adapted the idea of noiseless subsystems with micro-symmetries from Quantum Information.
Therefore, we expect that the symmetry of the braid excitations will induce emergent symme-
tries, including time and space translation invariance, in the low energy effective description of
the braids.
Particle identif ication and mass
The ultimate physical content of the 4-valent scheme is not fully comprehensible at this stage.
In the trivalent scheme, [13] tentatively maps the trivalent braids to SM particles. Whether such
a map exists in the 4-valent scheme is as yet obscure. One reason is that although the dynamics
of 4-valent braids strongly constrains the defining 8-tuple of a stable braid (in particular the
actively interacting braids), the closed form of this constraint is still missing. Consequently,
we lack a means to pick out the 4-valent braids that may be mapped to the SM particles.
Nonetheless, we are inclined to consider the prospect that braid excitations are fundamental
matter whose low energy effective theory yields the SM particles.
If the latter is true, the potential instability and non-locality of stable braids may not be an
issue because only the low energy effective counterparts of the braids are physically relevant.
We may also ask how mass arises. There appear to be two possibilities. First, a braid may
acquire zero or nonzero mass from some of its intrinsic attributes. Second, mass is not well-
defined at the level of spin networks but is emergent in the low energy limit, directly or via
certain symmetry breaking. The latter requires working out the effective theory, which is our
future work. As to the former, the number of crossings in a braid can be a candidate for its
mass (this is also conjectured in the trivalent scheme [13, 15]). Here is the logic. The number of
crossings of an actively interacting braid in its unique representation uniquely characterizes the
CPT-multiplet the braid belongs to; hence, this number cannot be the charge (already mapped
to the braid’s effective twists) or 3-momentum of the braid but probably related to the energy
of the braid. Besides, actively interacting braids are equivalent to trivial braids, whereas non-
actively interacting ones are not. If we associate a braid’s mass to its number of crossings, all
actively interacting braids seem massless, consistent with their analogue with (gauge) bosons.
and most non-actively interacting braids are massive because they are not fully reducible.
Further questions
• At the level of spin networks, is there a quantum statistics of braids that can turn the
analogy between actively interacting braids and bosons affirmative? If true, are there
anyonic braid states?
• Both trivalent and 4-valent schemes need a mechanism to create nontrivial braids on
unbraided networks20. Section 3.7.1 discusses a possible way out.
• Our ansatz that spacetime is fundamentally discrete is partly inspired by the LQG area
and volume operators with discrete spectra. Nevertheless, whether these area and volume
operators are physical is still under debate [26, 80]. On the one hand, these operators are
not gauge invariant. On the other hand, the areas and volumes that we routinely measure
are associated to spatial regions determined by matter [25, 84] but LQG was devised to
be a theory of gravity only. Now that our program of emergent matter shows that matter
may be encoded in LQG as emergent braid excitations of spin networks, it may help to
settle the debate.
20Trivial braids B˘0 in equation (3.5) can be otherwise created and annihilated, in the 4-valent scheme.
34 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
• Our program of emergent matter is also related to Quantum Graphity, a class of general
theories of background independent quantum gravity based on graphs [53, 54]. [43] finds
the speed of light21 as a Lieb–Robinson bound [60] in certain Quantum Graphity models.
As both trivalent and 4-valent braids can propagate, does a Lieb–Robinson bound of braid
propagation exist? The 4-valent scheme expects that actively interacting braids saturate
the Lieb–Robinson bound of the system but non-actively interacting ones do not, such
that they are respectively massless and massive.
3.7.1 Future directions
We now sketch our plan of reformulating the 4-valent scheme or even our whole program of
emergent matter in other frameworks of mathematical physics, such as GFT, Tensor Category
Theory, and so on.
Group field theories with braids. GFTs22 consider d-dimensional simplicies the funda-
mental building blocks of pd`1q-dimensional spacetime and treat them as fields whose variables
are elements in the group defining the simplicies. That is, a GFT is a “local, covariant quantum
field theory of universes” in terms of the fields associated with the fundamental building blocks.
It would produce a transition amplitude between quantum “universes” by summing over the
Feynman diagrams of this transition, i.e., summing over all triangulations and topologies as the
histories built from the evolution of the fundamental d-simplicies. These Feynman diagrams can
also be viewed as spin networks and dual to pd ` 1q-simplicies. Group Field Theories encom-
pass most of the other approaches to non-perturbative quantum gravity, such as Loop Quantum
Gravity and Spin Foam models, provide a link between them, and go beyond the limitations of
them [73].
We name two viable routes of formulating a GFT of 4-valent braids. Spin networks are purely
combinatoric and unembedded in GFTs, so the first strategy is to enlarge the configuration space
of a certain p3 ` 1q GFT by adding to its fundamental field group variables that characterize
a 4-valent braid.
Inspired by constructing theories of collective modes in condensed matter physics, our second,
simpler strategy is to devise a braid field as a composite field of a pair of fundamental group fields
whose group variables are identified in a braided way, and then integrate out the fundamental
fields to obtain an effective theory of the composite fields in certain backgrounds given by the
fundamental ones.
Both ways combine spin network labels automatically and are expected to result in a low
energy effective theory of braid excitations in a background spacetime. The former sounds more
fundamental and should be able to solve the issue that nontrivial braids cannot be created from
spin networks initially free of braids. The latter is what we are currently taking, by which we
found it is likely to construct a toy GFT with only certain trivial braids, whose effective theory
is a scalar φ4 theory.
Tensor categorical methods. Braided tensor categories [48] appear to be another elegant
and unified way to resolve many aforementioned issues once and for all. In fact, the connection
between LQG and SF Models and Tensor Categories has been recognized for decades [22, 49, 50].
Note that the string-net condensate due to Wen et al. [35, 36] also illuminates that tensor
categories may be the language underlying a unification of gravity and matter. Braided tensor
categories can free our program from embedding by casting both trivalent and 4-valent braids
combinatorially [49], which is beyond the context of LQG.
21This is understood as the maximum speed at which information can propagate in a system.
22The first GFT – the Boulatov model – originated as a generalization of the Matrix Models of 2D gravity to
3D [17]. GFTs in 3D and 4D were realised to be generating theories of Spin Foam models [32, 71]. Later, GFTs
are suggested to be fundamental formulations of quantum gravity [73]. [72] presents an extensive review on the
subject.
Emergent Braided Matter of Quantum Geometry 35
A twist of a strand of a braid can be interpreted as characterizing a non-trivial isomorphism
from Up1q to Up1q. Nonetheless, the concept of twist can be generalized to any vector spaces,
which is how it is defined in braided tensor categories. In this manner, we may view spin network
labels as if they represent generalized framing of spin networks other than the Up1q framing we
have just studied, such that generalized twists can arise, which may offer a unification of our
twists and spin network labels, as well as of internal symmetries and spacetime symmetries.
The end-nodes and external edges of 4-valent braids may exert further constraints on what
tensor categories are at our disposal or even motivate new types of tensor categories. Tensor-
categorized 4-valent braids and evolution moves may be evaluated by the relevant techniques
already defined in theories of tensor categories or new techniques adapted to our case.
3.7.2 Relation to topological quantum computing
One should not be surprised to notice that the 4-valent scheme of emergent matter is related
to Topological Quantum Computing (TQC). This relation has three facets. Firstly, though
seemingly superficial, braids and their algebra are present in both disciplines. A major difference
is that each 4-valent braid have two end-nodes and has only three strands, which is not the case
in TQC.
Secondly, as aforementioned, being a concept rooted in Quantum Computing/Information
and adapted to models of quantum gravity, noiseless subsystems are a key underlying notion of
the program of emergent matter. Furthermore, [56, 64] suggest that background independent
quantum gravity is a quantum information processing system. On the other hand, in [37, 58, 59]
topological quantum phase transitions have proven to give rise to emergent gauge and linearized
gravitons.
Thirdly, one of our future directions is to employ tensor categories – in particular braided
ribbon categories – to make an elegant reformulation of the 4-valent scheme, while TQC is also
naturally described in the language of tensor categories [86, 104] and related to framed spin
networks [50].
Therefore, it is interesting to study TQC from the viewpoint of quantum gravity and vice
versa, which may shed new light on both disciplines. For example, we may interpret each 4-
valent braid as representing a process of quantum computation, with an end-node of the braid
as a fusion rule of anyons or a quantum gate in TQC. We wonder if the interactions of 4-
valent braids can be introduced to TQC to study how two quantum processes can join, how one
quantum process can split, and when two sequences of quantum processes can be equivalent.
Conversely, TQC may be useful in understanding the significance of the conserved quantities of
4-valent braids.
4 A unified formalism
Recently in [39] the trivalent nodes were recast into the tetravalent scheme, giving a consistent
footing to study which results from each scheme could be transferred over to the other. Here we
reproduce the unified definition of braided ribbon networks of valence n (with n ě 3) as follows:
• We begin by considering an n-valent graph embedded in a compact 3 dimensional manifold.
We construct a 2-surface from this by replacing each node by a 2-sphere with n punctures
(1-sphere boundaries on the 2-sphere), and each edge by a tube which is then attached to
each of the nodes that it connects to by connecting the tube to one of the punctures on
the 2-sphere corresponding to the node.
• Lastly we add to each tube n´1 curves from one puncture to the other and then continue
these curves across the sphere in such a way that each of the n tubes connected to a node
shares a curve with each of the other tubes.
36 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
• We will freely call the tubes between spheres edges, the spheres nodes and the curves on
the tubes racing stripes or less formally stripes.
• We will call a braided ribbon network the equivalence class of smooth deformations of such
an embedding that do not involve intersections of the edges or the racing stripes.
We immediately face the following consequence: under this definition23 there are only braided
ribbon networks of valence 2, 3 or 4 (with valence 2 being a collection of framed loops). To
see this fact we consider a 5-valent node – a 2-sphere with 5 punctures, with each puncture
connected to each other puncture by a non-intersecting curve. Taking each puncture as a node,
and the curves as edges, we then get that these objects would constitute the complete graph
on 5 nodes and as they lie in the surface of a 2-sphere, such a graph would have to be planar.
This is impossible by Kuratowski’s theorem [57]: the complete graph on 5 nodes is non-planar.
Likewise, we have for any higher valence n that the graph that would be constructed would have
the complete graph on 5 nodes as a subgraph, and so they too can not be planar. If the reader
desires an intuition for this, it may be instructive to recall that these statements follow from the
four colour theorem – the existence of such a node would imply the existence of a map requiring
five (or more) colours.
(a) (b)
Figure 27. (a) Trivalent node. (b) Four-valent node.
We can also introduce a modification to the framework that allows for higher valence vertices.
To do this we first make a few definitions.
Definition 4.1. We define the natural valence of a braided ribbon network to be the number
of racing stripes on each edge.
Definition 4.2. We say that a node is natural if each of the tubes which intersect share a racing
stripe with each of the other tubes. Otherwise we will say that a node is composite.
We can then define a n-valent BRN with natural valence m (here n can take values of
n “ km´ 2pk ´ 1q for any integer k) as a braided ribbon network where each of the nodes has
n tubes which intersect it but where each of the tubes has m ´ 1 racing stripes. Likewise we
can define a multi-valent BRN with natural valence m in a similar manner but without fixing
the value of k for all nodes. We then construct composite nodes by connecting natural nodes in
series by simple edges and shortening the edges which connect them internally until all of these
nodes combine into a single sphere with the appropriate number of punctures (see Fig. 28). As
these combined nodes are simply glued they are then dual to gluings of simplices which when
grouped together would be equivalent to a polygon (for a natural valence of 3) or a polyhedron
with triangular faces(for natural valence of 4).
4.1 Relating to the ribbon pictures
We can construct from the previous form of trivalent braided ribbon graphs a braided ribbon
network as we’ve now defined them as follows: for each node of the network we consider a closed
ball in the embedding space which has the node on its surface but which has an empty intersection
23See [11] for another point of view.
Emergent Braided Matter of Quantum Geometry 37
Figure 28. Forming composite nodes.
Figure 29. From BRN to trinions.
with the rest of the ribbon graph. These spheres then define the nodes of the braided ribbon
network. The edges of the braided ribbon networks are then defined by similarly constructing
tubes between these spheres so that the boundaries of the edges of the ribbon graph coincide
with the boundary of the tubes. The boundaries of the surface of the ribbon graph then become
the racing stripes of the braided ribbon network.
Likewise we can construct a traditional braided ribbon network from a 3-valent braided
ribbon network by making the following observation: at each node the racing stripes divide
the sphere into two parts, likewise along each edge the tube is divided in two by the racing
stripes. Provided each closed path through the network encounters and even number of twists
through ˘π the network will split along the racing stripes into two orientable surfaces. We can
consistently choose one side or the other and identify this as the surface of a traditional braided
ribbon network (alternatively we can think of ‘squishing’ the two halves into a single surface, in
a way defining one side to be the ‘front’ and the other the ‘back’).
4.2 Applications of the unified formalism
In [38] and [39] this formalism was used to demonstrate several general results for braided ribbon
networks and embedded spin networks. We shall not reproduce these results here, but instead
direct the reader to those papers for demonstrations of:
• The generalization of the reduced link to the unified formalism, and hence to 4-valent
BRNs.
• The demonstration of the conservation of the 4-valent reduced link.
• The Construction of maps between BRNs and Spin Networks.
• The demonstration that the reduced link is a conserved quantity for Spin Networks.
These results give us a new use for braided ribbon networks: they have become an effective
tool for understanding the information in the embedding of Spin Networks. They also demon-
strate that a great deal of the structures that we study in BRNs also exist and are conserved in
embedded spin networks.
38 S. Bilson-Thompson, J. Hackett, L. Kauffman and Y. Wan
4.3 Correspondence between the trivalent and tetravalent cases
The natural formulation of framed tetravalent networks, as mentioned in Section 3.1, is as
tubular links between spherical nodes. It is in fact quite easy to see that such a network can
be matched up to a framed trivalent (ribbon) network, by simply “slicing” a tubular link down
opposite sides, as discussed in Section 4.1.
Given any framed trivalent network, we can always combine adjacent nodes to create com-
posite tetravalent nodes. Likewise, the tetravalent nodes of the ribbon networks obtained by the
splitting process described above can be decomposed into pairs of trivalent nodes. This allows
us to switch between braids in the framed trivalent and tetravalent cases.
Suppose we begin with a trivalent framed braid. We are always able to reduce this braid to
its pure twist form, as noted above. Once in this form, in which all crossings have been removed,
it is always possible to rotate the node at the top of the braid in such a manner that all the
twisting on one strand (say, the rightmost strand) is removed, and extra twists and crossings
are induced on the other two strands. We thereby arrive at a braid on three strands in which
a single strand does not carry any twisting or crossing. The node at the bottom of this strand
may then be freely combined with the node at the top of the braid to form a single tetravalent
node. Likewise the nodes at the bottom of the two twisted strands may be combined to form a
single tetravalent node. This process is illustrated in Fig. 30. By this process we obtain a braid
located between two tetravalent nodes, just as occur in the framed tetravalent case (Section 3.1).
The braid obtained is, of course, embedded in a ribbon network, but it can always be used to
reconstruct a tube-and-sphere framed tetravalent BRN.
Figure 30. Forming composite nodes allows us to convert between trivalent and tetravalent style braids.
The significance of this construction is that it allows us to associate framed tetravalent
networks with structures occurring in the helon model (and hence with SM fermions), and allows
the structures in the helon model to interact via the results on framed tetravalent networks [93].
We thereby obtain a model which allows us to reproduce both kinematic and dynamical aspects
of the Standard Model.
5 Conclusions
While the idea of matter emerging from spacetime as topological substructures is an old one, it
is only recently that our understanding of the subatomic structure of matter has made models
of such emergent matter viable. In this review article we have discussed two parallel approaches,
the trivalent and tetravalent scheme, which grew out of the suggestion that the most basic level
of substructure within matter may be modelled by braided ribbons. The tetravalent scheme
has proven to embody a rich dynamical theory of braid interactions and propagation ruled by
topological conservation laws, but has until now not been able to construct a direct mapping
to the particle states of the SM, instead producing a seemingly infinite range of equivalence
classes of braid states that fall into two types respectively analogous to bosons and fermions.
Emergent Braided Matter of Quantum Geometry 39
The trivalent scheme has been unable to model interactions, but has been quite successful at
taming the profusion of braid states present by constructing equivalence classes of braids, each
equivalence class being mapped to a single type of particle. The unification of trivalent and
tetravalent approaches we suggest here promises to allow the development of a fully dynamical
theory of interacting particles, to restrict the range of particle states existing within the theory,
and to provide a Rosetta stone that allows trivalent braids, tetravalent braids, and the particles
of the SM to be equated in a satisfying manner. If successful, this will be a compelling theory
of quantum spacetime and emergent matter.
Acknowledgements
SBT is grateful to the Ramsay family for their support through the Ramsay Postdoctoral Fellow-
ship. JH is grateful to his Thesis Advisor Lee Smolin for his discussion and critical comments.
YW is in debt to his Supervisor Mikio Nakahara for his constant support and generosity. YW
is also supported by “Open Research Center” Project for Private Universities: matching fund
subsidy from MEXT, Japan.
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1 Introduction
1.1 An invitation to emergent matter of quantum geometry
1.2 Noiseless subsystems
1.3 Spin networks
1.3.1 Penrose's spin networks
1.3.2 Spin networks from LQG
1.4 Three Ansatzes
2 The trivalent scheme
2.1 The helon model
2.2 Topological invariants of trivalent braids
2.3 Quantum numbers of particle states
2.4 Interactions and embedding in trivalent networks
3 The 4-valent scheme
3.1 4-valent braided ribbon networks
3.2 Braids
3.3 Equivalence moves
3.3.1 /3-rotations: generators of rotations
3.3.2 Braid representations
3.4 Dynamics: evolution moves
3.5 Dynamics: propagation, direct and exchange interaction of braids
3.6 Dynamics: CPT and braid Feynman diagrams
3.6.1 C, P, and T
3.6.2 Asymmetry of braid interaction
3.6.3 Braid Feynman diagrams
3.7 Discussions and outlook
3.7.1 Future directions
3.7.2 Relation to topological quantum computing
4 A unified formalism
4.1 Relating to the ribbon pictures
4.2 Applications of the unified formalism
4.3 Correspondence between the trivalent and tetravalent cases
5 Conclusions
References
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