A Cable Knot and BPS-Series
A series invariant of the complement of a knot was introduced recently. The invariants for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable...
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| Дата: | 2023 |
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Інститут математики НАН України
2023
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| Цитувати: | A Cable Knot and BPS-Series. John Chae. SIGMA 19 (2023), 002, 12 pages |
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| citation_txt | A Cable Knot and BPS-Series. John Chae. SIGMA 19 (2023), 002, 12 pages |
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| description | A series invariant of the complement of a knot was introduced recently. The invariants for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of the integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the figure eight knot. This relation provides an alternative, simple method of finding the former series invariant.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 002, 12 pages
A Cable Knot and BPS-Series
John CHAE
Department of Mathematics, Univeristy of California Davis, Davis, USA
E-mail: yjchae@ucdavis.edu
Received August 03, 2022, in final form January 05, 2023; Published online January 13, 2023
https://doi.org/10.3842/SIGMA.2023.002
Abstract. A series invariant of a complement of a knot was introduced recently. The
invariant for several prime knots up to ten crossings have been explicitly computed. We
present the first example of a satellite knot, namely, a cable of the figure eight knot, which
has more than ten crossings. This cable knot result provides nontrivial evidence for the
conjectures for the series invariant and demonstrates the robustness of integrality of the
quantum invariant under the cabling operation. Furthermore, we observe a relation between
the series invariant of the cable knot and the series invariant of the figure eight knot. This
relation provides an alternative simple method of finding the former series invariant.
Key words: knot complement; quantum invariant; q-series; Chern–Simons theory; categori-
fication
2020 Mathematics Subject Classification: 57K10; 57K16; 57K31; 81R50
1 Introduction
Inspired by a categorification of the Witten–Reshitikhin–Turaev invariant of a closed oriented
3-manifold [28, 29, 36] in [15, 16], a two variable series invariant FK(x, q) for a complement
of a knot M3
K was introduced in [13]. Although its rigorous definition is yet to be found, it
possesses various properties such as the Dehn surgery formula and the gluing formula. This
knot invariant FK takes the form1
FK(x, q) =
1
2
∞∑
m≥1
m odd
(
xm/2 − x−m/2
)
fm(q) ∈ 1
2c
q∆Z
[
x±1/2
][[
q±1
]]
, (1.1)
where fm(q) are Laurent series with integer coefficients,2 c ∈ Z+ and ∆ ∈ Q. Moreover, x-
variable is associated to the relative Spinc
(
M3
K , T 2
)
-structures, which is affinely isomorphic to
H2
(
M3
K , T 2;Z
) ∼= H1
(
M3
K ;Z
) ∼= Z. It is infinite cyclic, which is reflected as a series in FK . The
rational constant ∆ was investigated in [14], which elucidated its intimate connection to the
d-invariant (or the correction term) in certain versions of the Heegaard Floer homology
(
HF±)
for rational homology spheres. The physical interpretation of the integer coefficients in fm(q)
are number of BPS states of 3d N = 2 supersymmetric quantum field theory on M3
K together
with boundary conditions on ∂M3
K . Furthermore, it was conjectured that FK also satisfies the
Melvin–Morton–Rozansky conjecture [23, 30, 31] (proven in [1]):
Conjecture 1.1 ([13, Conjecture 1.5]). For a knot K ⊂ S3, the asymptotic expansion of the
knot invariant FK
(
x, q = eℏ
)
about ℏ = 0 coincides with the Melvin–Morton–Rozansky (MMR)
expansion of the colored Jones polynomial in the large color limit:
FK
(
x, q = eℏ
)
x1/2 − x−1/2
=
∞∑
r=0
Pr(x)
∆K(x)2r+1
ℏr, (1.2)
1Implicitly, there is a choice of group; originally, the group used is SU(2).
2They can be polynomials for monic Alexander polynomial of K, see Section 3.2.
mailto:yjchae@ucdavis.edu
https://doi.org/10.3842/SIGMA.2023.002
2 J. Chae
where x = enℏ is fixed, n is the color of K, Pr(x) ∈ Q
[
x±1
]
, P0(x) = 1 and ∆K(x) is the
(symmetrized) Alexander polynomial of K.
Additionally, motivated by the quantum volume conjecture/AJ-conjecture [7, 11] (explained
in Section 2.2), it was conjectured that FK-series is q-holonomic:
Conjecture 1.2 ([13, Conjecture 1.6]). For any knot K ⊂ S3, the normalized series fK(x, q)
satisfies a linear recursion relation generated by the quantum A-polynomial of K ÂK(q, x̂, ŷ):
ÂK(q, x̂, ŷ)fK(x, q) = 0, (1.3)
where fK := FK(x, q)/
(
x1/2 − x−1/2
)
.
The actions of x̂ and ŷ are
x̂fK(x, q) = xfK(x, q), ŷfK(x, q) = fK(xq, q).
FK-series has been computed for several prime knots up to ten crossings in [12, 13, 20, 27]. They
include the torus knots, the figure eight knot in [13, 26, 27]. Positive braid knots (10139, 10152),
strongly quasipositive braids knots (m(10145), 10154, 10161), double twist knots (m(52), m(73),
m(74)), and a few more prime knots (m(75), m(815)) were examined in [27]. Furthermore, the
series for 52 and 62 were calculated in [6].
In this paper, we verify the above conjectures by computing the FK-series for (9, 2)-cabling
of the figure eight knot and we compare our result to that of the figure eight knot. Furthermore,
we conjecture the form of the FK-series for a family of a cable knot of the figure eight.
The rest of the paper is organized as follows. In Section 2, we review the satellite operation on
a knot and the recursion ideal of the quantum torus. In Section 3, we analyze knot polynomials
of the cable knot of the figure eight. In Section 4, we derive the recursion relation for the cable
knot. Then we deduce ℏ expansion from the recursion in Section 5. In Section 6, consequences
of the cabling operation are discussed and we propose a conjecture about a family of a cable
knot. Finally, in Section 7, we state a relation between the series invariant of the cable knot
and the series invariant of the figure eight knot and conjecture about other cabling of the figure
eight knot.
2 Background
2.1 Satellites
The satellite operation consists of a pattern knot P in the interior of the solid torus S1 ×D2,
a companion knot K ′ in the S3 and an canonical identification hK′
hK′ : S1 ×D2 −→ ν(K ′) ⊂ S3, (2.1)
where ν(K ′) is the tubular neighborhood of K ′. A well-known example of satellite knots is
a cable knot hK′(P ) = C(r,s)(K
′) that is obtained by choosing P to be the (r, s)-torus knot
pushed into the interior of the S1 ×D2. This map hK′ has been investigated in [22, 24, 25].
2.2 Quantum torus and recursion ideal
Let T be a quantum torus
T := C
[
t±1
]〈
M±1, L±1
〉
/
(
LM − t2ML
)
.
A Cable Knot and BPS-Series 3
Figure 1. A companion K ′ (left), pattern knot P (center) and satellite knot P (K ′) (right).
The generators of the noncommutative ring T acts on a set of discrete functions, which are
colored Jones polynomials JK,n ∈ Z
[
t±1
]
in our context, as
MJK,n = t2nJK,n, LJK,n = JK,n+1.
The recursion (annihilator) ideal AK of JK,n is the left ideal AK in T consisting of operators
that annihilates JK,n:
AJK,n
:=
{
αK ∈ T |αKJK,n = 0
}
.
It turns out thatAK is not a principal ideal in general. However, by adding inverse of polynomials
of t and M to T [7], we obtain a principal ideal domain T̃
T̃ :=
{∑
j∈Z
aj(M)Lj
∣∣ aj(M) ∈ C
[
t±1
]
(M), aj = almost always 0
}
Using T̃ we get a principal ideal ÃK := T̃ AK generated by a single polynomial ÂK
ÂK(t,M,L) =
d∑
j=0
aj(t,M)Lj .
This ÂK polynomial is a noncommutative deformation of a classical A-polynomial of a knot [3]
(see also [4]). Alternative approaches to obtain ÂK(t,M,L) are by quantizing the classical
A-polynomial curve using a twisted Alexander polynomial or applying the topological recur-
sion [17]. A conjecture called AJ conjecture/quantum volume conjecture was proposed in [7, 11]
via different approaches:
Conjecture 2.1. For any knot K ⊂ S3, ÂK (t = −1, L,M) reduces to the (classical) A-poly-
nomial curve AK(L,M) up to a solely M -dependent overall factor.
In other words, JK,n(t) satisfies a linear recursion relation generated by ÂK(t,M,L). This
property of JK,n is often called q-holonomic [9]. The conjecture was confirmed for a variety of
knots [5, 7, 8, 10, 19, 21, 33, 35].
3 Knot polynomials
In this section we will analyze the colored Jones polynomial and the Alexander polynomial of
a cable knot to show that the former satisfies the MMR expansion and the latter is monic.
Furthermore, the MMR expansion enables us to read off the initial condition that is needed in
Section 5.
4 J. Chae
3.1 The colored Jones polynomial
For (r, 2)-cabling of the figure eight knot 41, we set P = T (r, 2) andK ′ = 41 in (2.1). The cabling
formula for an unnormalized sl2(C) colored Jones polynomial of a (r, 2)-cabling of a knot K ′
in S3 is [34]
J̃C(r,2)(K
′),n(q) = q−
r
2
(n2−1)
n∑
w=1
(−1)r(n−w)q
r
2
w(w−1)J̃K′,(2w−1)(q), |r| > 8 and odd.
r
half
twists
Figure 2. (r, 2)-cable of the figure eight knot.
Its application to K = C(9,2)(41),
3 whose diagram has 25 crossings, is
J̃K,n(q) = q−
9
2
(n2−1)
n∑
w=1
[
(−1)(n−w)q
9
2
w(w−1)[2w − 1]
×
2w−2∑
r=0
r∏
k=1
(
−q−k − qk + q1−2w + q2w−1
)]
.
Using the (0-framed) unknot U value
J̃U,n(t) =
t2n − t−2n
t2 − t−2
,
together with q = t4, the first few normalized polynomials JK,n(q) can be written as
JK,1(q) = 1,
JK,2(q) = q2 − q +
1
q4
+
1
q6
− 1
q7
+
1
q8
− 1
q9
+
1
q12
− 1
q13
,
JK,3(q) = q12 − q11 − q10 + q9 − q8 + q7 + q6 − q5 + q2 − 1 +
1
q8
+
1
q11
− 1
q13
+
1
q14
− 1
q16
+
1
q17
− 1
q18
− 1
q19
+
2
q20
− 1
q21
+
1
q23
− 1
q24
+
1
q25
+
1
q26
− 2
q27
− 1
q28
+
1
q29
− 1
q30
+
2
q32
− 1
q33
− 1
q34
+
1
q35
.
Proposition 3.1. The ℏ expansion of the above JK,n(q) is given by
JK,n
(
eℏ
)
= 1 +
(
6− 6n2
)
ℏ2 +
(
−42 + 42n2
)
ℏ3 +
(
801
2
− 462n2 +
123
2
n4
)
ℏ4
3This cabling parameters correspond to 91 for the pattern knot. We assume 0-framing for 41.
A Cable Knot and BPS-Series 5
+
(
−8451
2
+ 5173n2 − 1895
2
n4
)
ℏ5 +
(
3111491
60
− 132779
2
n2 + 14986n4
− 27281
60
n6
)
ℏ6 +
(
−14631401
20
+
19399417
20
n2 − 3028829
12
n4 +
840097
60
n6
)
ℏ7
+
(
39069313501
3360
− 950122877
60
n2 +
54585517
12
n4 − 1725671
5
n6
+
13273763
3360
n8
)
ℏ8 + · · · . (3.1)
We see that, at each ℏ order, the degree of the polynomial in n is at most the order of ℏ,
which is an equivalent characterization of the MMR expansion of the colored Jones polynomial
of a knot. Secondly, as a consequence of the cabling, odd powers of ℏ appear in the expansion,
even though they are absent in the case of the figure eight knot [13].
3.2 The Alexander polynomial
The cabling formula for the Alexander polynomial of a knot K is [18]
∆C(p,q)(K)(t) = ∆K(tp)∆T(p,q)
(t), 2 ≤ p < |q|, gcd(p, q) = 1,
where ∆(t) is the symmetrized Alexander polynomial and T(p,q) is the (p, q) torus knot. Note
that our convention for the parameters of the torus knot are switched (i.e., p ≡ 2, q ≡ r).
Lemma 3.2. The symmetrized Alexander polynomial of C(9,2)(41) is as follows:
∆C(9,2)(41)(x) = ∆41
(
x2
)
∆T(2,9)
(x)
= −x6 − 1
x6
+ x5 +
1
x5
+ 2x4 +
2
x4
− 2x3 − 2
x3
+ x2 +
1
x2
− x− 1
x
+ 1.
From this Alexander polynomial its symmetric expansion about x = 0 (in x) and x = ∞
(in 1/x) in the limit of ℏ → 0 can be computed:
lim
q→1
2FK(x, q) = 2 s. e.
(
x1/2 − x−1/2
∆K(x)
)
= x11/2 − 1
x11/2
+ 2x15/2 − 2
x15/2
+ 5x19/2 − 5
x19/2
+ 13x23/2 − 13
x23/2
+ 34x27/2 − 34
x27/2
− x29/2 +
1
x29/2
+ 89x31/2 − 89
x31/2
− 2x33/2 +
2
x33/2
+ 233x35/2 − 233
x35/2
− 5x37/2 +
5
x37/2
+ 610x39/2 − 610
x39/2
+ · · ·
∈ Z
[[
x±1/2
]]
. (3.2)
The coefficients in the expansions are integers and hence the Alexander polynomial is monic,
which is a necessary condition for fm(q)’s in (1.1) to be polynomials.
4 The recursion relation
The quantum (or noncommutative) A-polynomial of a class of cable knot C(r,2)(41) in S3 having
minimal L-degree is given by [32]
ÂK(t,M,L) = (L− 1)B(t,M)−1Q(t,M,L)
(
M rL+ t−2rM−r
)
∈ ÃK , (4.1)
6 J. Chae
where
Q(t,M,L) = Q2(t,M)L2 +Q1(t,M)L+Q0(t,M), B(t,M) :=
2∑
j=0
cjb
(
t, t2j+2M2
)
,
b(t,M) =
M
(
1 + t4M2
)(
−1 + t4M4
)(
−t2 + t14M4
)
t2 − t−2
,
c0 = P̂0
(
t, t4M2
)
P̂1
(
t, t6M2
)
, c1 = −P̂1
(
t, t2M2
)
P̂1
(
t, t6M2
)
,
c2 = P̂1
(
t, t2M2
)
P̂2
(
t, t4M2
)
.
The definitions of the operators P̂i are written in Appendix A.1. For K = C(9,2)(41), apply-
ing (4.1) to fK(x, q) together with x = qn yields via (1.3)
α(x, q)FK(x, q) + β(x, q)FK(xq, q) + γ(x, q)FK
(
xq2, q
)
+ δ(x, q)FK
(
xq3, q
)
+ FK
(
xq4, q
)
= 0, (4.2)
where α, β, γ, δ functions and their ℏ series are documented in [2]. From (4.2) we find the
recursion relation for fm.
Theorem 4.1. The recursion relation for fm(q) ∈ Z[q±1] of the above FK(x, q) is given by
fm+98(q) =
−1
q
109+m
2
(
1− q
87+m
2
)[t2fm+94 + t4fm+90 + t6fm+86 + t8fm+82 + t9fm+80
+ t10fm+78 + t11fm+76 + t12fm+74 + t13fm+72 + t14fm+70 + t15fm+68
+ t16fm+66 + t17fm+64 + t18fm+62 + t19fm+60 + t20fm+58 + t21fm+56
+ t22fm+54 + t23fm+52 + t24fm+50 + t25fm+48 + t26fm+46 + t27fm+44
+ t28fm+42 + t29fm+40 + t30fm+38 + t31fm+36 + t32fm+34 + t33fm+32
+ t34fm+30 + t35fm+28 + t36fm+26 + t37fm+24 + t38fm+22 + t39fm+20 (4.3)
+ t40fm+18 + t41fm+16 + t43fm+12 + t45fm+8 + t47fm+4 + t49fm
]
∈ Z
[
q±1
]
,
where tv = tv(q, q
m)’s are listed in [2]. The initial data for (4.3) were found using the ℏ-ex-
pansions of (4.2). An example of the expansion is written in Section 5 for FK(x, q). Using the
recursion relation (4.3) and the initial data documented in [2], FK(x, q) can be obtained to any
desired order in x.
5 An expansion of a knot complement
We next compute a series expansion of the FK of complement of the cable knot K. Specifi-
cally, a straightforward computation from (4.2) yields an ordinary differential equation (ODE)
for Pm(x) at each ℏ order. Using the initial conditions for the ODEs obtained from (3.1)
P1(1) = 0, P2(1) = 6, P3(1) = −42, P4(1) =
801
2
, P5(1) = −8451
2
, . . . ,
we find that
P1(x) = 5x12 +
5
x12
− 10x11 − 10
x11
− 13x10 − 13
x10
+ 36x9 +
36
x9
− 10x8 − 10
x8
− 16x7
− 16
x7
+ 15x6 +
15
x6
− 14x5 − 14
x5
+ 16x4 +
16
x4
− 18x3 − 18
x3
+ 19x2 +
19
x2
− 20x− 20
x
+ 20,
A Cable Knot and BPS-Series 7
P2(x) =
25x24
2
+
25
2x24
− 50x23 − 50
x23
− 14x22 − 14
x22
+ 306x21 +
306
x21
− 641x20
2
− 641
2x20
− 448x19 − 448
x19
+
2011x18
2
+
2011
2x18
− 358x17 − 358
x17
− 522x16 − 522
x16
+ 612x15
+
612
x15
− 589x14
2
− 589
2x14
+ 508x13 +
508
x13
− 3325x12
2
− 3325
2x12
+ 1648x11 +
1648
x11
+ 1538x10 +
1538
x10
− 3932x9 − 3932
x9
+ 1574x8 +
1574
x8
+ 1670x7 +
1670
x7
− 1798x6
− 1798
x6
+ 396x5 +
396
x5
− 1521x4
2
− 1521
2x4
+ 4082x3 +
4082
x3
− 6541x2
2
− 6541
2x2
− 8334x− 8334
x
+ 16831.
Substituting them into (1.2) results in
2F
(
x, eℏ
)
=
(
x11/2 − 1
x11/2
+ 2x15/2 − 2
x15/2
+ 5x19/2 − 5
x19/2
+ 13x23/2 − 13
x23/2
+ 34x27/2 − 34
x27/2
− x29/2 +
1
x29/2
+ 89x31/2 − 89
x31/2
− 2x33/2 +
2
x33/2
+ 233x35/2 − 233
x35/2
− 5x37/2 +
5
x37/2
+ · · ·
)
+ ℏ
(
5x11/2 − 5
x11/2
+ 12x15/2 − 12
x15/2
+ 35x19/2 − 35
x19/2
+ 104x23/2
− 104
x23/2
+ 306x27/2 − 306
x27/2
− 15x29/2 +
15
x29/2
+ 890x31/2 − 890
x31/2
− 36x33/2 +
36
x33/2
+ 2563x35/2 − 2563
x35/2
− 105x37/2 +
105
x37/2
+ · · ·
)
+ ℏ2
(25
2
x11/2 − 25
2
1
x11/2
+ 36x15/2 − 36
x15/2
+
247
2
x19/2 − 247
2
1
x19/2
+ 426x23/2 − 426
x23/2
+ 1441x27/2 − 1441
x27/2
− 225
2
x29/2 +
225
2
1
x29/2
+ 4781x31/2
− 4781
x31/2
− 324x33/2 +
324
x33/2
+ 2563x35/2 − 2563
x35/2
− 2207
2
x37/2
+
2207
2
1
x37/2
+ · · ·
)
.
Comparing to the series of the figure eight knot [13], we notice that every order of ℏ appears
in the above series whereas the series corresponding to the figure eight knot consists of only
even powers of ℏ (i.e., Pi(x) = 0 for i odd). This difference is an effect of the torus knot whose
expansion involve all powers of ℏ [13]. Furthermore, the x-terms begin from m = 11 instead
of m = 1 and there are gaps in their powers. Specifically, x±13/2, x±17/2, x±21/2 and x±25/2
are absent. This is a consequence of the structure of (3.2). A distinctive feature of the cable
knot is that from x±29/2 the coefficients are negative. Moreover, the positive and the negative
coefficients alternate from that x-power for all ℏ powers. These differences persist in the higher
ℏ-orders. We will see these differences in a manifest way in the next section.
6 Effects of the cabling
Since the initial data plays a core role in the recursion relation method, we discuss their features
for the cable knot and then propose conjectures about it, which can be a useful guide for finding
initial data for a family of the cable knots.
8 J. Chae
In the initial data (see [2]) for the recursion relation (4.3), we notice several differences from
that of the figure eight knot [13]. Before discussing them, let us begin with the properties of
the FK that are preserved by the cabling. The initial data consists of an odd number of terms
and power of q increases by one between every consecutive terms in a fixed fm for all m’s, which
are also true for f99 and f101. Additionally, the reflection symmetry of coefficients is retained up
to f43 for positive coefficients and up to f61 for the negative ones but of course, those fm’s do
not have the complete amphichiral structure. These invariant properties are a remnant feature
of the amphichiral property of the figure eight knot.
A difference is that the nonzero initial data begins from f11 and the gaps between the pow-
ers of x is four up to x27/2, which is in the accordance with fm’s. These features are direct
consequences of the symmetric expansion of the Alexander polynomial of the cable knot (3.2).
In the case of the figure eight its coefficient functions start from f1 and there are no such
gaps. Another distinctive difference is that fm’s containing negative coefficients appear from
m = 29. Moreover, the positive and negative coefficient fm’s alternate from f27 (i.e., positive
coefficients for f27, f31, . . . and negative coefficients for f29, f33, . . . ). Furthermore, from f47 the
reflection symmetry of the positive coefficients in the appropriate fm’s is broken. This phe-
nomenon also occurs for the negative coefficient fm’s from m = 65. Breaking of the symmetry
is expected since the cable knot of the figure eight is not amphichiral. The next difference is
that the maximum power of q in the positive coefficient fm’s for m ≥ 15, the powers increase
by +2,+2,+3,+3,+4,+4, . . . ,+11,+11. For example, for f15, f19, f23, f27, and f31, their max-
imum powers are q6, q8, q10, q13 and q16, respectively [2]. For the negative coefficient case, the
changes in maximum powers are 4, 4, 5, 5, 6, 6, . . . , 11, 11 from m = 33. The minimum powers
of fm’s having positive coefficients exhibit their changes as 0, 0,−1,−1,−2,−2,−3,−3, . . . and
for those with negative coefficients the pattern is +2,+2,+1,+1, 0, 0,−1,−1,−2,−2, . . . .
An universal feature of the negative coefficient fm’s in the initial data is that their coefficient
modulo sign is determined by the positive coefficient fm. For example, the absolute value of the
coefficients of f29 is same as that of f11; f33’s coefficients come from that of f15 up to sign and
so forth. Hence coefficients of fm having negative coefficients are determined by fm−18. In fact,
this peculiar coefficient correlation also exists in the non-initial data f101 whose coefficients are
correlated with that of f83.
Conjecture 6.1. For a class of a cable knot of the figure eight Kr = C(r,2)(41) ⊂ S3, r > 8
and odd, having monic Alexander polynomial, the coefficient functions
{
fm(q) ∈ Z
[
q±1
]}
of
FKr(x, q) can be classified into two (disjoint) subsets: one of them consists of elements having
all positive coefficients
{
f+
t (q)
}
t∈I+ and the other subset contains elements whose coefficients are
all negative
{
f−
w (q)
}
w∈I−. Furthermore, for every element in
{
f−
w (q)
}
, its coefficients coincide
with that of an element in
{
f+
t (q)
}
up to a sign.
7 A relation to the figure eight knot
In this section, we observe an interesting relation between FC(9,2)(41)(x, q) and F41(x, q). The
latter was computed in [13]. The relation enable us to circumvent the recursion method and
hence provides an alternative and efficient method for computing FC(9,2)(41)(x, q).
Proposition 7.1. The positive and negative subsets of the initial data for (4.3) of FC(9,2)(41)(x, q)
are related to F41(x, q) as follows, respectively.
f11(q) = h1(q)q
5,
f15(q) = h3(q)q
6,
f19(q) = h5(q)q
7,
A Cable Knot and BPS-Series 9
...
f43(q) = h17(q)q
13,
f47(q) = h19(q)q
14 + h1(q)q
34,
f51(q) = h21(q)q
15 + h3(q)q
39,
f55(q) = h23(q)q
16 + h5(q)q
44,
...
f79(q) = h35(q)q
22 + h17(q)q
74,
f83(q) = h37(q)q
23 + h19(q)q
79 + h1(q)q
99,
f87(q) = h39(q)q
24 + h21(q)q
84 + h3(q)q
108,
f91(q) = h41(q)q
25 + h23(q)q
89 + h5(q)q
117,
f95(q) = h43(q)q
26 + h25(q)q
94 + h7(q)q
126,
f29(q) = −h1(q)q
15,
f33(q) = −h3(q)q
18,
f37(q) = −h5(q)q
21,
...
f61(q) = −h17(q)q
39,
f65(q) = −h19(q)q
42 − h1(q)q
62,
f69(q) = −h21(q)q
45 − h3(q)q
69,
f73(q) = −h23(q)q
48 − h5(q)q
76,
...
f97(q) = −h35(q)q
66 − h17(q)q
118,
where hs(q) are the coefficient functions of F41(x, q) (see Appendix A.2).
We note that f1 = f3 = f5 = f7 = f9 = 0 as written in [2]. We emphasize that the initial
data for (4.3) in [2] were found from (1.2). The data turns out to be related to that of the figure
eight knot. The above relation persists for fm(q) that are in the complement of the initial data
set, namely, for m > 97. For example,
f99(q) = h45(q)q
27 + h27(q)q
99 + h9(q)q
135,
f101(q) = −h37(q)q
69 − h19(q)q
125 − h1(q)q
145.
They are in agreement with that obtained from (4.3). We state the following conjectures.
Conjecture 7.2. For a (r, 2)-cabling of 41 (|r| ≥ 2, gcd(r, 2) = 1) in ZHS3, the coefficient func-
tions {fm(q)} of the series invariant FC(r,2)(41)(x, q) are determined by the coefficient functions
{hs(q)} of F41(x, q) as
fm(q) = ±
(
hs1(q)q
w1 + · · ·+ hsj (q)q
wj
)
, wi ∈ Z,
for some j ∈ Z+ (j implicitly depends on m).
More generally, we propose that
10 J. Chae
Conjecture 7.3. For a (r, 2)-cabling of any (prime) hyperbolic knot K (|r| ≥ 2, gcd(r, 2) = 1)
in ZHS3, the coefficient functions {fm(q)} of the series invariant FC(r,2)(K)(x, q) are determined
by the coefficient functions {hs(q)} of FK(x, q) as
fm(q) = ±
(
hs1(q)q
w1 + · · ·+ hsj (q)q
wj
)
, wi ∈ Z,
for some j ∈ Z+.
We finish by listing two applications of our result for future work. First, we can use
FC(p,2)(41)(x, q) to find Ẑ associated with a closed oriented 3-manifold obtained by the Dehn
surgery on the cable knot using the surgery formula in [13, 26]. This would in turn enable us to
find the WRT invariant of the manifold using the result in [15]. In both applications, it would
extend Ẑ and the WRT invariant to broader classes of 3-manifolds.
A Appendix
A.1 The definitions of the operators
We list the definitions of the operators in the Â-polynomial (4.1):
Q2(t,M) = P̂2(t, t
4M2)P̂1
(
t, t2M2
)
P̂0
(
t, t6M2
)
,
Q1(t,M) = P̂0(t, t
4M2)P̂1
(
t, t6M2
)
P̂2(t, t
2M2)− P̂1
(
t, t6M2
)
P̂1
(
t, t2M2
)
P̂1
(
t, t4M2
)
+ P̂2
(
t, t4M2
)
P̂1
(
t, t2M2
)
P̂0
(
t, t6M2
)
,
Q0(t,M) = P̂0
(
t, t4M2
)
P̂1
(
t, t6M2
)
P̂0
(
t, t2M2
)
,
P̂0(t,M) := t6M4
(
−1 + t12M4
)
,
P̂1(t,M) := −
(
−1 + t4M2
)(
1 + t4M2
)(
1− t4M2 − t4M4 − t12M4 − t12M4
− t12M6 + t16M8
)
,
P̂2(t,M) := t10M4
(
−1 + t4M4
)
.
A.2 The data for the figure eight knot
We record the initial data and the recursion relation for the figure eight knot from [13]:
h1(q) = 1,
h3(q) = 2,
h5(q) =
1
q
+ 3 + q,
h7(q) =
2
q2
+
2
q
+ 5 + 2q + 2q2,
h9(q) =
1
q4
+
3
q3
+
4
q2
+
5
q
+ 8 + 5q + 4q2 + 3q3 + q4,
h11(q) =
2
q6
+
2
q5
+
6
q4
+
7
q3
+
10
q2
+
10
q
+ 15 + 10q + 10q2 + 7q3 + 6q4 + 2q5 + 2q6,
h13(q) =
1
q9
+
3
q8
+
4
q7
+
7
q6
+
11
q5
+
15
q4
+
18
q3
+
21
q2
+
23
q
+ 27 + 23q + 21q2 + 18q3 + 15q4
+ 11q5 + 7q6 + 4q7 + 3q8 + q9,
hm+14(q) = − q−
m
2
− 11
2
q
m
2
+ 13
2 − 1
[
hm
(
q
m
2
+ 17
2 − qm+9
)
+ hm+2
(
q
m
2
+ 15
2 − q
m
2
+ 17
2 + qm+9 − qm+10
)
A Cable Knot and BPS-Series 11
+ hm+4
(
−q
m
2
+ 11
2 − q
m
2
+ 17
2 − q
m
2
+ 19
2 + q
3m
2
+ 21
2 + qm+8 + qm+9 + qm+12 − q7
)
+ hm+6
(
−q
m
2
+ 9
2 + q
m
2
+ 11
2 − q
m
2
+ 15
2 − q
m
2
+ 17
2 + q
3m
2
+ 25
2 + qm+9 + qm+10
− qm+12 + qm+13 − q5
)
+ hm+8
(
q
m
2
+ 11
2 + q
m
2
+ 13
2 − q
m
2
+ 17
2 + q
m
2
+ 19
2 − q
3m
2
+ 31
2 − qm+8 + qm+9
− qm+11 − qm+12 + q2
)
+ hm+10
(
q
m
2
+ 9
2 + q
m
2
+ 11
2 + q
m
2
+ 17
2 − q
3m
2
+ 35
2 − qm+9 − qm+12 − qm+13 + 1
)
+ hm+12
(
q
m
2
+ 11
2 − q
m
2
+ 13
2 + qm+11 − qm+12
)]
.
Acknowledgements
I would like to thank Sergei Gukov, Thang Lê and Laura Starkston for helpful conversations.
I am grateful to Ciprian Manolescu for valuable suggestions on a draft of this paper. I am also
grateful to Colin Adams for valuable comments. I would like to thank to the referees for the
suggestions that led to an improvement of my manuscript.
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1 Introduction
2 Background
2.1 Satellites
2.2 Quantum torus and recursion ideal
3 Knot polynomials
3.1 The colored Jones polynomial
3.2 The Alexander polynomial
4 The recursion relation
5 An expansion of a knot complement
6 Effects of the cabling
7 A relation to the figure eight knot
A Appendix
A.1 The definitions of the operators
A.2 The data for the figure eight knot
References
|
| id | nasplib_isofts_kiev_ua-123456789-211882 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T18:56:37Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chae, John 2026-01-14T09:56:54Z 2023 A Cable Knot and BPS-Series. John Chae. SIGMA 19 (2023), 002, 12 pages 1815-0659 2020 Mathematics Subject Classification: 57K10; 57K16; 57K31; 81R50 arXiv:2101.11708 https://nasplib.isofts.kiev.ua/handle/123456789/211882 https://doi.org/10.3842/SIGMA.2023.002 A series invariant of the complement of a knot was introduced recently. The invariants for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of the integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the figure eight knot. This relation provides an alternative, simple method of finding the former series invariant. I would like to thank Sergei Gukov, Thang Lê, and Laura Starkston for helpful conversations. I am grateful to Ciprian Manolescu for valuable suggestions on a draft of this paper. I am also grateful to Colin Adams for valuable comments. I would like to thank the referees for the suggestions that led to an improvement of my manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Cable Knot and BPS-Series Article published earlier |
| spellingShingle | A Cable Knot and BPS-Series Chae, John |
| title | A Cable Knot and BPS-Series |
| title_full | A Cable Knot and BPS-Series |
| title_fullStr | A Cable Knot and BPS-Series |
| title_full_unstemmed | A Cable Knot and BPS-Series |
| title_short | A Cable Knot and BPS-Series |
| title_sort | cable knot and bps-series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211882 |
| work_keys_str_mv | AT chaejohn acableknotandbpsseries AT chaejohn cableknotandbpsseries |