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On the Total CR Twist of Transversal Curves in the 3-Sphere

We investigate the total CR twist functional on transversal curves in the standard CR 3-sphere S3⊂C2. The question of the integration by quadratures of the critical curves and the problem of the existence and properties of closed critical curves are addressed. A procedure for the explicit integratio...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2023
Main Authors: Musso, Emilio, Nicolodi, Lorenzo
Format: Article
Language:English
Published: Інститут математики НАН України 2023
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/212030
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:On the Total CR Twist of Transversal Curves in the 3-Sphere. Emilio Musso and Lorenzo Nicolodi. SIGMA 19 (2023), 101, 36 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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author Musso, Emilio
Nicolodi, Lorenzo
author_facet Musso, Emilio
Nicolodi, Lorenzo
citation_txt On the Total CR Twist of Transversal Curves in the 3-Sphere. Emilio Musso and Lorenzo Nicolodi. SIGMA 19 (2023), 101, 36 pages
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container_title Symmetry, Integrability and Geometry: Methods and Applications
description We investigate the total CR twist functional on transversal curves in the standard CR 3-sphere S3⊂C2. The question of the integration by quadratures of the critical curves and the problem of the existence and properties of closed critical curves are addressed. A procedure for the explicit integration of general critical curves is provided, and a characterization of closed curves within a specific class of general critical curves is given. Experimental evidence of the existence of an infinite countable number of closed critical curves is provided.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 101, 36 pages On the Total CR Twist of Transversal Curves in the 3-Sphere Emilio MUSSO a and Lorenzo NICOLODI b a) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy E-mail: emilio.musso@polito.it b) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy E-mail: lorenzo.nicolodi@unipr.it Received July 11, 2023, in final form November 26, 2023; Published online December 21, 2023 https://doi.org/10.3842/SIGMA.2023.101 Abstract. We investigate the total CR twist functional on transversal curves in the stan- dard CR 3-sphere S3 ⊂ C2. The question of the integration by quadratures of the critical curves and the problem of existence and properties of closed critical curves are addressed. A procedure for the explicit integration of general critical curves is provided and a characteri- zation of closed curves within a specific class of general critical curves is given. Experimental evidence of the existence of infinite countably many closed critical curves is provided. Key words: CR 3-sphere; transversal curves; CR invariants; total CR twist; Griffiths’ for- malism; Lax formulation of E-L equations; integration by quadratures; closed critical curves 2020 Mathematics Subject Classification: 53C50; 53C42; 53A10 Dedicated to Peter Olver on the occasion of his 70th birthday 1 Introduction The present paper finds its inspiration and theoretical framework in the subjects of moving frames, differential invariants, and invariant variational problems, three of the many research topics to which Peter Olver has made lasting contributions. Among the many publications of Peter Olver dedicated to these subjects, we like to mention [16, 17, 34, 35] as the ones that most influenced our research activity. More specifically, in this paper we further develop some of the themes considered in [25, 31, 32] concerning the Cauchy–Riemann (CR) geometry of transversal and Legendrian curves in the 3- sphere. In three dimensions, a CR structure on a manifold is defined by an oriented contact distribution equipped with a complex structure. While the automorphism group of a contact manifold is infinite dimensional, that of a CR threefold is finite dimensional and of dimension less or equal than eight [5, 6, 7]. The maximally symmetric CR threefold is the 3-sphere S3, realized as a real hyperquadric of CP2 acted upon transitively by the Lie group G ∼= SU(2, 1). This homogeneous model allows the application of differential-geometric techniques to the study of transversal and Legendrian curves in S3. Since the seminal work of Bennequin [1], the study of the topological properties of transversal and Legendrian knots in 3-dimensional contact manifolds has been an important area of research (see, for instance, [10, 12, 13, 14, 15, 18] and the literature therein). Another reason of interest for 3-dimensional contact geometry comes from This paper is a contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver. The full collection is available at https://www.emis.de/journals/SIGMA/Olver.html mailto:emilio.musso@polito.it mailto:lorenzo.nicolodi@unipr.it https://doi.org/10.3842/SIGMA.2023.101 https://www.emis.de/journals/SIGMA/Olver.html 2 E. Musso and L. Nicolodi its applications to neuroscience. In fact, as shown by Hoffman [20], the visual cortex can be modeled as a bundle equipped with a contact structure. For more details, the interested reader is referred to the monograph [36, Section 5]. Recently, the CR geometry of Legendrian and transversal curves in S3 has also found interesting applications in the framework of integrable system [4]. Let us begin by recalling some results from the CR geometry of transversal curves in S3. Ac- cording to [31], away from CR inflection points, a curve transversal to the contact distribution of S3 can be parametrized by a natural pseudoconformal parameter s and in this parametriza- tion it is uniquely determined, up to CR automorphisms, by two local CR invariants: the CR bending κ and the CR twist τ . This was achieved by developing the method of moving frames and by constructing a canonical frame field along generic1 transversal curves. Moreover, for closed transversal curves, we defined three discrete global invariants, namely, the wave number, the CR spin, and the CR turning number. Next, we investigated the total strain functional, defined by integrating the strain element ds. We proved that the corresponding critical curves have constant bending and twist, and hence arise as orbits of 1-parameter groups of CR auto- morphisms. Finally, closed critical curves are shown to be transversal positive torus knots with maximal Bennequin number. In the present paper, we consider the CR invariant variational problem for generic transversal curves in S3 defined by the total CR twist functional, W(γ) = ∫ γ τ ds. Our purpose is to address both the question of the explicit integration of critical curves and the problem of existence and properties of closed critical curves of W. We now give a brief outline of the content and results of this paper. In Section 2, we shortly describe the standard CR structure of the 3-sphere S3, viewed as a homogeneous space of the group G, and collect some preliminary material. We then recall the basic facts about the CR geometry of transversal curves in S3 as developed in [31] (see the description above). Moreover, besides the already mentioned discrete global invariants for a closed transversal curve, we introduce a fourth global invariant, the trace of the curve with respect to a spacelike line. In Section 3, we apply the method of moving frames and the Griffiths approach to the calculus of variations [19, 21, 26] to compute the Euler–Lagrange equations of the total CR twist functional. We construct the momentum space of the corresponding variational problem and find a Lax pair formulation for the Euler–Lagrange equations satisfied by the critical curves. This is the content of Theorem A, the first main result of the paper, whose proof occupies the whole Section 3. As a consequence of Theorem A, to each critical curve we associate a momentum operator, which is a fixed element of the G-module h of traceless selfadjoint endomorphisms of C2,1. From the conservation of the momentum along a critical curve, we derive two conservation laws, involving two real parameters c1 and c2. The pair c = (c1, c2) is referred to as the modulus of the critical curve. In Section 4, we introduce the phase type of the modulus of a critical curve. We then define the phase curve of a given modulus and the associated notion of signature of a critical curve with that given modulus. For a generic modulus c, the phase type of c refers to the properties of the roots of the quintic polynomial in principal form given by Pc(x) = x5 + 3 2 c2x 2 + 27c1x− 27 2 c21. The phase curve of the modulus c is the real algebraic curve defined by the equation y2 = Pc(x). The signature of a critical curve γ with modulus c and nonconstant twist provides a parametriza- tion of the connected components of the phase curve of c by the twist of γ. Importantly, the 1I.e., with no CR inflection points. On the Total CR Twist of Transversal Curves in the 3-Sphere 3 periodicity of the twist of γ amounts to the compactness of the image of the signature of γ. This will play a role in Sections 5 and 6, where the closedness question for critical curves is addressed. Using the Klein formulae for the icosahedral solutions of the quintic [23, 33, 38], the roots of Pc can be evaluated in terms of hypergeometric functions. As a byproduct, we show that the twist and the bending of a critical curve can be obtained by inverting incomplete hyperelliptic integrals of the first kind. We further specialize our analysis by introducing the orbit type of the modulus c of a critical curve γ. The orbit type of c refers to the spectral properties of the momentum associated to γ. Depending on the phase type, the number of connected components of the phase curves, and the orbit type, the critical curves are then divided into twelve classes. The critical curves of only three of these classes have periodic twist. In Section 5, we show that a general critical curve (cf. Definition 5.1) can be integrated by quadratures using the momentum of the curve. This is the content of Theorem B, the second main result of the paper. Theorem B is then specialized to one of the twelve classes of critical curves, the class characterized by the compactness of the connected component of the phase curve and by the existence of three distinct real eigenvalues of the momentum. Theorem C, the third main result, shows that the critical curves of this specific class can be explicitly written by inverting hyperelliptic integrals of the first and third kind. We then examine the closure conditions and prove that a critical curve in this class is closed if and only if certain complete hyperelliptic integrals depending on the modulus of the curve are rational. Finally, the relations between these rational numbers and the global CR invariants mentioned above are discussed. In the last section, Section 6, we develop convincing heuristic and numerical arguments to support the claim that there exist infinite countably many distinct congruence classes of closed critical curves. These curves are uniquely determined by the four discrete geometric invariants: the wave number, the CR spin, the CR turning number, and the trace with respect to the spacelike λ1-eigenspace of the momentum. Using numerical tools, we construct and illustrate explicit examples of approximately closed critical curves. 2 Preliminaries 2.1 The standard CR structure on the 3-sphere Let C2,1 denote C3 with the indefinite Hermitian scalar product of signature (2, 1) given by ⟨z,w⟩ = tzhw, h = (hij) =  0 0 i 0 1 0 −i 0 0  . (2.1) Following common terminology in pseudo-Riemannian geometry, a nonzero vector z ∈ C2,1 is spacelike, timelike or lightlike, depending on whether ⟨z, z⟩ is positive, negative or zero. By N we denote the nullcone, i.e., the set of all lightlike vectors. Let S = P(N ) be the real hypersurface in CP 2 defined by S = { [z] ∈ CP 2 | ⟨z, z⟩ = i(z1z3 − z3z1) + z2z2 = 0 } . The restriction of the affine chart s : C2 ∋ (z1, z2) 7−→ [ t ( 1 + z1 2 , i z2√ 2 , i 1− z1 2 )] ∈ S ⊂ CP 2 to the unit sphere S3 of C2 defines a smooth diffeomorphism between S3 and S. For each p = [z] ∈ S, the differential (1, 0)-form ζ̃ ∣∣ p = − i⟨z,dz⟩ z tz ∣∣∣∣ p ∈ Ω1,0 ( CP 2 )∣∣ p 4 E. Musso and L. Nicolodi is well defined. In addition, the null space of the imaginary part of ζ̃|p is T (S)|p, namely the tangent space of S at p. Thus, the restriction of ζ̃ to T (S) is a real-valued 1-form ζ ∈ Ω1(S). Since the pullback of ζ by the diffeomorphism s : S3 → S is the standard contact form iz · dz|S3 of S3, then ζ is a contact form whose contact distribution D is, by construction, a complex subbundle of T ( CP 2 )∣∣ S . Therefore, D inherits from T ( CP 2 )∣∣ S a complex structure J . This defines a CR structure on S. Let e1, e2, e3 denote the standard basis of C3. Consider P0 = [e1] ∈ S and P∞ = [e3] ∈ S as the origin and the point at infinity of S. Then, Ṡ := S \ {P∞} can be identified with Euclidean 3-space with its standard contact structure dz−ydx+xdy by means of the Heisenberg projection2 πH : Ṡ ∋ [z] 7−→ t(Re(z2/z1), Im(z2/z1),Re(z3/z1)) ∈ R3. The inverse of the Heisenberg projection is the Heisenberg chart jH : R3 ∋ t(x, y, z) 7−→ [ t ( 1, x+ iy, z + i 2 ( x2 + y2 ))] ∈ Ṡ. The Heisenberg chart can be lifted to a map whose image is a 3-dimensional closed subgroup H3 of G, which is isomorphic to the 3-dimensional Heisenberg group [31]. Let G be the special pseudo-unitary group of (2.1), i.e., the 8-dimensional Lie group of unimodular complex 3× 3 matrices preserving (2.1), G = { A ∈ SL(3,C) | tĀhA = h } ∼= SU(2, 1), and let g denote the Lie algebra of G, g = { X ∈ sl(3,C) | tX̄h+ hX = 0 } . The Maurer–Cartan form of the group G takes the form ϑ = A−1dA = α1 1 + iβ11 −iα2 3 − β23 α1 3 α2 1 + iβ21 −2iβ11 α2 3 + iβ23 α3 1 iα2 1 + β21 −α1 1 + iβ11  , where the 1-forms ( α1 1, β 1 1 , α 2 1, β 2 1 , α 3 1, α 2 3, β 2 3 , α 1 3 ) form a basis of the dual Lie algebra g∗. The center of G is Z = { ϖI3 | ϖ ∈ C, ϖ3 = 1 } ∼= Z3, where I3 denotes the 3 × 3 identity matrix. Let [G] denote the quotient Lie group G/Z and for A ∈ G let [A] denote its equivalence class in [G]. Thus [A] = [B] if and only if B = ϖA, for some cube root of unity ϖ. For any A ∈ G, the column vectors (A1, A2, A3) of A form a basis of C2,1 satisfying ⟨Ai, Aj⟩ = hij and det(A1, A2, A3) = 1. Such a basis is referred to as a lightcone basis. On the other hand, a basis (u1,u2,u3) of C2,1, such that det(u1,u2,u3) = 1 and ⟨ui,uj⟩ = δijϵj , where ϵ1 = −1, ϵ2 = ϵ3 = 1, is referred to as a unimodular pseudo-unitary basis. The group G acts transitively and almost effectively on the left of S by A[z] = [Az], ∀A ∈ G, ∀ [z] ∈ S. This action descends to an effective action of [G] = G/Z on S. It is a classical result of E. Cartan [5, 6, 7] that [G] is the group of CR automorphisms of S. If we choose [e1] = [ t(1, 0, 0) ] ∈ S as an origin of S, the natural projection πS : G ∋ A 7→ A[e1] = [A1] ∈ S 2This map is the analogue of the stereographic projection in Möbius (conformal) geometry. On the Total CR Twist of Transversal Curves in the 3-Sphere 5 makes G into a (trivial) principal fiber bundle with structure group G0 = {A ∈ G | A[e1] = [e1]} . The elements of G0 consist of all 3× 3 unimodular matrices of the form X(ρ, θ, v, r) = ρeiθ −iρe−iθv̄ eiθ(r − i 2ρ|v| 2) 0 e−2iθ v 0 0 ρ−1eiθ,  , (2.2) where v ∈ C, r ∈ R, 0 ≤ θ < 2π, and ρ > 0. Remark 2.1. The left-invariant 1-forms α2 1, β 2 1 , α 3 1 are linearly independent and generate the semi-basic 1-forms for the projection πS : G → S. So, if s : U ⊆ S → G is a local cross section of πS , then ( s∗α3 1, s ∗α2 1, s ∗β21 ) defines a coframe on U and s∗α3 1 is a positive contact form. 2.2 Transversal curves Definition 2.2. Let γ : J → S be a smooth immersed curve. We say that γ is transversal (to the contact distribution D) if the tangent vector γ′(t) ̸∈ D|γ(t), for every t ∈ J . The parametrization γ is said to be positive if ζ(γ′(t)) > 0, for every t and for every positive contact form compatible with the CR structure. From now on, we assume that the parametrization of a transversal curve is positive. Definition 2.3. Let γ : J → S be a smooth curve. A lift of γ is a map Γ: J → N into the nullcone N ⊂ C2,1, such that γ(t) = [Γ(t)], for every t ∈ J . If Γ is a lift, any other lift is given by rΓ, where r is a smooth complex-valued function, such that r(t) ̸= 0, for every t ∈ J . From the definition of the contact distribution, we have the following. Proposition 2.4. A parametrized curve γ : J → S is transversal and positively oriented if and only if −i⟨Γ,Γ′⟩|t > 0, for every t ∈ J and for every lift Γ. Definition 2.5. A frame field along γ : J → S is a smooth map A : J → G such that πS ◦A = γ. Since the fibration πS is trivial, there exist frame fields along every transversal curve. If A = (A1, A2, A3) is a frame field along γ, A1 is a lift of γ. Let A be a frame field along γ. Then A−1A′ = a11 + ib11 −ia23 − b23 a13 a21 + ib21 −2ib11 a23 + ib23 a31 ia21 + b21 −a11 + ib11  , where a31 is a strictly positive real-valued function. Any other frame field along γ is given by à = AX(ρ, θ, v, r), where ρ (ρ > 0), θ, r : J → R, v = p+ iq : J → C are smooth functions and X(ρ, θ, v, r) : J → G0 is as in (2.2). If we let Ã−1Ã′ = ã11 + ib̃11 −iã23 − b̃23 ã13 ã21 + ib̃21 −2ib̃11 ã23 + ib̃23 ã31 iã21 + b̃21 −ã11 + ib̃11  , then Ã−1Ã′ = X−1A−1A′X +X−1X ′, 6 E. Musso and L. Nicolodi which implies ã31 = ρ2a31, ã21 + ib̃21 = ρe3iθ ( a21 + ib21 ) − ρ2e2iθ(p+ iq)a31. From this it follows that along any parametrized transversal curve there exists a frame field A for which a21 + ib21 = 0. Such a frame field is said to be of first order. Definition 2.6. Let Γ be a lift of a transversal curve γ : J → S. If det(Γ,Γ′,Γ′′)|t0 = 0, for some t0 ∈ J , then γ(t0) is called a CR inflection point. The notion of CR inflection point is independent of the lift Γ. A transversal curve with no CR inflection points is said to be generic. The notion of a CR inflection point is invariant under reparametrizations and under the action of the group of CR automorphisms. Remark 2.7. If A : J → G is a frame field along a transversal curve γ, then γ(t0) is a CR inflection point if det(A1, A ′ 1, A ′′ 1) ∣∣ t0 = 0. A transversal curve all of whose points are CR inflection points is called a chain. The notion of chain on a CR manifold goes back to Cartan [5, 6] (see also [22] and the literature therein). If γ is transversal and Γ is one of its lifts, then the complex plane [Γ ∧ Γ′]t is of type (1, 1) and the set of null complex lines contained in [Γ ∧ Γ′]t is a chain which is independent of the choice of the lift Γ. This chain, denoted by Cγ |t, is called the osculating chain of γ at γ(t). By construction, Cγ |t is the unique chain passing through γ(t) and tangent to γ at the contact point γ(t). For more details on the CR-geometry of transversal curves in the 3-sphere, we refer to [31]. As a basic reference for transversal knots and their topological invariants in the framework of 3-dimensional contact geometry, we refer to [14] and the literature therein. 2.3 The canonical frame and the local CR invariants In the following, we will consider generic transversal curves. Definition 2.8. Let γ be a generic transversal curve. A lift Γ of γ, such that det(Γ,Γ′,Γ′′) = −1, is said to be a Wilczynski lift (W-lift) of γ. If Γ is a Wilczynski lift, any other is given by ϖΓ, where ϖ ∈ C is a cube root of unity. The function aγ = i⟨Γ,Γ′⟩−1 is smooth, real-valued, and independent of the choice of Γ. We call aγ the strain density of the parametrized transversal curve γ. The linear differential form ds = aγdt is called the infinitesimal strain. Proposition 2.9 ([31]). The strain density and the infinitesimal strain are invariant under the action of the CR transformation group. In addition, if h : I → J is a change of parameter, then the infinitesimal strains ds and ds̃ of γ and γ̃ = γ ◦ h, respectively, are related by ds̃ = h∗(ds). Proof. This proof corrects a few misprints contained in the original one. If A ∈ G and if Γ is a Wilczynski lift of γ, then Γ̂ = AΓ is a Wilczynski lift of γ̂ = Aγ. This implies that aγ = aγ̂ . Next, consider a reparametrization γ̃ = γ ◦ h of γ. Then, Γ∗ = Γ ◦ h is a lift of γ̃, such that det ( Γ∗, (Γ∗)′, (Γ∗)′′ ) = −(h′)3. On the Total CR Twist of Transversal Curves in the 3-Sphere 7 This implies that Γ̃ = (h′)−1Γ∗ is a Wilczynski lift of γ̃. Hence〈 Γ̃, ( Γ̃ )′〉 = (h′)−1⟨Γ,Γ′⟩ ◦ h. Therefore, the strain densities of γ and γ̃ are related by aγ̃ = h′aγ ◦ h. Consequently, we have h∗(ds) = h′aγ ◦ hdt = aγ̃dt = ds̃. ■ As a straightforward consequence of Proposition 2.9, we have the following. Corollary 2.10. A generic transversal curve γ can be parametrized so that aγ = 1. Definition 2.11. If aγ = 1, we say that γ : J → S is a natural parametrization, or a parametriza- tion by the pseudoconformal strain or pseudoconformal parameter. In the following, the natural parameter will be denoted by s. We can state the following. Proposition 2.12 ([31]). Let γ : J → S be a generic transversal curve, parametrized by the natural parameter. There exists a (first order) frame field F = (F1, F2, F3) : J → G along γ, such that F1 is a W-lift and F−1F ′ = iκ −i τ 0 −2iκ 1 1 0 iκ  =: Kκ,τ (s), (2.3) where κ, τ : J → R are smooth functions, called the CR bending and the CR twist, respectively. The frame field F is called a Wilczynski frame. If F is a Wilczynski frame, any other is given by ϖF , where ϖ is a cube root of unity. Thus, there exists a unique frame field [F ] : J → [G] along γ, called the canonical frame of γ. Remark 2.13. Given two smooth functions κ, τ : J → R, there exists a generic transversal curve γ : J → S, parametrized by the natural parameter, whose bending is κ and whose twist is τ . The curve γ is unique up to CR automorphisms of S. Remark 2.14 (cf. [31]). (1) Let γ : J → S be as above and F = (F1, F2, F3) : J → G be a Wilczynski frame along γ. Then, γ# : J ∋ s 7→ [F3(s)]C ∈ S is an immersed curve, called the dual of γ. The dual curve is Legendrian (i.e., tangent to the contact distribution) if and only if τ = 0. Thus, the twist can be viewed as a measure of how the dual curve differs from being a Legendrian curve. (2) Generic transversal curves with constant bending and twist have been studied by the au- thors in [31]. In the following we will consider generic transversal curves with nonconstant CR invariant functions. Remark 2.15. Regarding the CR 3-sphere S3 ∼= S with its standard pseudo-hermitian (PSH) structure (J, ζ), Chiu and Ho (cf. [8]) obtained a complete set of local PSH invariants for hori- zontally regular curves in S3 parametrized by the horizontal arc length w, namely the p-curvature kpsh(w) and the T -variation τpsh(w). The canonical PSH frame field producing the PSH invari- ants originates a CR frame field which can be further adapted to a canonical CR frame following the reduction procedure developed in [31]. From this one can read the CR invariants. Thus, in principle, the CR bending κ(s) and the CR twist τ(s) can be expressed in terms of the PSH invariants kpsh(w(s)), τpsh(w(s)) and their derivatives with respect to s. 8 E. Musso and L. Nicolodi 2.4 Discrete CR invariants of a closed transversal curve Referring to [31, 32], we briefly recall some CR invariants for closed transversal curves, namely the notions of wave number, CR spin, and CR turning number (or Maslov index). These invariants will be used in Sections 5 and 6. The wave number is the ratio between the least period ωγ of γ and the least period ω of the functions (κ, τ). The CR spin is the ratio between ωγ and the least period of a Wilczynski lift of γ. The CR turning number is the degree (winding number) of the map F1−iF3 : R/ωγZ ∼= S1 → Ċ = C\{0}, where F = (F1, F2, F3) is a Wilczynski frame along γ. We will also make use of another invariant. Definition 2.16. Let [z] ∈ CP2 be a spacelike line. Denote by C[z] the chain of all null lines orthogonal to [z], equipped with its positive orientation. Consider a closed generic transversal curve γ with its positive orientation. Since γ is closed and generic, the intersection of γ with C[z] is either a finite set of points, or the empty set. The trace of γ with respect to [z], denoted by tr[z](γ), is the integer defined as follows: (1) if γ ∩ C[z] ̸= ∅, then tr[z](γ) counts the number of intersection points of γ with C[z] (since γ is not necessarily a simple curve, the intersection points are counted with their multiplicities); (2) otherwise, tr[z](γ) = Lk(γ, C[z]), the linking number of γ with C[z]. The trace of γ is a G-equivariant map, that is, tr[z](γ) = trA[z](Aγ), for every A ∈ G. 3 The total CR twist functional Let T be the space of generic transversal curves in S, parametrized by the natural parameter. We consider the total CR twist functional W : T → R, defined by W[γ] = ∫ Jγ τγηγ , where Jγ is the domain of definition of the transversal curve γ, τγ is its twist, and ηγ = dsγ is the infinitesimal strain of γ (cf. Section 2.3). A curve γ ∈ T is said to be a critical curve in S if it is a critical point of W when one considers compactly supported variations through generic transversal curves. The main result of this section is the following. Theorem A. Let γ : J → S be a generic transversal curve parametrized by the natural param- eter. Then, γ is a critical curve if and only if 3 L′(s) = [L(s),Kκ,τ (s)], (3.1) where L = 0 iτ ′ + 3(1− τκ) 2iτ τ 0 τ ′ + 3i(1− τκ) 3i −iτ 0  (3.2) and Kκ,τ is defined as in (2.3). Proof. The proof of Theorem A is organized in four steps and three lemmas. Step 1. We show that generic transversal curves are in 1-1 correspondence with the integral curves of a suitable Pfaffian differential system. 3As usual, we write [L,K] = LK −KL for the commutator of L and K. On the Total CR Twist of Transversal Curves in the 3-Sphere 9 Let γ : J → S be a generic transversal curve parametrized by the natural parameter. Ac- cording to Proposition 2.12, the canonical frame of γ defines a unique lift [F ] : J → [G]. The map f : J ∋ s 7−→ ([F(s)], κ(s), τ(s)) ∈ [G]× R2 is referred to as the extended frame of γ. The product space M := [G] × R2 is called the configuration space. The coordinates on R2 will be denoted by (κ, τ). With some abuse of notation, we use α1 1, β 1 1 , α 2 1, β 2 1 , α 3 1, α 2 3, β 2 3 , α 1 3 to denote the entries of the Maurer–Cartan form of [G] as well as their pull-backs on the configuration space M . By Proposition 2.12, the extended frames of γ are the integral curves of the Pfaffian differential system (A, η) on M generated by the linearly independent 1-forms µ1 = α2 1, µ2 = β21 , µ3 = α2 3 − α3 1, µ4 = β23 , µ5 = α1 1, µ6 = β11 − κα3 1, µ7 = α1 3 − τα3 1, with the independence condition η := α3 1. If f = ([F ], κ, τ) : J →M is an integral curve of (A, η), then γ = [F1] : J → S defines a generic transversal curve, such that [F ] is its canonical frame, κ its bending and τ its twist. Accordingly, the integral curves of (A, η) are the extended frames of generic transversal curves in S. Thus, generic transversal curves are in 1-1 correspondence with the integral curves of the Pfaffian system (A, η) on the configuration space M . If we put π1 = dκ, π2 = dτ, the 1-forms ( η, µ1, . . . , µ7, π1, π2 ) define an absolute parallelism on M . Exterior differentiation and use of the Maurer–Cartan equations of G yield the following structure equations for the coframe ( η, µ1, . . . , µ7, π1, π2 ) :{ dη = 2µ1 ∧ µ2 + 2µ5 ∧ η, dπ1 = dπ2 = 0, (3.3) dµ1 = −µ1 ∧ µ5 + 3µ2 ∧ µ6 + ( 3κµ2 − µ3 ) ∧ η, dµ2 = −3µ1 ∧ µ6 − µ2 ∧ µ3 − ( 3κµ1 + µ4 ) ∧ η, dµ3 = −2µ1 ∧ µ2 − µ1 ∧ µ7 + µ3 ∧ µ5 + 3µ4 ∧ µ6 − ( τµ1 − 3κµ4 + 3µ5 ) ∧ η, dµ4 = −µ2 ∧ µ7 − 3µ3 ∧ µ6 + µ4 ∧ µ5 − ( τµ2 + 3κµ3 − 3µ6 ) ∧ η, dµ5 = −µ1 ∧ µ4 + µ2 ∧ µ3 + ( µ2 − µ7 ) ∧ η, dµ6 = −2κµ1 ∧ µ2 − µ1 ∧ µ3 − µ2 ∧ µ4 − ( µ1 + 2κµ5 ) ∧ η − π1 ∧ η, dµ7 = −2τµ1 ∧ µ2 − 2µ3 ∧ µ4 − 2µ5 ∧ µ7 + ( 2µ4 − 2τµ5 ) ∧ η − π2 ∧ η. (3.4) Remark 3.1. From the structure equations it follows that the derived flag of (A, η) is given by A(4) ⊂ A(3) ⊂ A(2) ⊂ A(1), where A(4) = {0}, A(3) = span { µ1 } , A(2) = span { µ1, µ2, µ3 } , A(1) = span { µ1, µ2, µ3, µ4, µ5 } . Thus, all the derived systems of (A, η) have constant rank. For the notion of derived flag, see [19]. Step 2. We develop a construction due to Griffiths [19] on an affine subbundle of T ∗(M) (cf. also [2, 21, 26]) in order to derive the Euler–Lagrange equations. Let Z ⊂ T ∗(M) be the affine subbundle defined by the 1-forms µ1, . . . , µ7 and λ := τη, namely Z = λ+ span { µ1, . . . , µ7 } ⊂ T ∗(M). 10 E. Musso and L. Nicolodi We call Z the phase space of the Pfaffian system (A, η). The 1-forms ( µ1, . . . , µ7, λ ) induce a global affine trivialization of Z, which may be identified with M × R7 by the map M × R7 ∋ (([F ], κ, τ), p1, . . . , p7) 7−→ λ|([F],κ,τ) + 7∑ j=1 pjµ j |([F],κ,τ) ∈ Z, where p1, . . . , p7 are the fiber coordinates of the bundle map Z → M with respect to the trivialization. Under this identification, the restriction to Z of the Liouville (canonical) 1-form of T ∗(M) takes the form ξ = τη + 7∑ j=1 pjµ j . Exterior differentiation and use of the quadratic equations (3.3) and (3.4) yield dξ ≡ π2 ∧ η + 2τµ5 ∧ η + 7∑ j=1 dpj ∧ µj + p1 ( 3κµ2 − µ3 ) ∧ η − p2 ( 3κµ1 + µ4 ) ∧ η − p3 ( τµ1 − 3κµ4 + 3µ5 ) ∧ η − p4 ( τµ2 + 3κµ3 − 3µ6 ) ∧ η + p5 ( µ2 − µ7 ) ∧ η − p6 ( π1 + µ1 + 2κµ5 ) ∧ η − p7 ( π2 − 2µ4 + 4τµ5 ) ∧ η, where the sign ‘≡’ denotes equality modulo the span of {µi ∧ µj}i,j=1,...,7. The Cartan system (C(dξ), η) of the 2-form dξ is the Pfaffian system on Z generated by the 1-forms {X⌟dξ | X ∈ X(Z)} ⊂ Ω1(Z), with independence condition η ̸= 0. By Step 1, generic transversal curves are in 1-1 correspondence with the integral curves of the Pfaffian system (A, η). Let f : J →M be the extended frame corresponding to the generic transversal curve γ : J → S parametrized by the natural parameter. According to Griffiths approach to the calculus of variations (cf. [2, 19, 21, 26]), if the extended frame f admits a lift y : J → Z to the phase space Z which is an integral curve of the Cartan system (C(dξ), η), then γ is a critical curve of the total twist functional with respect to compactly supported variations. As observed by Bryant [2], if all the derived systems of (A, η) are of constant rank, as in the case under discussion (cf. Remark 3.1), then the converse is also true. Hence all extremal trajectories arise as projections of integral curves of the Cartan system (C(dξ), η). Next, we compute the Cartan system (C(dξ), η). Contracting the 2-form dξ with the vector fields of the tangent frame (∂η, ∂µ1 , . . . , ∂µ7 , ∂π1 , ∂π2 , ∂p1 , . . . , ∂p7) on Z, dual to the coframe ( η, µ1, . . . , µ7, π1, π2, dp1, . . . ,dp7 ) , yields the 1-forms ∂pj⌟ dξ ≡ µj , j = 1, . . . , 7, −∂π1⌟ dξ ≡ p6η =: π̇1, −∂π2⌟ dξ ≡ (p7 − 1)η =: π̇2, −∂η⌟ dξ ≡ (1− p7)π 2 =: η̇, (3.5) On the Total CR Twist of Transversal Curves in the 3-Sphere 11 −∂µ1⌟ dξ ≡ dp1 + (3κp2 + τp3 + p6)η =: µ̇1, −∂µ2⌟ dξ ≡ dp2 − (3κp1 − τp4 + p5)η =: µ̇2, −∂µ3⌟ dξ ≡ dp3 + (p1 + 3κp4)η =: µ̇3, −∂µ4⌟ dξ ≡ dp4 + (p2 − 3κp3 − 2p7)η =: µ̇4, −∂µ5⌟ dξ ≡ dp5 − (2τ − 3p3 − 2κp6 − 4τp7)η =: µ̇5, −∂µ6⌟ dξ ≡ dp6 − 3p4η =: µ̇6, −∂µ7⌟ dξ ≡ dp7 + p5η =: µ̇7. We have proved the following. Lemma A1. The Cartan system (C(dξ), η) is the Pfaffian system on Z ∼=M ×R7 generated by the 1-forms{ µ1, . . . , µ7, π̇1, π̇2, η̇, µ̇ 1, . . . , µ̇7 } and with independence condition η ̸= 0. Now, the Cartan system (C(dξ), η) is reducible, i.e., there exists a nonempty submanifold Y ⊆ Z, called the reduced space, such that: (1) at each point of Y there exists an integral element of (C(dξ), η) tangent to Y; (2) if X ⊆ Z is any other submanifold with the same property of Y, then X ⊆ Y. The reduced space Y is called the momentum space of the variational problem. Moreover, the restriction of the Cartan system (C(dξ), η) to Y is called the Euler–Lagrange system of the variational problem, and will be denoted by (J , η). A basic result states that the Pfaffian systems (C(dξ), η) and (J , η) have the same integral curves (cf. [19, 26]). The system (J , η) can be constructed by an algorithmic procedure (cf. [19]). Lemma A2. The momentum space Y is the 11-dimensional submanifold of Z defined by the equations p7 = 1, p6 = p5 = p4 = 0, p3 = −2 3 τ, p2 = 2(1− τκ). The Euler–Lagrange system (J , η) is the Pfaffian system on Y ∼= M × R, with independence condition η ̸= 0, generated by the 1-forms µ1|Y , . . . , µ 7 |Y , σ1 = dp1 + 6κ(1− τκ)η − 2 3τ 2η, σ2 = −2τdκ− 2κdτ − 3kp1η, σ3 = −2dτ + 3p1η. (3.6) Proof of Lemma A2. Let V1(dξ) ↪→ P(T (Z)) → Z be the totality of 1-dimensional integral elements of (C(dξ), η). In view of (3.5), we find that V1(dξ)|(([F],κ,τ);p1,...,p7) ̸= ∅ ⇐⇒ p6 = 0, p7 = 1. Thus, the image Z1 ⊂ Z of V1(dξ) with respect to the natural projection V1(dξ) → Z, is given by Z1 = {(([F ], κ, τ); p1, . . . , p7) ∈ Z | p6 = 0, p7 = 1} . Next, the restriction of µ̇6 and µ̇7 to Z1 take the form µ̇6 = −3p4η and µ̇7 = p5η. Thus, the second reduction Z2 is given by Z2 = {(([F ], κ, τ); p1, . . . , p7) ∈ Z1 | p4 = p5 = 0} . 12 E. Musso and L. Nicolodi Considering the restriction of µ̇4 and µ̇5 to Z2 yields the equations p2 = 2(1− τκ), p3 = −2 3 τ, which define the third reduction Z3. Now, the restriction C3(dξ) to Z3 of the Cartan system C(dξ) is generated by the 1-forms µ1, . . . , µ7 and σ1 = dp1 + 6κ(1− τκ)η − 2 3τ 2η, σ2 = dp2 − 3kp1η = −2τdκ− 2κdτ − 3kp1η, σ3 = −2dτ + 3p1η. This implies that there exists an integral element of V1(dξ) over each point of Z3, i.e., for each p ∈ Z3, V1(dξ)|p ̸= ∅. Hence, Y := Z3 is the momentum space and (J , η) := (C3(dξ), η) is the reduced system of (C(dξ), η). ■ Step 3. We derive the Euler–Lagrange equations. By the previous discussion, all the extremal trajectories of S arise as projections of the integral curves of the Euler–Lagrange system. If y : J → Y is an integral curve of the Euler–Lagrange system (J , η) and pr : Y → S is the natural projection of Y onto S, then γ = pr ◦ y : J → S is a critical curve of the total twist functional with respect to compactly supported variations. We can prove the following. Lemma A3. A curve y : J → Y is an integral curve of the Euler–Lagrange system (J , η) if and only if the bending κ and the twist τ of the transversal curve γ = pr ◦ y : J → S satisfy the equations 2κτ ′ + τκ′ = 0, (3.7) τ ′′ + 9κ(1− τκ)− τ2 = 0. (3.8) Proof of Lemma A3. If y = (([F ], κ, τ); p1) : J → Y is an integral curve of the Euler–Lagrange system (J , η), the projection γ = pr ◦ y is the smooth curve γ(s) = [F1(s)], where F1 is the first column of F . The equations µ1 = · · · = µ7 = 0 together with the independence condition η ̸= 0 tell us that ([F ], κ, τ) is an integral curve of the Pfaffian system (A, η) on the configuration space M . Hence γ is a generic transversal curve with bending κ, twist τ and F is a Wilczynski frame along γ. Next, for the smooth function κ, τ : J → R, let κ′, κ′′ and τ ′, τ ′′, etc., be defined by dκ = κ′η, dκ′ = κ′′η, dτ = τ ′η, dτ ′ = τ ′′η. With reference to (3.6), equation σ3 = 0 implies p1 = 2 3 τ ′. Further, σ2 = 0 gives 2κτ ′ + τκ′ = 0. Finally, equation σ1 = 0 yields τ ′′ + 9κ(1− τκ)− τ2 = 0. On the Total CR Twist of Transversal Curves in the 3-Sphere 13 Conversely, let γ : J → S be a generic transversal curve, parametrized by the natural param- eter, satisfying (3.7) and (3.8) and let [F ] its canonical frame. Then, y(s) = ( ([F ], κ, τ); 2 3 τ ′ ) is, by construction, an integral curve of the Euler–Lagrange system (J , η). ■ Step 4. We eventually provide a Lax formulation for the Euler–Lagrange equations (cf. (3.7) and (3.8)) of a critical curve γ : J → S. Using the Killing form of g, the dual Lie algebra g∗ can be identified with h = ig, the G- module of traceless selfadjoint endomorphisms of C2,1. Under this identification, the restriction to Y of the tautological 1-form ξ goes over to an element of h which originates the h-valued function L : J → h given by L(s) = 0 iτ ′ + 3(1− τκ) 2iτ τ 0 τ ′ + 3i(1− τκ) 3i −iτ 0  . (3.9) A direct computation shows that the Euler–Lagrange equations (3.7) and (3.8) of the critical curve γ are satisfied if and only if L′(s) = [L(s),Kκ,τ (s)], where Kκ,τ is given by (2.3). This concludes the proof of Theorem A. ■ As a consequence of Theorem A, we have the following. Corollary 3.2. Let γ : J → S be a generic transversal curve parametrized by the natural pa- rameter. Let [F ] : J → [G] be the canonical frame of γ and let L : J → h be as in (3.9). If γ is a critical curve, the Lax equation (3.1) implies that F(s)L(s)F−1(s) = M, ∀ s ∈ J, where M is a fixed element of h corresponding to a chosen value L(s0) of L(s). Definition 3.3. The element M ∈ h is called the momentum of the critical curve γ. The characteristic polynomial of the momentum M is −x3 − 6κτ2x+ 54κτ − 27κ2τ2 + 2τ3 − 3τ ′2 − 27. The conservation of the momentum along γ yields the two conservation laws κτ2 = c1, −18κτ + 9κ2τ2 − 2 3 τ3 + τ ′2 = C2 − 9, for real constants c1 and C2. We let c2 := C2 − 9. Using this notation, the (opposite of the) characteristic polynomial of the momentum is Q(x) = x3 + 6c1x+ (27 + 3c2). If c1 ̸= 0, the twist and the bending are never zero and the conservation laws can be rewritten as κ = c1τ −2, 3 2 τ2τ ′2 = τ5 + 3 2 c2τ 2 + 27c1τ − 27 2 c21. (3.10) If c1 = 0, it can be easily proved that κ = 0 and the second conservation law takes the form τ ′2 = 2 3 τ3 + c2. 14 E. Musso and L. Nicolodi Definition 3.4. The pair of real constants c = (c1, c2) is called the modulus of the critical curve γ. Remark 3.5. For the application of Griffiths’ approach to other geometric variational problems, the reader is referred to [9, 11, 19, 26, 27, 28, 30]. 4 The CR twist of a critical curve 4.1 Phase types For c = (c1, c2) ∈ R2, we denote by Pc the quintic polynomial in principal form given by Pc(x) = x5 + 3 2 c2x 2 + 27c1x− 27 2 c21 and by Qc the cubic polynomial given by Qc(x) = x3 + 6c1x+ (27 + 3c2). (4.1) Excluding the case c = 0, Pc possesses at least a pair of complex conjugate roots. Definition 4.1. We adopt the following terminology. � c ∈ R2 is of phase type A if Pc has four complex roots aj ± ibj , j = 1, 2, 0 < b1 < b2, and a simple real root e1; � c ∈ R2 is of phase type B if Pc has two complex roots a± ib, b > 0, and three simple real roots e1 < e2 < e3; � c ∈ R2 is of phase type C if Pc has a multiple real root. In the latter case, two possibilities may occur: (1) Pc has a double real root and a simple real root; or (2) Pc has a real root of multiplicity 5. By the same letters, we also denote the corresponding sets of moduli of phase types A, B, and C, respectively. Next, we give a more detailed description of the sets A, B, and C. To this end, we start by defining the separatrix curve. Let (m,n) be the homogeneous coordinates of RP1 and let [(m∗, n∗)] be the point of RP1 such that 3m3 ∗ + 6m2 ∗n∗ + 4m∗n 2 ∗ + 2n3∗ = 0 (i.e., m∗ = 1 and n∗ ≈ −0.72212). Definition 4.2. The separatrix curve Ξ ⊂ R2 is the image of the parametrized curve ξ = (ξ1, ξ2) : RP1 \ {[(m∗, n∗)]} → R2, defined by ξ1([(m,n)]) = 6 3 √ 2mn4/3 ( 3m2 + 2mn+ n2 )4/3( 3m3 + 6m2n+ 4mn2 + 2n3 )5/3 , ξ2([(m,n)]) = − 36n ( 3m2 + 2mn+ n2 )( 4m3 + 3m2n+ 2mn2 + n3 )( 3m3 + 6m2n+ 4mn2 + 2n3 )2 . Remark 4.3. The map ξ is injective and Ξ has a cusp at ξ([(1, 1)]) = ( 4 5 ( 6 5 )2/3 ,−48 5 ) . It is regular elsewhere. In addition, Ξ has a horizontal inflection point at ξ([(0, 1)]) = (0,−9). Let Jξ be the interval (arctan(n∗), arctan(n∗) + π) ≈ (−0.625418, 2.51617). Then, ξ̃ : t ∈ Jξ → ξ(cos(t), sin(t)) ∈ Ξ is another parametrization of Ξ. The inflection point is ξ̃(π/2). The “negative part” Ξ− = Ξ ∩ { c ∈ R2 | c1 < 0 } of Ξ is parametrized by the restriction of ξ̃ to Ĵξ = (π/2, π + arctan(n∗)). The left picture of Figure 1 reproduces the separatrix curve (in black); the negative part of the separatrix curve is highlighted in dashed-yellow. The cusp is the red point and the horizontal inflection point is coloured in green. On the Total CR Twist of Transversal Curves in the 3-Sphere 15 Figure 1. On the left: the separatrix curve (in black), the upper domain M+ (coloured in three orange tones) and the lower domain M− (coloured in two brown tones). On the right: the curve C and its parametrization obtained by intersecting C with lines through the origin. Definition 4.4. The (open) upper and lower domains bounded by the separatrix curve Ξ are denoted by M±. In Figure 1, the upper domain M+ is coloured in three orange tones: orange, dark-orange and light-orange; the lower domain M− is coloured in two brown tones: light-brown and brown. Proposition 4.5. The polynomial Pc has multiple roots if and only if c ∈ Ξ ∪ Oy, has four complex roots if and only if c ∈ M− \ (Oy ∩M−), and has three distinct real roots if and only if c ∈ M+ \ (Oy ∩M+). Equivalently, A = M− \ (Oy ∩M−), B = M+ \ (Oy ∩M+), C = Ξ ∪Oy. Proof. First, we prove the following claim. Claim. Pc has a double root a3 ̸= 0 if and only if c belongs to the separatrix curve minus the cusp. Note that c1 ̸= 0 (otherwise the double root would be 0). Let a4 be the other simple real root and b1 + ib2, b1 − ib2, b2 > 0, be the two complex conjugate roots. Since the sum of the roots of Pc is zero, we have b1 = −1 2(2a3 + a4). Since the coefficient of x3 is zero and b2 > 0, we get b2 = √ 2a23 + a3a4 + 3a24/4. Expanding (x− a3) 2(x− a4)(x− b1 − ib2)(x− b1 + ib2) and comparing the coefficients of the monomials xn, n = 1, . . . , 4, with the coefficients of Pc we may write c1 and c2 as functions of a3 and a4, c1 = 1 27 ( 3a43 + 6a33a4 + 4a23a 2 4 + 2a3a 3 4 ) , c2 = −2 3 ( 4a33 + 3a23a4 + 2a3a 2 4 + a34 ) . (4.2) In addition, c21 = 2 27 ( 3a43a4 + 2a33a 2 4 + a23a 3 4 ) . Taking into account that a3 ̸= 0, it follows that (a3, a4) belongs to the algebraic curve C (the black curve on the right picture in Figure 1) defined by the equation 54y ( 3x2 + 2xy + y2 ) − ( 3x3 + 6x2y + 4xy2 + 2y3 )2 = 0. 16 E. Musso and L. Nicolodi Now, consider the line ℓm,n through the origin, with homogeneous coordinates (m,n), i.e., the line with parametric equations pm,n(t) = (mt, nt). If (m,n) ̸= (1, 0) and 3m3+6m2n+4mn2+2n3 ̸= 0 (we are excluding the two red lines on the right picture in Figure 1), ℓm,n intersects C when t = 0 and t = tm,n, where tm,n = 3 3 √ 2 3 √ n ( 3m2 + 2mn+ n2 ) 3 √( 3m3 + 6m2n+ 4mn2 + 2n3 )2 . If (m,n) = (1, 0) or 3m3 + 6m2n+ 4mn2 + 2n3 = 0, ℓm,n intersects C only at the origin (see the right picture in Figure 1). Hence β : [(m,n)] → tm,n · (m,n), [(m,n)] ̸= [(1, 0)], 3m3 + 6m2n + 4mn2 + 2n3 ̸= 0, is a parametrization of C \ {(0, 0)}. Thus, using (4.2), the map [(m,n)] → (c1(β([(m,n])), c2(β([(m,n])) ∈ R2 is a parametrization of the set of all c, c1 ̸= 0, such that Pc has multiple roots. It is now a computational matter to check that (c1(β([(m,n])), c2(β([(m,n])) = ξ([(m,n)]). This proves the claim. It also shows that Pc has multiple roots if and only if c ∈ Ξ ∪Oy. To prove the other assertions, we begin by observing that the discriminant of the derived polynomial P′ c is negative. Hence P′ c has two distinct real roots and a pair of complex conjugate roots. Denote by x′c and x′′c the real roots of P′ c, ordered so that x′c < x′′c. Observe that x′c and x′′c are differentiable functions of c. Then, Pc possesses three distinct real roots if and only if x′c · x′′c < 0, one simple real root if and only if x′c · x′′c > 0, and a multiple root if and only if x′c · x′′c = 0. From the first part of the proof, the set of all c ∈ R2, such that Pc has only simple roots is the complement of Ξ ∪Oy. This set has five connected components: M′ + = {c ∈ M+ \ (Oy ∩M+) | c1 < 0}, M′′ + = {c ∈ M+ \ (Oy ∩M+) | c1 > 0 and c2 > 0}, M′′′ + = {c ∈ M+ \ (Oy ∩M+) | c1 > 0 and c2 < 0}, M′ − = {c ∈ M− \ (Oy ∩M−) | c1 < 0}, M′′ − = {c ∈ M− \ (Oy ∩M−) | c1 > 0}. (4.3) Referring to the left picture in Figure 1, M′ + is the orange domain, M′′ + is the dark-orange domain, M′′′ + is the light-orange domain, M′ − is the light-brown domain, and M′′ − is the brown domain. Consider the following points (the black points in Figure 1): c1 = (−2, 1) ∈ M′ +, c2 = (1/6, 8) ∈ M′′ +, c3 = (1/6,−8) ∈ M′′′ +, c4 = (−6,−9) ∈ M′ −, c5 = (4,−9) ∈ M′′ −. Using Klein’s formulas for the icosahedral solution of a quintic polynomial in principal form (cf. [23, 33, 38]),4 we find that the polynomials Pcj , j = 1, 2, 3, have three distinct real roots and that Pcj , j = 4, 5, have one real root. The domain M′ + is connected and the function M′ + ∋ c 7→ x′c · x′′c is differentiable and nowhere zero. Since x′c1 · x′′c1 < 0, it follows that c 7→ x′c · x′′c is strictly negative. Then, Pc has three distinct real roots, for every c ∈ M′ +. Similarly, Pc has three distinct real roots, for every c ∈ M′′ + ∪M′′′ + and a unique real root for every c ∈ M′ − ∪M′′ −. This concludes the proof. ■ 4We used the Trott and Adamchik code (cf. [38]) implementing Klein’s formulas in the software Mathematica. On the Total CR Twist of Transversal Curves in the 3-Sphere 17 Figure 2. On the left: the phase curve of c ∈ A. On the right: the phase curve of c ∈ B. Remark 4.6. The real roots of Pc1 are e1 = −2.44175 < e2 = −0.9904 < 0 < e3 = 2.87645 and those of Pc2 are e1 = −2.14118 < e2 = −0.448099 < 0 < e3 = 0.0701938. Instead, the roots of Pc3 are 0 < e1 = 0.12498 < e2 = 0.250656 < 2.15383. Since the product e2(c)e3(c) is a continuous function on the connected components M′ +, M′′ +, and M′′′ +, we deduce that the lowest roots of Pc are negative if c ∈ M′ + ∪M′′ + and positive if c ∈ M′′′ +. 4.2 Phase curves and signatures Definition 4.7. Let Σc be the real algebraic curve defined by y2 = Pc(x). We call Σc the phase curve of c. If c ∈ A ∪ B, Σc is a smooth real cycle of a hyperelliptic curve of genus 2. If c ∈ C, and c ̸= 0, Σc is a singular real cycle of an elliptic curve. If c = 0, Σc is a singular rational curve. The following facts can be easily verified: � if c ∈ A, Σc is connected, unbounded, and intersects the Ox-axis at (e1, 0) (see Figure 2); � if c ∈ B, Σc has two smooth connected components, one is compact and the other is unbounded. Let Σ′ c be the compact connected component and Σ′′ c be the noncompact one. Σ′ c intersects the Ox-axis at (e1, 0) and (e2, 0), while Σ′′ c intersects the Ox-axis at (e3, 0) (see Figure 2); � if c ∈ C and c1 ̸= 0 , Σc has a smooth, unbounded connected component Σ′′ c and an isolated singular point (e1, 0), where e1 = e2 is the double real root of Pc(x). The unbounded connected component intersects the Ox-axis at (e3, 0), where e3 is the simple real root of Pc(x) (see Figure 3). If c1 = 0 and c2 ̸= 0, Σc is connected, with an ordinary double point (see Figure 3). If c = 0, Σc is connected with a cusp at the origin (see Figure 3). Definition 4.8. Let γ be a critical curve with nonconstant twist and modulus c. Let Jγ ⊂ R be the maximal interval of definition of γ. With reference to (3.10), we adapt to our context the terminology used in [3, 24, 29] and call σγ : Jγ → R2, s 7→ ( τ(s), √ 3/2τ(s)τ ′(s) ) the signature of γ. Remark 4.9. From the Poincaré–Bendixson theorem, it follows that the twist of γ is periodic if and only if σγ(Jγ) is compact. Observing that σγ(Jγ) is one of the 1-dimensional connected 18 E. Musso and L. Nicolodi Figure 3. On the left: the phase curve of c ∈ C, c1 ̸= 0. On the center: the phase curve of c ∈ C, c1 = 0 and c2 ̸= 0. On the right: the phase curve of c = (0, 0). components of Σc, we can conclude that the twist is a periodic function if and only if c ∈ B and σγ(Jγ) = Σ′ c. Definition 4.10. A critical curve γ with modulus c is said to be of type B′ if c ∈ B and σγ(Jγ) = Σ′ c; it is said to be of type B′′ if c ∈ B and σγ(Jγ) = Σ′′ c. 4.3 The twist of a critical curve 4.3.1 The twist of a critical curve of type A Let γ be a critical curve of type A, i.e., with modulus c ∈ A. Then Pc has a unique real root e1. The polynomial Pc(x) is positive if x > e1 and is negative if x < e1. Since Pc(0) = −27c21/2 < 0, the root is positive. Let ωc > 0 be the improper hyperelliptic integral of the first kind defined by ωc = √ 3 2 ∫ +∞ e1 τdτ√ Pc(τ) > 0. The incomplete hyperelliptic integral hc(τ) = √ 3 2 ∫ τ e1 udu√ Pc(u) , u ≥ e1 is a strictly increasing diffeomorphism of [e1,+∞) onto [0, ωc) (see Figure 4). The twist is the unique even function τc : (−ωc, ωc) → R, such that τc = h−1 c on [0, ωc). The maximal domain of definition is Jc = (−ωc, ωc). τc is strictly positive, with vertical asymptotes as s→ ∓ω± c (see Figure 4). Note that τc is the solution of the Cauchy problem τ ′′ = τ2 − 9c1τ −2 ( 1− c1τ −1 ) , τ(0) = e1, τ ′(0) = 0. 4.3.2 The twist of a critical curve of type B′ Let e1 < e2 < e3 be the simple real roots of Pc. The highest root e3 is positive. The lower roots e1 and e2 are either both negative or both positive and Pc is positive on (e1, e2). Let ωc > 0 be the complete hyperelliptic integral of the first kind ωc = sign(e1) √ 3 2 ∫ e1 e2 τdτ√ Pc(τ) > 0. (4.4) On the Total CR Twist of Transversal Curves in the 3-Sphere 19 Figure 4. On the left: the graph of the function hc, c = (1/2,−4.8) ∈ A. The red line is the horizontal asymptote y = e1. On the right: the graph of the twist. The red lines are the vertical asymptotes x = ±ωc. Let hc be the incomplete hyperelliptic integrals of the first kind hc(τ) =  √ 3 2 ∫ τ e2 udu√ Pc(u) , τ ∈ [e1, e2], e1 < e2 < 0,√ 3 2 ∫ τ e1 udu√ Pc(u) , τ ∈ [e1, e2], 0 < e1 < e2. The function hc is a diffeomorphism of [e1, e2] onto [0, ωc], strictly decreasing if e1 < e2 < 0 and strictly increasing if 0 < e1 < e2 (see Figure 5). The twist τc is the even periodic function with least period 2ωc, obtained by extending periodically the function τ(s) = h−1 c (s) defined on [0, ωc] and on [−ωc, 0], respectively. � If e1 < e2 < 0, then τc is strictly negative with minimum value e1 and maximum value e2, attained, respectively, at s ≡ ωc mod 2ωc and at s ≡ 0 mod 2ωc (see Figure 5). � If 0 < e1 < e2, then τc is strictly positive, with minimum value e1 and maximum value e2, attained, respectively, at s ≡ 0 mod 2ωc and at s ≡ ωc mod 2ωc. Observe that τc is the solution of the Cauchy problem (i) τ ′′ = τ2 − 9c1τ −2 ( 1− c1τ −1 ) , τ(0) = e2, τ ′(0) = 0 if e1 < e2 < 0, (ii) τ ′′ = τ2 − 9c1τ −2 ( 1− c1τ −1 ) , τ(0) = e1, τ ′(0) = 0 if 0 < e1 < e2. (4.5) 4.3.3 The twist of a critical curve of type B′′ The twist of a critical curve of type B′′ can be constructed as in the case of a critical curve of type A. More precisely, let e3 > 0 be the highest real root of Pc and ωc be the improper hyperelliptic integral of the first kind given by ωc = √ 3 2 ∫ +∞ e3 τdτ√ Pc(τ) > 0. Let hc(τ) be the incomplete hyperelliptic integral hc(τ) = √ 3 2 ∫ τ e3 udu√ Pc(u) , τ ≥ e3. 20 E. Musso and L. Nicolodi Figure 5. On the left: the graph of the function hc for a critical curve of type B′, with modulus c ≈ (−0.828424,−8.349417) ∈ B. On the right: the graph of the twist, an even periodic function with least period 2ωc. The lowest roots e1 and e2 are negative. Then, hc is a strictly increasing diffeomorphism of [e3,+∞) onto [0, ωc). The twist is the unique even function τc : (−ωc, ωc) → R, such that τc = h−1 c on [0, ωc). The maximal interval of definition of τc is Jc = (−ωc, ωc). The function τc is positive, with vertical asymptotes as s→ ∓ω± c , and is the solution of the Cauchy problem τ ′′ = τ2 − 9c1τ −2 ( 1− c1τ −1 ) , τ(0) = e3, τ ′(0) = 0. 4.3.4 The twist of a critical curve of type C with c1 ̸= 0 The twist of a critical curve of type C, with c1 ̸= 0, can be constructed as for curves of types A or B′′. Let e3 > 0 be simple real root of Pc and ωc be the improper elliptic integral of the first kind ωc = √ 3 2 ∫ +∞ e3 τdτ√ Pc(τ) > 0. Let hc(τ) be the incomplete elliptic integral hc(τ) = √ 3 2 ∫ τ e3 udu√ Pc(u) , τ ≥ e3. Then, hc is a strictly increasing diffeomorphism of [e3,+∞) onto [0, ωc). The twist τc is the unique even function τc : (−ωc, ωc) → R, such that τc = h−1 c on [0, ωc). The maximal interval of definition of τc is Jc = (−ωc, ωc). The twist is positive, with vertical asymptotes as s → ∓ω± c . Note that τc is the solution of the Cauchy problem τ ′′ = τ2 − 9c1τ −2 ( 1− c1τ −1 ) , τ(0) = e3, τ ′(0) = 0. 4.3.5 The twist of a critical curve with c1 = 0 If c1 = 0, the bending vanishes identically and the twist is a solution of the second order ODE τ ′′ − τ2 = 0. Then, τ(s) = 3 √ 6℘ ( s+ a 3 √ 6 ∣∣∣∣0, g3), g3 = − 3 √ 2/243c2, κ(s) = 0, where a is an unessential constant and ℘(−, g2, g3) is the Weierstrass function with invari- ants g2, g3. On the Total CR Twist of Transversal Curves in the 3-Sphere 21 Figure 6. The nine regions Aj , Bj and Cj , j = 1, 2, 3. 4.4 Orbit types and the twelve classes of critical curves with nonconstant twist The moduli of the critical curves can be classified depending on the properties of the eigenvalues of the momenta. Definition 4.11. For c = (c1, c2) ∈ R2, let ∆1(c) = −27 ( 32c31+9(9+ c2) 2 ) be the discriminant of the cubic polynomial Qc (cf. (4.1)). We say that c ∈ R2 is 1. Of orbit type 1 (in symbols, c ∈ OT1) if ∆1(c) > 0; the momentum of a critical curve with modulus c ∈ OT1 has three distinct real eigenvalues: λ1 = −(λ2 + λ3) < 0 < λ2 < λ3. 2. Of orbit type 2 (in symbols, c ∈ OT2) if ∆1(c) < 0; the momentum of a critical curve with modulus c ∈ OT2 has a real eigenvalue λ1 and two complex conjugate roots: λ2, with positive imaginary part, and λ3 = λ2. 3. Of orbit type 3 (in symbols, c ∈ OT3) if ∆1(c) = 0; the momentum of a critical curve with modulus c ∈ OT3 has an eigenvalue with algebraic multiplicity greater than one (> 1). Correspondingly, R2 is partitioned into nine regions (see Figure 6): Aj = A ∩OTj , Bj = B ∩OTj , Cj = C ∩OTj , j = 1, 2, 3. Definition 4.12. Let γ be a critical curve with modulus c and j ∈ {1, 2, 3}. We say that γ is of type Aj if c ∈ Aj ; of type B′ j if c ∈ Bj and the image of its signature σγ is compact; of type B′′ j if c ∈ Bj and the image of σγ is unbounded; and of type Cj if c ∈ Cj , j = 1, 2, 3. Remark 4.13. The only critical curves with periodic twist are those of the types B′ j , j = 1, 2, 3. Consequently, critical curves of the other types cannot be closed. Remark 4.14. B1 lies in the half-plane {(c1, c2) | c1 < 0}; it is bounded below by Ξ′ = {c ∈ Ξ | c1 < 0} and above by ∆′ = { c ∈ R2 | ∆1(c) = 0, c2 > −9 } . The curves Ξ′ and ∆′ intersect each other tangentially at c′ = (c′1, c ′ 2) ≈ (−11.339754, 63.004420) (see Figure 7). Thus, B1 has two connected components: B− 1 = {c ∈ B1 | c1 ∈ (c′1,−9)}, B+ 1 = {c ∈ B1 | c1 > c′1}. Referring to Remark 4.3, Ξ′ is parametrized by the restriction of ξ̃ to the interval Ĵξ = (π/2, π+ arctan(n∗)). Let t ′ be the point of Ĵξ such that ξ̃(t′) = c′, (t′ ≈ 2.3008). Put Ĵ−ξ = (π/2, t′) and 22 E. Musso and L. Nicolodi Figure 7. On the left: the connected component B− 1 (dark brown) of B1. The point coloured in cyan is the inflection point of Ξ (the union of the arcs coloured in black and magenta and of the two points) and the cusp of ∆1 = 0 (the union of the yellow and red arcs and of the two points). The point coloured in purple is the point of tangential contact. On the right: the connected component B+ 1 (dark brow) of B1. Ĵ+ξ = (t′, π + arctan(n∗)). The restriction of ξ̃ to Ĵ−ξ is a parametrization of Ξ− = {c ∈ Ξ′ | c1 ∈ (c′1, 0)} and the restriction to Ĵ+ξ is a parametrization of Ξ+ = {c ∈ Ξ′ | c1 < c′1}. Consequently, B± 1 are parametrized by ψ± : K± ∋ (t, s) 7−→ ( ξ̃(t)− p(t) ) s+ p(t), (4.6) where K± are the rectangles Ĵ±ξ × (0, 1) and p(t) = ( ξ̃1(t), 1 3 ( 4 √ −2ξ̃1(t)3 − 27 )) . 5 Integrability by quadratures 5.1 Integrability by quadratures of general critical curves Definition 5.1. Let ∆2 be the polynomial ∆2(c) = 9c31 ( c31 + 216 ) + 6c31c2(c2 + 36) + (c2 + 9)(c2 + 18)3. A critical curve γ with modulus c is said to be general if ∆1(c)∆2(c) ̸= 0. Since ∆1(c) ̸= 0, the momentum Mγ of a general critical curve γ has three distinct eigenvalues λ1, λ2, λ3, sorted as in Definition 4.11. Let J be the maximal interval of definition of the twist (it can be computed in terms of the modulus). Define yj : J → C1,2, j = 1, 2, 3, by yj = t ( τ(3− iτ ′)− λ2j − 3c1, 9− 9c1 τ − λjτ − 3iτ ′, i ( τ2 − 3λj )) . (5.1) Let V: J → gl(3,C) be the matrix-valued map with column vectors y1, y2 and y3. Let D(z1, z2, z3) denote the diagonal matrix with zj as the jth element on the diagonal. Recall that, if c1 ̸= 0, then τ is nowhere zero. We can prove the following. On the Total CR Twist of Transversal Curves in the 3-Sphere 23 Theorem B. Let γ : J → S be a general critical curve. The functions det(V) and τ2 − 3λj, j = 1, 2, 3, are nowhere zero. Let rj be continuous determinations of √ τ2 − 3λj and let ϕj be the functions defined by5 ϕj(s) = ∫ s 0 3c1λj − ( 4c1 + λ2j ) τ2(u) + 3τ3(u) τ2(u) ( 3λj − τ2(u) ) du. (5.2) Then, γ is congruent to J ∋ s 7−→ [ MD ( r1e iϕ1 , r2e iϕ2 , r3e iϕ3 ) V−1e1 ] ∈ S, where M = V(0)D(r1(0), r2(0), r3(0)) −1. Proof. The proof of Theorem B is organized into three lemmas. Lemma B1. The following statements hold true: (1) if the momentum has three distinct real eigenvalues, then ± √ 3λ2 and ± √ 3λ3 cannot be roots of Pc; (2) if the momentum has two complex conjugate eigenvalues and a positive real eigenvalue λ1, then ± √ 3λ1 cannot be roots of Pc. Proof of Lemma B1. First, note that the image of the parametrized curve α(t) = ( −t ( t3 3 + √ 3 ) , t3 ( t3 3 + 2 √ 3 ) − 9 ) is contained in the zero locus of ∆2. This can be proved by a direct computation. Secondly, from the expression of Qc, it follows that c1 = −1 6 ( λ22 + λ2λ3 + λ23 ) = −1 6 ( λ21 + λ1λ2 + λ22 ) , c2 = 1 3 ( λ22λ3 + λ2λ 2 3 − 27 ) = 1 3 ( λ21λ2 + λ1λ 2 2 − 27 ) . (5.3) 1. Suppose that the momentum has three distinct real eigenvalues. By contradiction, suppose that √ 3λ2 is a root of Pc. Then 0 = −8 3 Pc (√ 3λ2 ) = λ43 + 2λ2λ 3 3 − ( λ22 − 12 √ 3λ 1/2 2 ) λ23 − 2 ( λ32 − 6 √ 3λ 3/2 2 ) λ3 + ( λ42 − 12 √ 3λ 5/2 2 + 108λ2 ) . Solving this equation with respect to λ3, taking into account that λ3 > 0, we obtain λ3 = 1 2 ( −λ2 + √ 5λ22 − 24 √ 3λ2 ) . Substituting into (5.3), we find c1 = √ 3λ2 − 1 3 λ22, c2 = −9− 2 √ 3λ 3/2 2 + 1 3 λ32. Then, c = α ( − √ λ2 ) . This implies that c belongs to the zero locus of ∆2, which is a contra- diction. By an analogous argument, we prove that also − √ 3λ2 cannot be a root of Pc. By 5If c1 ̸= 0, the denominator of the integrand in nowhere zero and the ϕj are real-analytic. If c1 = 0, the integrand reduces to ( 3τ − λ2 j )( 3λj − τ2 )−1 . Thus, also in this case the functions ϕj are real-analytic. 24 E. Musso and L. Nicolodi interchanging the role of λ2 and λ3 and arguing as above, it follows that also ± √ 3λ3 cannot be roots of Pc. 2. Next, suppose that the momentum has two complex conjugate eigenvalues and a nonneg- ative real eigenvalue λ1. Recall that the eigenvalues are sorted so that the imaginary part of λ2 is positive. By contradiction, suppose that √ 3λ1 is a root of Pc. Then, 0 = −8 3 Pc (√ 3λ1 ) = λ42 + 2λ1λ 3 2 − ( λ21 − 12 √ 3λ 1/2 1 ) λ22 − 2 ( λ31 − 6 √ 3λ 3/2 1 ) λ2 + ( λ41 − 12 √ 3λ 5/2 1 + 108λ1 ) . Solving this equation with respect to λ2, taking into account that the imaginary part of λ2 is positive, we find λ2 = 1 2 ( −λ1 + √ 5λ21 − 24 √ 3λ1 ) . Substituting into (5.3) yields c = α ( − √ λ1 ) . Thus, c is a root of ∆2, which is a contradiction. An analogous argument shows that − √ 3λ1 cannot be a root of Pc. This concludes the proof of the lemma. ■ Lemma B2. det(V)(s) ̸= 0, for every s ∈ Jγ. Proof of Lemma B2. Let Lj be the 1-dimensional eigenspaces of the momentum Mγ relative to the eigenvalues λj . Let L be as in (3.2). By Corollary 3.2 of Theorem A, we have FLF−1 = M, where F is a Wilczynski frame field along γ. Then, L(s) andM have the same eigenvalues. Next, consider the line bundles Λj = { (s,y) ∈ Jγ × C1,2 | L(s)y = λjy } , j = 1, 2, 3. Note that (s,y) ∈ Λj if and only if F(s)y ∈ Lj . Let yj , j = 1, 2, 3, be as in (5.1). A direct computation shows that Lyj = λjyj . Thus, yj is a cross section of the eigenbundle Λj . Hence, det(V)(s) ̸= 0 if and only if yj(s) ̸= 0⃗, for every s. Case I. The eigenvalues of the momentum are real and distinct. Let yij , i = 1, 2, 3, denote the components of yj . Since λ1 is negative, it follows from (5.1) that y31(s) ̸= 0, for every s, and hence y1(s) ̸= 0⃗. We prove that y2(s) ̸= 0⃗. Suppose, by contradiction, that y2(s∗) = 0⃗, for some s∗ ∈ Jγ . From y12(s∗) = y22(s∗) = 0, it follows that τ ′(s∗) = 0. Hence e := τ(s∗) is a root of Pc. From y32(s∗) = 0, it follows that e = ± √ 3λ2, which contradicts Lemma B1. An analogous argument leads to the conclusion that y3(s) ̸= 0⃗, for every s ∈ Jγ . Case II. The momentum has a real eigenvalue λ1 and two complex conjugate eigenval- ues λ2, λ3 (λ2 with positive imaginary part). Since λ2 and λ3 have nonzero imaginary parts and τ is real valued, y32(s) ̸= 0 and y33(s) ̸= 0, for every s. If λ1 < 0, then y31(s) ̸= 0, for every s. If λ1 ≥ 0, suppose, by contradiction, that y1(s∗) = 0⃗. From y11(s∗) = y21(s∗) = 0, we infer that τ ′(s∗) = 0. Hence e = τ(s∗) is a root of Pc. From y31(s∗) = 0, we have e = ± √ 3λ1, which contradicts Lemma B1. ■ We are now in a position to conclude the proof. For j = 1, 2, 3, let wj be defined by wj = Fyj : Jγ → C1,2. Then, wj(s) ∈ Lj and wj(s) ̸= 0⃗, for every s. Thus, there exist smooth functions Φj : Jγ → C, such that w′ j = Φjwj . From (2.3), we have Φjyj = y′ j +Kyj , j = 1, 2, 3, (5.4) On the Total CR Twist of Transversal Curves in the 3-Sphere 25 where K = ic1τ −2 −i τ 0 −2ic1τ −2 1 1 0 ic1τ −2  . Then, the third component of y′ j +Kyj is equal to 3τ + 3c1λjτ −2 − ( λ2j + 4c1 ) + iττ ′. Hence, using (5.4) we obtain Φj = − ττ ′ 3λj − τ2 + i 3c1λj − ( 4c1 + λ2j ) τ2 + 3τ3 τ2 ( 3λj − τ2 ) . (5.5) Lemma B3. The functions 3λj − τ2, j = 1, 2, 3, are nowhere zero. Proof of Lemma B3. The statement is obvious if λj is real and negative or complex, with nonzero imaginary part. If λj is real non-negative, the smoothness of Φj implies that ττ ′ ( 3λj − τ2 )−1 is differentiable. Then ( 3λj − τ2 ) (s) ̸= 0, for every s, such that τ(s)τ ′(s) ̸= 0. If τ(s)τ ′(s) = 0, it follows that τ(s) is a root of the polynomial Pc. Therefore, by Lemma B1, we have that ( 3λj − τ2 ) (s) ̸= 0. ■ From (5.5), we have∫ s 0 Φjdu = log (√ τ2 − 3λj ) + iϕj + bj , j = 1, 2, 3, where bj is a constant of integration, √ τ2 − 3λj is a continuous determination of the square root of τ2 − 3λj and log (√ τ2 − 3λj ) is a continuous determination of the logarithm of √ τ2 − 3λj . Since w′ j = Φjwj , we obtain Fyjr −1 j e−iϕj = mj , j = 1, 2, 3, where mj is a constant vector belonging to the eigenspace Lj of M. This implies F = MD ( r1e iϕ1 , r2e iϕ2 , r2e iϕ2 ) V−1, where M is an invertible matrix such that M−1MM = D(λ1, λ2, λ3). By possibly replacing γ with a congruent curve, we may suppose that F(0) = I3. Then, since ϕj(0) = 0, we have M = V(0)D(r1(0), r2(0), r3(0)) −1. This concludes the proof of Theorem B. ■ 5.2 Integrability by quadratures of general critical curves of type B′ 1 We now specialize the above procedure to the case of general critical curves of type B′ 1 (i.e., general critical curves with modulus c ∈ B1 and with periodic twist). Let M′ + be as in (4.3). Since B1 is contained in M′ +, the lowest roots e1 and e2 of Pc are negative, for every c ∈ B1 (cf. Remark 4.6). Lemma 5.2. Let γ be a general critical curve of type B′ 1. The λ1-eigenspace of the momentum is spacelike. 26 E. Musso and L. Nicolodi Proof. Let yj be as in (5.1). Then F(s)yj(s) belongs to the λj-eigenspace of M, for every s ∈ R. Using the conservation law 3 2τ 2(τ ′)2 = Pc(τ) (cf. (3.10)) and taking into account that λ3j + 6c1λj + 3(9 + c2) = 0, we compute ⟨Fyj ,Fyj⟩ = ⟨yj ,yj⟩ = 3 ( τ2 − 3λj )( 2c1 + λ2j ) . Moreover, since λ1 = −(λ2 + λ3) and c1 = − ( λ22 + λ2λ3 + λ23 ) /6, we have 2c1 + λ21 = 1 3 (2λ2 + λ3)(λ2 + 2λ3) > 0, 2c1 + λ23 = 1 3 (λ3 − λ2)(λ2 + 2λ3) > 0, 2c1 + λ22 = −1 3 (λ3 − λ2)(2λ2 + λ3) < 0. From the fact that λ1 < 0, it follows that ⟨y1,y1⟩ > 0. This proves that the λ1-eigenspace of the momentum is spacelike. ■ Definition 5.3. There are two possible cases: either the λ3-eigenspace of M is spacelike, or else is timelike. In the first case, we say that γ is positively polarized, while in the second case, we say that γ is negatively polarized. Remark 5.4. In view of the above lemma, γ is positively polarized if and only if e21 − 3λ3 > 0 and is negatively polarized if and only if e22 − 3λ3 < 0. It is a linear algebra exercise to prove the existence of A ∈ G, such that A−1MA = Mλ1,λ2,λ3 , where Mλ1,λ2,λ3 =  1 2(λ2 + λ3) 0 εi 2 (λ2 − λ3) 0 λ1 0 − εi 2 (λ2 − λ3) 0 1 2(λ2 + λ3)  , (5.6) where ε = ±1 accounts for the polarization of γ (see below). It is clear that any critical curve of type B′ 1 is congruent to a critical curve whose momentum is in the canonical form Mλ1,λ2,λ3 . Definition 5.5. A critical curve of type B′ 1 is said to be in a standard configuration if its momentum is in the canonical form (5.6). Two standard configurations with the same twist are congruent with respect to the left action of the maximal compact abelian subgroup T2 = {A ∈ G | Ae2 ∧ e2 = 0}. Let c ∈ B1, such that ∆1(c)∆2(c) ̸= 0. Let e1 < e2 < e3 be the real roots of Pc and let λ1 = −(λ2 + λ3) < 0 < λ2 < λ3 be the roots of Qc. Let τ be the periodic function defined as in the first of the (4.5) and ϕj , j = 1, 2, 3, be as in (5.2). Let ρj be the constants ρ1 = 1√ (2λ2 + λ3)(λ2 + 2λ3) , ρ2 = 1√ 2(λ3 − λ2)(2λ2 + λ3) , ρ3 = 1√ 2(λ3 − λ2)(λ2 + 2λ3) and zj be the functions z1 = ρ1 √ 3(λ2 + λ3) + τ2eiϕ1 , z2 = ρ2 √ 3λ2 − τ2eiϕ2 , z3 = ρ3 √ 3λ3 − τ2eiϕ3 . (5.7) Let ε = −sign ( e22 − 3λ3 ) . We can state the following. On the Total CR Twist of Transversal Curves in the 3-Sphere 27 Theorem C. A general critical curve of type B′ 1 with modulus c is congruent to γ : R ∋ s 7−→ [ t(z2 + z3, εiz1,−εi(z2 − z3)) ] ∈ S. (5.8) In addition, γ is in a standard configuration. Proof. Let γ̃ be a critical curve of type B′ 1 with modulus c. Let F be a Wilczynski frame along γ̃. Suppose ε = 1 (i.e., ⟨y3,y3⟩ < 0). Let uj be the maps defined by u1 = 1 √ 3 √ 2c1 + λ21 √ τ2 − 3λ1 y1, u2 = 1 √ 3 √ − ( 2c1 + λ22 )√ τ2 − 3λ2 y2, u3 = 1 √ 3 √ 2c1 + λ23 √ τ2 − 3λ3 y3. Consider the map U = (u3,u2,u1) : R → GL(3,C). From Theorem B and Lemma 5.2, we have � ⟨u1,u1⟩ = ⟨u2,u2⟩ = −⟨u3,u3⟩ = 1, and ⟨ui,uj⟩ = 0, for i ̸= j, that is, U(s) is a pseudo- unitary basis of C1,2, for every s ∈ R; � U−1LU = D(λ3, λ2, λ1). Using again Theorem B, we obtain FUD ( e−iϕ3 , e−iϕ2 , e−iϕ1 ) = MD (√ 3 ( 2c1 + λ23 ) , √ −3 ( 2c1 + λ22 ) , √ 3 ( 2c1 + λ21 ))−1 , (5.9) where the matrix M ∈ GL(3,C) diagonalizes the momentum of γ̃, i.e., M−1M̃M = D(λ3, λ2, λ1). In particular, the column vectors of the right hand side of (5.9), denoted by B̃, constitutes a pseudo-unitary basis. Let ϵ be the inverse of a cubic root of B̃. Then, the column vectors of ϵB̃ constitute a unimodular pseudo-unitary basis. Therefore, there exists a unique A ∈ G, such that ϵAB̃ = B, where B =  1√ 2 − 1√ 2 0 0 0 i i√ 2 i√ 2 0  . (5.10) Then AF = ϵ−1BD ( eiϕ3 , eiϕ2 , eiϕ1 ) U−1 = ϵ−1BD ( eiϕ3 , eiϕ2 , eiϕ1 ) D(−1, 1, 1)tŪh. (5.11) It is now a computational matter to check that the first column vector of the right hand side of (5.11) is ϵ−1 t(z2 + z3, iz1,−i(z2 − z3)). This implies γ̃ = A−1γ (i.e., γ̃ and γ are congruent to each other). Taking into account that U−1LU = D(λ3, λ2, λ1) and using (5.9), the momentum of γ̃ is M̃ = B̃D(λ3, λ2, λ1)B̃ −1. Therefore, the momentum of γ is M = AB̃D(λ3, λ2, λ1)B̃ −1A−1 = BD(λ3, λ2, λ1)B −1 = Mλ1,λ2,λ3 . This proves that γ is in standard configuration. If ε = −1 (i.e., ⟨y3,y3⟩ > 0), considering U = (u2,u3,u1) and arguing as above, we get the same conclusion. ■ Remark 5.6. Theorem C implies that a standard configuration γ does not pass through the pole [e3] of the Heisenberg projection πH . Thus γ̌ := πH ◦ γ is a transversal curve of R3, which does not intersect the Oz-axis. 28 E. Musso and L. Nicolodi Remark 5.7. Breaking the integrands into partial fractions, the integrals fj(τ) = √ 3 2 ∫ τ e2 3c1λj − ( 4c1 + λ2j ) + 3τ3 τ ( 3λj − τ2 )√ Pc(τ) dτ, j = 1, 2, 3, can be written as linear combinations of standard hyperelliptic integrals of the first and third kind. Then ϕj is the odd quasi-periodic function with quasi-period 2ω such that ϕj(s) = fj [τ(s)]. In practice, we compute τ and ϕj , j = 1, 2, 3, by numerically solving the following system of ODE, τ ′′ = τ2 − 9c1τ −2 ( 1− c1τ −1 ) , ϕ′j = 3c1λj − ( 4c1 + λ2j ) τ2 + 3τ3 τ2 ( 3λj − τ2 ) , j = 1, 2, 3, (5.12) with initial conditions τ(0) = e2, τ ′(0) = 0, ϕj(0) = 0, j = 1, 2, 3. (5.13) 5.3 Closing conditions From Theorem C, it follows that a critical curve of type B′ 1 is closed if and only if Pj = 1 2π ϕj(2ω) ∈ Q, j = 1, 2, 3. On the other hand, 1 2π ϕj(2ω) = 1 π ∫ e1 e2 √ 3 ( 3c1λj − ( 4c1 + λ2j ) τ2 + 3τ3 ) √ 2τ ( 3λj − τ2 )√ Pc(τ) dτ. (5.14) Thus, γ is closed if and only if the complete hyperelliptic integrals on the right hand side of (5.14) are rational. For a closed critical curve γ, we put Pj = qj = mj/nj , where nj > 0 and gcd(mj , nj) = 1. We call qj , the quantum numbers of γ. By construction, ei2πP1 , ei2πP2 and ei2πP3 are the eigenvalues of the monodromy Mγ = F(2ω)F(0)−1 of γ. Since det(Mγ) = 1, we have 3∑ j=1 Pj ≡ 0, mod Z. Then, γ is closed if and only if two among the integrals Pj , j = 1, 2, 3, are rational. Remark 5.8. The closing conditions can be rephrased as follows. Consider the even quasi- periodic functions ϕ1, ϕ3. Then, the critical curve is closed if and only if the jumps ϕj |2ω0 , j = 1, 3, are rational. Example 5.9. We now consider an example, which will be taken up again in the last section. Choose c ≈ (−0.8284243304411575,−8.349417691746162) ∈ B− 1 . The real roots of the quintic polynomial are e1 ≈ −0.931924 < e2 ≈ −0.678034 < 0 < e3 ≈ 2.79051 and the eigenvalues of the momentum are λ1 ≈ −2.40462 < 0 < λ2 ≈ 0.40614 < λ3 ≈ 1.99848. On the Total CR Twist of Transversal Curves in the 3-Sphere 29 Figure 8. The graph of ϕ1 for c ≈ (−0.8284243304411575,−8.349417691746162) ∈ B− 1 . The half-period of the twist is computed by numerically evaluating the hyperelliptic inte- gral (4.4). We evaluate τ , ϕ1, ϕ2, ϕ3 by solving numerically the system (5.12), with initial conditions (5.13) on the interval [−4ω, 4ω]. Figure 8 reproduces the graph of the quasi-periodic function ϕ1 on the interval [−4ω, 4ω] (the graph of the twist was depicted in Figure 5). The red point on the Ox-axis is 2ω and the length of the arrows is the jump ϕ1|2ω0 . In this example,∣∣∣∣− 1 2π 2 15 − ϕ1|2ω0 ∣∣∣∣ = 1.6151 · 10−8, ∣∣∣∣− 1 2π 10 21 − ϕ3|2ω0 ∣∣∣∣ = 4.46887 · 10−8. So, modulo negligible numerical errors, the corresponding critical curves are closed, with quan- tum number q1 = −2/15 and q3 = −10/21. In the last section, we will explain how we computed the modulus. A standard configuration of a curve with modulus c is represented in Figure 13. 5.4 Discrete global invariants of a closed critical curve Consider a closed general critical curve γ of type B′ 1, with modulus c and quantum numbers q1 = m1/n1, q2 = m2/n2, q3 = m3/n3, q1+q2+q3 ≡ 0 mod Z. The half-period ω of the twist is given by the complete hyperelliptic integral (4.4). Let Mγ = F(ω)F(0)−1 be the monodromy of γ. The monodromy does not depend on the choice of the canonical lift. It is a diagonalizable element of G with eigenvalues e2πiq1 , e2πiq2 , and e2πiq3 . Thus, Mγ has finite order n = lcm(n1, n3). The momentum Mγ has three distinct real eigenvalues, so its stabilizer is a maximal compact abelian subgroup T2 γ ∼= S1 × S1 of G ( if γ is a standard configuration, T2 γ = T2 ) . Since [Mγ ,Mγ ] = 0, Mγ ∈ T2 γ . Let s1, s3 be the integers defined by n = s1n1 = s3n3. The CR spin of γ is 1/3 if and only if n ≡ 0 mod 3 and m1s1 ≡ m3s3 ̸≡ 0 mod 3. The wave number nγ of γ is n if the spin is 1 and n/3 if the spin is 1/3. Let |[γ]| denote the trajectory of γ. The stabilizer Ĝγ = {[A] ∈ [G] | [A] · |[γ]| = |[γ]|} is spanned by [Mγ ] and is a cyclic group of order nγ . Geometrically, Ĝγ is the symmetry group of the critical curve γ. The CR turning number wγ is the degree of the map R/2nωZ ∋ s 7→ F1−iF3 ∈ Ċ := C\{0}, where the Fj ’s are the components of a Wilczynski frame along γ. Without loss of generality, we may suppose that γ is in a standard configuration. From (5.8), it follows that wγ is the degree of R/2nωZ ∋ s 7→ z3 ∈ Ċ, if εγ = 1, and is the degree of R/2nωZ ∋ s 7→ z2 ∈ Ċ, if εγ = −1. Therefore, wγ = { s3m3 if εγ = 1, s2m2 if εγ = −1. A closed critical curve γ has an additional discrete CR invariant, denoted by tr∗(γ), the trace of γ with respect to the spacelike λ1-eigenspace of the momentum. To clarify the geometrical meaning of the trace, it is convenient to consider a standard configuration. In this case, L1 is 30 E. Musso and L. Nicolodi spanned by e2 ∈ C1,2 and the corresponding chain is the intersection of S with the projective line z2 = 0. The Heisenberg projection of this chain is the upward oriented Oz-axis. Thus, tr∗(γ) is the linking number Lk ( γ̌, Oz↑ ) of the Heisenberg projection of γ with the upward oriented Oz-axis. Proposition 5.10. Let γ be as above. Then tr∗(γ) = { (q1 − q3)nγ if εγ = 1, (q1 − q2)nγ if εγ = −1. Proof. Without loss of generality, we may assume that γ is in standard configuration. The Heisenberg projection of γ is γ̌ = t ( Re ( εiz1, z2 + z3 ) , Im ( εiz1, z2 + z3 ) ,Re ( −εi(z2 − z3) z2 + z3 )) . Since γ̌ does not intersect the Oz-axis, the linking number Lk ( γ̌, Oz↑ ) is the degree of R/2nγωZ ∋ s 7−→ z1 z2 + z3 ∈ Ċ. From (5.7), it follows that this degree is the degree of f : R/2nγωZ ∋ s 7−→ ρ1 √ 3(λ2 + λ3) + τ2(s)eiϕ1 ρ2 √ 3λ2 − τ2(s)eiϕ2 + ρ3 √ 3λ3 − τ2(s)eiϕ3 . Suppose that γ is negatively polarized. Then, τ2 − 3λ3 < τ2 − 3λ2 < 0 and 0 < τ2 < 3λ2. Therefore, 0 < ρ2 √ 3λ2 − τ2 ρ3 √ 3λ3 − τ2 = √( 3λ2 − τ2 ) (λ2 + 2λ3)( 3λ3 − τ2 ) (2λ2 + λ3) ≤ √ λ2(λ2 + 2λ3) λ3(2λ2 + λ3) < 1. Thus f = ρ1 √ 3(λ2 + λ3) + τ2 ρ3 √ 3λ3 − τ2 ei(ϕ1−ϕ3) 1 + hei(ϕ2−ϕ3) , where h = ρ2 √ 3λ2 − τ2 ρ3 √ 3λ3 − τ2 . Since 0 < h < 1, the image of 1+hei(ϕ2−ϕ3) is a curve contained in a disk of radius < 1 centered at (1, 0). Hence 1 + hei(ϕ2−ϕ3) is null-homotopic in Ċ. This implies deg(f) = 1 2π (ϕ1 − ϕ3) ∣∣∣2nγω 0 = nγ(q1 − q3). Suppose that γ is positively polarized. Then, τ2 − 3λ2 > τ2 − 3λ3 > 0. In particular, τ2 > 3λ3 > 0 and 0 < ρ3 √ τ2 − 3λ3 ρ2 √ τ2 − 3λ2 = √ (τ2 − 3λ3)(2λ2 + λ3) (τ2 − 3λ2)(λ2 + 2λ3) < √ 2λ2 + λ3 λ2 + 2λ3 < 1. On the Total CR Twist of Transversal Curves in the 3-Sphere 31 Then f = −i ρ1 √ 3(λ2 + λ3) + τ2 ρ2 √ τ2 − 3λ2 ei(ϕ1−ϕ2) 1 + h̃ei(ϕ3−ϕ2) , where h̃ = ρ3 √ τ2 − 3λ3 ρ2 √ τ2 − 3λ2 . Since 0 < h̃ < 1, the image of 1+ h̃ei(ϕ3−ϕ2) is a curve contained in a disk of radius < 1 centered at (1, 0). Hence 1 + h̃ei(ϕ3−ϕ2) is null-homotopic in Ċ. This implies deg(f) = 1 2π (ϕ1 − ϕ2) ∣∣∣2nγω 0 = nγ(q1 − q2). ■ Summarizing: the quantum numbers of a closed critical curve are determined by the wave number, the CR spin, the CR turning number, and the trace. 6 Experimental evidence of the existence of infinite countably many closed critical curves of type B′ 1 and examples This section is of an experimental nature. We use numerical tools, implemented in the software Mathematica 13.3, to support the claim that there exist countably many closed critical curves of type B′ 1, with moduli belonging to the connected component B− 1 of B1 (cf. Remark 4.14). The same reasoning applies, as well, if the modulus belongs to the other connected component B+ 1 of B1. We parametrize B− 1 by the map ψ− : K− → B− 1 , defined in (4.6), whereK− is the rectangle Ĵ−ξ × (0, 1), Ĵ−ξ = (π/2, 2.3008). We take p = (p1, p2) ∈ K− as the fundamental parameters. The modulus c = (c1, c2), the roots e1 < e2 < 0 < e3 of the quintic polynomial, and the eigenvalues λ1 = −(λ2 + λ3) < 0 < λ2 < λ3 of the momentum are explicit functions of the parameters (p1, p2). Let K ∗ − be the open set of the general parameters, that is, K∗ − = {p ∈ K− | ∆1(ψ−(p))∆2(ψ−(p)) ̸= 0} . The complete hyperelliptic integrals Pj can be evaluated numerically as functions of p ∈ K∗ −. Consider the real analytic map P = (P1,P3) : K ∗ − → R2.6 Choose p∗ = (2, 1/2) ∈ K∗ − and plot the graphs of the functions f11(p1) = P1(p1, 1/2), f12(p2) = P1(2, p2), f31(p1) = P3(p1, 1/2), and f32(p2) = P3(2, p2) (see Figures 9 and 10). The function f11 is strictly increasing, while the other three functions are strictly decreasing. This implies thatP has maximal rank at p∗. Thus P− = P(K−) is a set with non empty interior. In particular Pr − := P− ∩ Q is an infinite countable set and, for every q = (q1, q2) ∈ Pr −, there exists a closed critical curve of type B′− 1 with quantum numbers q1 and q2. Figure 11 reproduces the plot of the map P, an open convex set. The mesh supports a stronger conclusion: the map P is 1-1. Therefore, one can assume that, for every rational point (q1, q3) ∈ P−, there exists a unique congruence class of closed critical curves with quantum numbers q1 and q3. The construction of a standard configuration of a critical curve associated to a rational point q ∈ P− can be done in three steps. Step 1. Choose a rational point q = (q1, q3) = (m1/n1,m3/n3) ∈ P−. To find the parameter p ∈ K−, such that P(p) = q, we may proceed as follows: plot the level curves Xq1 = P−1 1 (q1) 6Actually, P is real-analytic on all K−. Instead, P2 has a jump discontinuity at the exceptional locus. 32 E. Musso and L. Nicolodi Figure 9. On the left: the graph of the function f11. On the right: the graph of the function f12. Figure 10. On the left: the graph of the function f31. On the right: the graph of the function f32. and Yq3 = P−1 3 (q3) and choose a small rectangle R ⊂ K− containing Xq1 ∩ Yq2 (see Figure 12). Then we minimize numerically the function δq : R ∋ p 7−→ √ (P1(p)− q1)2 + (P3(p)− q3)2. We use the stochastic minimization method named “differential evolution” [37] implemented in Mathematica. Example 6.1. Let us revisit Example 5.9. Choose q = (−2/15,−10/21) ∈ P−. The plot of the level curves Xq1 and Yq3 is depicted in Figure 12. Minimizing δq on the rectangle R = [1.83, 1.86]× [0.65, 0.75] (depicted on the right picture in Figure 12) we obtain p = (1.84438, 0.719473) and δq(p) = 3.26867 · 10−9. So, up to negligible numerical errors, we may assume p = P−1(q). Computing ψ−(p), we find the modulus c = (c1, c2) of the curve, where c1 = −0.828424 and c2 = −8.349418. With the modulus at hand, we compute the lowest real roots of the quintic polynomial, e1 ≈ −0.931924 < e2 ≈ −0.678034, and the roots of the momentum, namely λ1 ≈ −2.40462 < 0 < λ2 ≈ 0.40614 < λ3 ≈ 1.99848. Step 2. We evaluate numerically the integral (4.4) and we get the half-period ω of the twist of the critical curve. In our example ω ≈ 0.732307. The next step is to evaluate the twist τ . This can be done by solving numerically the Cauchy problem (4.5) on the interval [0, 2nω], On the Total CR Twist of Transversal Curves in the 3-Sphere 33 Figure 11. The plot of the map P. Figure 12. On the left: the point q = (−2/15,−10/21) ∈ P−. On the right: the level curves X−2/15 and Y−10/21. The dotted curve is the exceptional locus. The green and the cyan domains are the two connected components of K∗ −. The brow rectangle is the one chosen for the numerical minimization of the function δq. n = lcm(n1, n2). The bending is given by κ = c1/τ 2. Next, we solve the Frenet type linear system (2.3), with initial condition F(0) = I3. Then, γ̃ : [0, 2nω] ∋ s 7−→ [F1(s)] ∈ S is a critical curve with quantum numbers q1 and q3 and F is a Wilczynski frame field along γ̃. However, γ̃ is not in a standard configuration. Step 3. The last step consists in building the standard configuration. The momentum M of γ̃ is L(0), where L is as in (3.2). Taking into account that τ(0) = e2, τ ′(0) = 0, and that κ(0) = c1/e 2 2, we get M =  0 3 ( 1− c1 e2 ) 2ie2 e2 0 3i ( 1− c1 e2 ) 3i −ie2 0  . The eigenspace of the highest eigenvalue is timelike (i.e., these critical curves are negatively polarized). We compute the eigenvectors and we build a unimodular pseudo-unitary basis A = (A1, A2, A3), such that A1 is an eigenvector of λ3, A2 is an eigenvector of λ2, and A3 is an eigenvector of λ1. Let B be as in (5.10). Consider M = BA−1 ∈ G. Then, γ = Mγ̃ is a standard configuration of a critical curve with quantum numbers q1 and q3. 34 E. Musso and L. Nicolodi Figure 13. The Heisenberg projection of a standard configuration of a critical curve with quantum numbers q1 = −2/15 and q2 = −10/21. The figure on the left reproduces the fundamental arc γ([0, 2ω)) (coloured in yellow). The curve can be constructed by acting with the monodromy on the fundamental arc. Figure 14. On the left: the Heisenberg projection of a standard configuration of a critical curve of type B′− 1 with quantum numbers q1 = −3/10, q2 = −9/25. On the right: the Heisenberg projection of a standard configuration of a critical curve of type B′− 1 with quantum numbers q1 = −1/5, q2 = −3/7. Remark 6.2. The curve γ does not pass through the pole of the Heisenberg projection πH . So, γ̂ = πH ◦ γ is a closed transversal curve of R3 which does not intersect the Oz-axis and tr∗(γ) = Lk(γ̂, Oz). Example 6.3. Applying Steps 2 and 3 to Example 6.1 and computing the Heisenberg projection, we obtain the transversal curve depicted in Figure 13, a non-trivial transversal knot. The quantum numbers are q1 = −2/15 and q2 = −10/21. Recalling what has been said about the discrete invariants of a critical curve (cf. Section 5.4), the spin is 1/3, the wave number is n = 35, the CR turning number is −50, and the trace is 12. Example 6.4. Figures 14 and 15 reproduce the Heisenberg projections of the standard config- urations of critical curves of type B′− 1 with quantum numbers (−3/10, −9/25), (−1/5,−3/7), (5/49,−4/7), and (−7/36,−23/54), respectively. All of them have spin 1. The first is a trivial torus knot with wave number n = 50, tr∗ = 3, and CR turning number w = −18; the second example is a nontrivial transversal knot with n = 35, tr∗ = 8, and w = −15. The third example On the Total CR Twist of Transversal Curves in the 3-Sphere 35 Figure 15. On the left: the Heisenberg projection of a standard configuration of a critical curve of type B′− 1 , with quantum numbers q1 = 5/49, q2 = −4/7. On the right: the Heisenberg projection of a standard configuration of a critical curve of type B′− 1 , with quantum numbers q1 = −7/36, q2 = −23/54. is a “tangled” transversal curve with n = 49, tr∗ = 33, and w = −28. The last example is a nontrivial transversal torus knot with wave number 108, tr∗ = 25, and w = −46. Remark 6.5. 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id nasplib_isofts_kiev_ua-123456789-212030
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2026-03-14T05:42:52Z
publishDate 2023
publisher Інститут математики НАН України
record_format dspace
spelling Musso, Emilio
Nicolodi, Lorenzo
2026-01-23T10:08:31Z
2023
On the Total CR Twist of Transversal Curves in the 3-Sphere. Emilio Musso and Lorenzo Nicolodi. SIGMA 19 (2023), 101, 36 pages
1815-0659
2020 Mathematics Subject Classification: 53C50; 53C42; 53A10
arXiv:2307.04763
https://nasplib.isofts.kiev.ua/handle/123456789/212030
https://doi.org/10.3842/SIGMA.2023.101
We investigate the total CR twist functional on transversal curves in the standard CR 3-sphere S3⊂C2. The question of the integration by quadratures of the critical curves and the problem of the existence and properties of closed critical curves are addressed. A procedure for the explicit integration of general critical curves is provided, and a characterization of closed curves within a specific class of general critical curves is given. Experimental evidence of the existence of an infinite countable number of closed critical curves is provided.
The authors were partially supported by PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” (protocollo 2017JZ2SW5-004) and by the GNSAGA of INdAM. The present research was also partially supported by MIUR grant “Dipartimenti di Eccellenza” 2018-2022, CUP: E11G18000350001, DISMA, Politecnico di Torino. The authors gratefully acknowledge the referees for their helpful comments and suggestions.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
On the Total CR Twist of Transversal Curves in the 3-Sphere
Article
published earlier
spellingShingle On the Total CR Twist of Transversal Curves in the 3-Sphere
Musso, Emilio
Nicolodi, Lorenzo
title On the Total CR Twist of Transversal Curves in the 3-Sphere
title_full On the Total CR Twist of Transversal Curves in the 3-Sphere
title_fullStr On the Total CR Twist of Transversal Curves in the 3-Sphere
title_full_unstemmed On the Total CR Twist of Transversal Curves in the 3-Sphere
title_short On the Total CR Twist of Transversal Curves in the 3-Sphere
title_sort on the total cr twist of transversal curves in the 3-sphere
url https://nasplib.isofts.kiev.ua/handle/123456789/212030
work_keys_str_mv AT mussoemilio onthetotalcrtwistoftransversalcurvesinthe3sphere
AT nicolodilorenzo onthetotalcrtwistoftransversalcurvesinthe3sphere

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