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 $\mathrm S^3 \subset \mathbb C^2$. 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 characterization 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.


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 3sphere.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 S 3 , realized as a real hyperquadric of CP 2 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 S 3 .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 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 S 3 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 S 3 .According to [31], away from CR inflection points, a curve transversal to the contact distribution of S 3 can be parametrized by a natural pseudoconformal parameter s and in this parametrization 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 automorphisms.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 S 3 defined by the total CR twist functional, 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 S 3 , 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 S 3 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 C 2,1 .From the conservation of the momentum along a critical curve, we derive two conservation laws, involving two real parameters c 1 and c 2 .The pair c = (c 1 , c 2 ) 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 The phase curve of the modulus c is the real algebraic curve defined by the equation y 2 = P c (x).The signature of a critical curve γ with modulus c and nonconstant twist provides a parametrization of the connected components of the phase curve of c by the twist of γ.Importantly, the 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 P c 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 λ 1 -eigenspace of the momentum.Using numerical tools, we construct and illustrate explicit examples of approximately closed critical curves.

Preliminaries
2.1 The standard CR structure on the 3-sphere Let C 2,1 denote C 3 with the indefinite Hermitian scalar product of signature (2,1) given by Following common terminology in pseudo-Riemannian geometry, a nonzero vector z ∈ C 2,1 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 The restriction of the affine chart to the unit sphere S 3 of C 2 defines a smooth diffeomorphism between S 3 and S. For each p = [z] ∈ S, the differential (1, 0)-form is well defined.In addition, the null space of the imaginary part of ζ| p is T (S)| p , namely the tangent space of S at p. Thus, the restriction of ζ to T (S) is a real-valued 1-form ζ ∈ Ω 1 (S).
Since the pullback of ζ by the diffeomorphism s : S 3 → S is the standard contact form iz • dz| S 3 of S 3 , then ζ is a contact form whose contact distribution D is, by construction, a complex subbundle of T CP2 S .Therefore, D inherits from T CP 2 S a complex structure J.This defines a CR structure on S.
Let e 1 , e 2 , e 3 denote the standard basis of C 3 .Consider P 0 = [e 1 ] ∈ S and P ∞ = [e 3 ] ∈ 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 projection 2 The inverse of the Heisenberg projection is the Heisenberg chart The Heisenberg chart can be lifted to a map whose image is a 3-dimensional closed subgroup H 3 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), and let g denote the Lie algebra of G, The Maurer-Cartan form of the group G takes the form where the 1-forms Such a basis is referred to as a lightcone basis.On the other hand, a basis (u is referred to as a unimodular pseudo-unitary basis.
The group G acts transitively and almost effectively on the left of S by 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 [e 1 ] = t (1, 0, 0) ∈ S as an origin of S, the natural projection makes G into a (trivial) principal fiber bundle with structure group The elements of G 0 consist of all 3 × 3 unimodular matrices of the form where v ∈ C, r ∈ R, 0 ≤ θ < 2π, and ρ > 0.

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.
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 π S • A = γ.Since the fibration π S is trivial, there exist frame fields along every transversal curve.
Let A be a frame field along γ.Then where a 3 1 is a strictly positive real-valued function.Any other frame field along γ is given by Ã = AX(ρ, θ, v, r), where ρ (ρ > 0), θ, r : J → R, v = p + iq : J → C are smooth functions and From this it follows that along any parametrized transversal curve there exists a frame field A for which a 2 1 + ib 2 1 = 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 : for some t 0 ∈ J, then γ(t 0 ) 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.
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 C γ | t , is called the osculating chain of γ at γ(t).By construction, C γ | 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.

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 is smooth, real-valued, and independent of the choice of Γ.We call a γ the strain density of the parametrized transversal curve γ.The linear differential form ds = a γ dt 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 ds of γ and γ = γ • h, respectively, are related by ds = 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 Γ = AΓ is a Wilczynski lift of γ = Aγ.This implies that a γ = a γ .Next, consider a reparametrization γ = γ • h of γ.Then, Therefore, the strain densities of γ and γ are related by As a straightforward consequence of Proposition 2.9, we have the following.
Definition 2.11.If a γ = 1, we say that γ : J → S is a natural parametrization, or a parametrization 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 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.
(1) Let γ : J → S be as above and F = (F 1 , F 2 , F 3 ) : J → G be a Wilczynski frame along γ.Then, 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 authors in [31].In the following we will consider generic transversal curves with nonconstant CR invariant functions.
Remark 2.15.Regarding the CR 3-sphere S 3 ∼ = S with its standard pseudo-hermitian (PSH) structure (J, ζ), Chiu and Ho (cf.[8]) obtained a complete set of local PSH invariants for horizontally regular curves in S 3 parametrized by the horizontal arc length w, namely the p-curvature k psh (w) and the T -variation τ psh (w).The canonical PSH frame field producing the PSH invariants 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 k psh (w(s)), τ psh (w(s)) and their derivatives with respect to s.

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 ω γ of γ and the least period ω of the functions (κ, τ ).The CR spin is the ratio between ω γ and the least period of a Wilczynski lift of γ.The CR turning number is the degree (winding number) of the map We will also make use of another invariant.

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 where J γ is the domain of definition of the transversal curve γ, τ γ is its twist, and η γ = ds γ 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 parameter.Then, γ is a critical curve if and only if3 where and K κ,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.Let γ : J → S be a generic transversal curve parametrized by the natural parameter.According to Proposition 2.12, the canonical frame of γ defines a unique lift [F] : is referred to as the extended frame of γ.The product space M := [G] × R 2 is called the configuration space.The coordinates on R 2 will be denoted by (κ, τ ).
With some abuse of notation, we use 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 with the independence condition η 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 the 1-forms η, µ 1 , . . ., µ 7 , π 1 , π 2 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, µ 1 , . . ., µ 7 , π 1 , π 2 : From the structure equations it follows that the derived flag of (A, η) is given by , where . Thus, all the derived systems of (A, η) have constant rank.For the notion of derived flag, see [19].
Let Z ⊂ T * (M ) be the affine subbundle defined by the 1-forms µ 1 , . . ., µ 7 and λ := τ η, namely We call Z the phase space of the Pfaffian system (A, η).The 1-forms µ 1 , . . ., µ 7 , λ induce a global affine trivialization of Z, which may be identified with M × R 7 by the map where p 1 , . . ., p 7 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 Exterior differentiation and use of the quadratic equations (3.3) and (3.4) yield where the sign '≡' denotes equality modulo the span of {µ i ∧ µ j } 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 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).
Lemma A1.The Cartan system (C(dξ), η) is the Pfaffian system on Z ∼ = M × R 7 generated by the 1-forms µ 1 , . . ., µ 7 , π1 , π2 , η, μ1 , . . ., μ7 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).
Lemma A2.The momentum space Y is the 11-dimensional submanifold of Z defined by the equations The Euler-Lagrange system (J , η) is the Pfaffian system on Y ∼ = M × R, with independence condition η ̸ = 0, generated by the 1-forms Next, the restriction of μ6 and μ7 to Z 1 take the form μ6 = −3p 4 η and μ7 = p 5 η.Thus, the second reduction Z 2 is given by Considering the restriction of μ4 and μ5 to Z 2 yields the equations which define the third reduction Z 3 .Now, the restriction C 3 (dξ) to Z 3 of the Cartan system C(dξ) is generated by the 1-forms µ 1 , . . ., µ 7 and This implies that there exists an integral element of V 1 (dξ) over each point of Z 3 , i.e., for each Hence, Y := Z 3 is the momentum space and (J , η) := (C 3 (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 Proof of Lemma A3.If y = (([F], κ, τ ); p 1 ) : J → Y is an integral curve of the Euler-Lagrange system (J , η), the projection γ = pr • y is the smooth curve γ(s) = [F 1 (s)], where F 1 is the first column of F. The equations 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 With reference to (3.6), equation σ 3 = 0 implies Further, σ 2 = 0 gives Finally, equation σ 1 = 0 yields Conversely, let γ : J → S be a generic transversal curve, parametrized by the natural parameter, satisfying (3.7) and (3.8) and let [F] its canonical frame.Then, 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 Gmodule of traceless selfadjoint endomorphisms of C 2,1 .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 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 where K κ,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 parameter.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 where M is a fixed element of h corresponding to a chosen value L(s 0 ) 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 The conservation of the momentum along γ yields the two conservation laws for real constants c 1 and C 2 .We let c 2 := C 2 − 9. Using this notation, the (opposite of the) characteristic polynomial of the momentum is If c 1 ̸ = 0, the twist and the bending are never zero and the conservation laws can be rewritten as If c 1 = 0, it can be easily proved that κ = 0 and the second conservation law takes the form Definition 3.4.The pair of real constants c = (c 1 , c 2 ) is called the modulus of the critical curve γ.

Phase types
For c = (c 1 , c 2 ) ∈ R 2 , we denote by P c the quintic polynomial in principal form given by and by Q c the cubic polynomial given by Excluding the case c = 0, P c possesses at least a pair of complex conjugate roots.
c ∈ R 2 is of phase type A if P c has four complex roots a j ± ib j , j = 1, 2, 0 < b 1 < b 2 , and a simple real root e 1 ; c ∈ R 2 is of phase type B if P c has two complex roots a ± ib, b > 0, and three simple real roots e 1 < e 2 < e 3 ; c ∈ R 2 is of phase type C if P c has a multiple real root.
In the latter case, two possibilities may occur: (1) P c has a double real root and a simple real root; or (2) P c 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.
Claim.P c has a double root a 3 ̸ = 0 if and only if c belongs to the separatrix curve minus the cusp.
Note that c 1 ̸ = 0 (otherwise the double root would be 0).Let a 4 be the other simple real root and b 1 + ib 2 , b 1 − ib 2 , b 2 > 0, be the two complex conjugate roots.Since the sum of the roots of P c is zero, we have b 1 = − 1 2 (2a 3 + a 4 ).Since the coefficient of x 3 is zero and b and comparing the coefficients of the monomials x n , n = 1, . . ., 4, with the coefficients of P c we may write c 1 and c 2 as functions of a 3 and a 4 , In addition, Taking into account that a 3 ̸ = 0, it follows that (a 3 , a 4 ) belongs to the algebraic curve C (the black curve on the right picture in Figure 1) defined by the equation Now, consider the line ℓ m,n through the origin, with homogeneous coordinates (m, n), i.e., the line with parametric equations p m,n (t) = (mt, nt).If (m, n) ̸ = (1, 0) and 3m 3 +6m 2 n+4mn 2 +2n 3 ̸ = 0 (we are excluding the two red lines on the right picture in Figure 1), ℓ m,n intersects C when t = 0 and t = t m,n , where .
If (m, n) = (1, 0) or 3m 3 + 6m 2 n + 4mn 2 + 2n 3 = 0, ℓ m,n intersects C only at the origin (see the right picture in Figure 1).Hence , is a parametrization of C \ {(0, 0)}.Thus, using (4.2), the map is a parametrization of the set of all c, c 1 ̸ = 0, such that P c has multiple roots.It is now a computational matter to check that (c From the first part of the proof, the set of all c ∈ R 2 , such that P c has only simple roots is the complement of Ξ ∪ Oy.This set has five connected components: 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): 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 P c j , j = 1, 2, 3, have three distinct real roots and that P c j , j = 4, 5, have one real root.The domain M ′ + is connected and the function x ′′ c is strictly negative.Then, P c has three distinct real roots, for every c ∈ M ′ + .Similarly, P c has three distinct real roots, for every c ∈ M ′′ + ∪ M ′′′ + and a unique real root for every c ∈ M ′ − ∪ M ′′ − .This concludes the proof.■

Phase curves and signatures
Definition 4.7.Let Σ c be the real algebraic curve defined by y 2 = P c (x).We call Σ c the phase curve of c.
If c ∈ A ∪ B, Σ c is a smooth real cycle of a hyperelliptic curve of genus 2. If c ∈ C, and c ̸ = 0, Σ c is a singular real cycle of an elliptic curve.If c = 0, Σ c is a singular rational curve.The following facts can be easily verified: if c ∈ A, Σ c is connected, unbounded, and intersects the Ox-axis at (e 1 , 0) (see Figure 2); if c ∈ B, Σ c 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 (e 1 , 0) and (e 2 , 0), while Σ ′′ c intersects the Ox-axis at (e 3 , 0) (see Figure 2); if c ∈ C and c 1 ̸ = 0 , Σ c has a smooth, unbounded connected component Σ ′′ c and an isolated singular point (e 1 , 0), where e 1 = e 2 is the double real root of P c (x).The unbounded connected component intersects the Ox-axis at (e 3 , 0), where e 3 is the simple real root of P c (x) (see Figure 3).If c 1 = 0 and c 2 ̸ = 0, Σ c is connected, with an ordinary double point (see Figure 3).If c = 0, Σ c 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 J γ ⊂ 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 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 σ γ (J γ ) is compact.Observing that σ γ (J γ ) is one of the 1-dimensional connected Definition 4.10.A critical curve γ with modulus c is said to be of type

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 P c has a unique real root e 1 .
The polynomial P c (x) is positive if x > e 1 and is negative if x < e 1 .Since P c (0) = −27c 2 1 /2 < 0, the root is positive.Let ω c > 0 be the improper hyperelliptic integral of the first kind defined by The incomplete hyperelliptic integral is a strictly increasing diffeomorphism of [e 1 , +∞) onto [0, ω c ) (see Figure 4).The twist is the unique even function The maximal domain of definition is J c = (−ω c , ω c ). τ c is strictly positive, with vertical asymptotes as s → ∓ω ± c (see Figure 4).Note that τ c is the solution of the Cauchy problem  Let h c be the incomplete hyperelliptic integrals of the first kind The function h c is a diffeomorphism of [e 1 , e 2 ] onto [0, ω c ], strictly decreasing if e 1 < e 2 < 0 and strictly increasing if 0 < e 1 < e 2 (see Figure 5).The twist τ c is the even periodic function with least period 2ω c , obtained by extending periodically the function τ (s) = h −1 c (s) defined on [0, ω c ] and on [−ω c , 0], respectively.If e 1 < e 2 < 0, then τ c is strictly negative with minimum value e 1 and maximum value e 2 , attained, respectively, at s ≡ ω c mod 2ω c and at s ≡ 0 mod 2ω c (see Figure 5).If 0 < e 1 < e 2 , then τ c is strictly positive, with minimum value e 1 and maximum value e 2 , attained, respectively, at s ≡ 0 mod 2ω c and at s ≡ ω c mod 2ω c .
Observe that τ c is the solution of the Cauchy problem (4.5)

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 e 3 > 0 be the highest real root of P c and ω c be the improper hyperelliptic integral of the first kind given by Let h c (τ ) be the incomplete hyperelliptic integral

4.3.4
The twist of a critical curve of type C with c 1 ̸ = 0 The twist of a critical curve of type C, with c 1 ̸ = 0, can be constructed as for curves of types A or B ′′ .Let e 3 > 0 be simple real root of P c and ω c be the improper elliptic integral of the first kind Let h c (τ ) be the incomplete elliptic integral Then, h c is a strictly increasing diffeomorphism of [e 3 , +∞) onto [0, ω c ).The twist τ c is the unique even function The twist is positive, with vertical asymptotes as s → ∓ω ± c .Note that τ c is the solution of the Cauchy problem

The twist of a critical curve with c 1 = 0
If c 1 = 0, the bending vanishes identically and the twist is a solution of the second order ODE τ ′′ − τ 2 = 0.Then, where a is an unessential constant and ℘(−, g 2 , g 3 ) is the Weierstrass function with invariants g 2 , g 3 .

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.
the momentum of a critical curve with modulus c ∈ OT 1 has three distinct real eigenvalues: 2. Of orbit type 2 (in symbols, c ∈ OT 2 ) if ∆ 1 (c) < 0; the momentum of a critical curve with modulus c ∈ OT 2 has a real eigenvalue λ 1 and two complex conjugate roots: λ 2 , with positive imaginary part, and λ 3 = λ 2 .
3. Of orbit type 3 (in symbols, c ∈ OT 3 ) if ∆ 1 (c) = 0; the momentum of a critical curve with modulus c ∈ OT 3 has an eigenvalue with algebraic multiplicity greater than one (> 1).
Let J be the maximal interval of definition of the twist (it can be computed in terms of the modulus).Define y j : J → C 1,2 , j = 1, 2, 3, by Let V : J → gl(3, C) be the matrix-valued map with column vectors y 1 , y 2 and y 3 .Let D(z 1 , z 2 , z 3 ) denote the diagonal matrix with z j as the jth element on the diagonal.Recall that, if c 1 ̸ = 0, then τ is nowhere zero.We can prove the following.
Theorem B. Let γ : J → S be a general critical curve.The functions det(V) and τ 2 − 3λ j , j = 1, 2, 3, are nowhere zero.Let r j be continuous determinations of τ 2 − 3λ j and let ϕ j be the functions defined by5 Then, γ is congruent to where 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 ± √ 3λ 2 and ± √ 3λ 3 cannot be roots of P c ; (2) if the momentum has two complex conjugate eigenvalues and a positive real eigenvalue λ 1 , then ± √ 3λ 1 cannot be roots of P c .
Proof of Lemma B1.First, note that the image of the parametrized curve is contained in the zero locus of ∆ 2 .This can be proved by a direct computation.Secondly, from the expression of Q c , it follows that 1. Suppose that the momentum has three distinct real eigenvalues.By contradiction, suppose that √ 3λ 2 is a root of P c .Then Solving this equation with respect to λ 3 , taking into account that λ 3 > 0, we obtain Substituting into (5.3),we find Then, c = α − √ λ 2 .This implies that c belongs to the zero locus of ∆ 2 , which is a contradiction.By an analogous argument, we prove that also − √ 3λ 2 cannot be a root of P c .By interchanging the role of λ 2 and λ 3 and arguing as above, it follows that also ± √ 3λ 3 cannot be roots of P c .
2. Next, suppose that the momentum has two complex conjugate eigenvalues and a nonnegative real eigenvalue λ 1 .Recall that the eigenvalues are sorted so that the imaginary part of λ 2 is positive.By contradiction, suppose that √ 3λ 1 is a root of P c .Then, Solving this equation with respect to λ 2 , taking into account that the imaginary part of λ 2 is positive, we find Substituting into (5.3)yields c = α − √ λ 1 .Thus, c is a root of ∆ 2 , which is a contradiction.An analogous argument shows that − √ 3λ 1 cannot be a root of P c .This concludes the proof of the lemma.■ Lemma B2. det(V)(s) ̸ = 0, for every s ∈ J γ .
Proof of Lemma B2.Let L j be the 1-dimensional eigenspaces of the momentum M γ relative to the eigenvalues λ j .Let L be as in (3.2).By Corollary 3.2 of Theorem A, we have where F is a Wilczynski frame field along γ.Then, L(s) and M have the same eigenvalues.Next, consider the line bundles Note that (s, y) ∈ Λ j if and only if F(s)y ∈ L j .Let y j , j = 1, 2, 3, be as in (5.1).A direct computation shows that Ly j = λ j y j .Thus, y j is a cross section of the eigenbundle Λ j .Hence, det(V)(s) ̸ = 0 if and only if y j (s) ̸ = ⃗ 0, for every s.
Proof of Lemma B3.The statement is obvious if λ j is real and negative or complex, with nonzero imaginary part.If λ j is real non-negative, the smoothness of Φ j implies that τ τ ′ 3λ j − τ 2 −1 is differentiable.Then 3λ j − τ 2 (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 P c .Therefore, by Lemma B1, we have that 3λ j − τ 2 (s) ̸ = 0. ■ From (5.5), we have where b j is a constant of integration, τ 2 − 3λ j is a continuous determination of the square root of τ 2 − 3λ j and log τ 2 − 3λ j is a continuous determination of the logarithm of τ 2 − 3λ j .Since w ′ j = Φ j w j , we obtain where m j is a constant vector belonging to the eigenspace L j of M. This implies where M is an invertible matrix such that M −1 MM = D(λ 1 , λ 2 , λ 3 ).By possibly replacing γ with a congruent curve, we may suppose that F(0) = I 3 .Then, since ϕ j (0) = 0, we have M = V(0)D(r 1 (0), r 2 (0), r 3 (0)) −1 .This concludes the proof of Theorem B. ■ 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 ∈ B 1 and with periodic twist).Let M ′ + be as in (4.3).Since B 1 is contained in M ′ + , the lowest roots e 1 and e 2 of P c are negative, for every c ∈ B 1 (cf.Remark 4.6).Lemma 5.2.Let γ be a general critical curve of type B ′ 1 .The λ 1 -eigenspace of the momentum is spacelike.
Remark 5.4.In view of the above lemma, γ is positively polarized if and only if e 2 1 − 3λ 3 > 0 and is negatively polarized if and only if e 2 2 − 3λ 3 < 0. It is a linear algebra exercise to prove the existence of A ∈ G, such that A −1 MA = M λ 1 ,λ 2 ,λ 3 , where 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 M λ 1 ,λ 2 ,λ 3 .
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 e 2 < e 3 be the real roots of P c and let λ 1 = −(λ 2 + λ 3 ) < 0 < λ 2 < λ 3 be the roots of Q c .Let τ be the periodic function defined as in the first of the (4.5) and ϕ j , j = 1, 2, 3, be as in (5.2).Let ρ j be the constants and z j be the functions Let ε = −sign e 2 2 − 3λ 3 .We can state the following.
Remark 5.7.Breaking the integrands into partial fractions, the integrals can be written as linear combinations of standard hyperelliptic integrals of the first and third kind.Then ϕ j is the odd quasi-periodic function with quasi-period 2ω such that ϕ j (s) = f j [τ (s)].

Closing conditions
From Theorem C, it follows that a critical curve of type B ′ 1 is closed if and only if On the other hand, (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 P j = q j = m j /n j , where n j > 0 and gcd(m j , n j ) = 1.We call q j , the quantum numbers of γ.By construction, e i2πP 1 , e i2πP 2 and e i2πP 3 are the eigenvalues of the monodromy M γ = F(2ω)F(0) −1 of γ.Since det(M γ ) = 1, we have Then, γ is closed if and only if two among the integrals P j , j = 1, 2, 3, are rational.
Remark 5.8.The closing conditions can be rephrased as follows.Consider the even quasiperiodic functions ϕ 1 , ϕ 3 .Then, the critical curve is closed if and only if the jumps ϕ j | 2ω 0 , j = 1, 3, are rational.
Example 5.9.We now consider an example, which will be taken up again in the last section.The half-period of the twist is computed by numerically evaluating the hyperelliptic integral (4.4).We evaluate τ , ϕ 1 , ϕ 2 , ϕ 3 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 ϕ 1 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 ϕ 1 | 2ω 0 .In this example, So, modulo negligible numerical errors, the corresponding critical curves are closed, with quantum number q 1 = −2/15 and q 3 = −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.

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 q 1 = m 1 /n 1 , q 2 = m 2 /n 2 , q 3 = m 3 /n 3 , q 1 + q 2 + q 3 ≡ 0 mod Z.The half-period ω of the twist is given by the complete hyperelliptic integral (4.4).Let M γ = 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 e 2πiq 1 , e 2πiq 2 , and e 2πiq 3 .Thus, M γ has finite order n = lcm(n 1 , n 3 ).The momentum M γ has three distinct real eigenvalues, so its stabilizer is a maximal compact abelian subgroup T and is a cyclic group of order n γ .Geometrically, Ĝγ is the symmetry group of the critical curve γ.The CR turning number w γ is the degree of the map R/2nωZ ∋ s → F 1 −iF 3 ∈ Ċ := C\{0}, where the F j '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 w γ is the degree A closed critical curve γ has an additional discrete CR invariant, denoted by tr * (γ), the trace of γ with respect to the spacelike λ 1 -eigenspace of the momentum.To clarify the geometrical meaning of the trace, it is convenient to consider a standard configuration.In this case, L 1 is spanned by e 2 ∈ C 1,2 and the corresponding chain is the intersection of S with the projective line z 2 = 0.The Heisenberg projection of this chain is the upward oriented Oz-axis.Thus, tr * (γ) is the linking number Lk γ, Oz ↑ of the Heisenberg projection of γ with the upward oriented Oz-axis.
The function f 11 is strictly increasing, while the other three functions are strictly decreasing.This implies that P has maximal rank at p * .Thus P − = P(K − ) is a set with non empty interior.In particular P r − := P − ∩ Q is an infinite countable set and, for every q = (q 1 , q 2 ) ∈ P r − , there exists a closed critical curve of type B ′ − 1 with quantum numbers q 1 and q 2 .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 (q 1 , q 3 ) ∈ P − , there exists a unique congruence class of closed critical curves with quantum numbers q 1 and q 3 .The construction of a standard configuration of a critical curve associated to a rational point q ∈ P − can be done in three steps.
We use the stochastic minimization method named "differential evolution" [37] implemented in Mathematica.
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],  n = lcm(n 1 , n 2 ).The bending is given by κ = c 1 /τ 2 .Next, we solve the Frenet type linear system (2.3), with initial condition F(0) = I 3 .Then, γ : [0, 2nω] ∋ s −→ [F 1 (s)] ∈ S is a critical curve with quantum numbers q 1 and q 3 and F is a Wilczynski frame field along γ.However, γ is not in a standard configuration.
Step 3. The last step consists in building the standard configuration.The momentum M of γ is L(0), where L is as in (3.2).Taking into account that τ (0) = e 2 , τ ′ (0) = 0, and that κ(0) = c 1 /e 2 2 , we get 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 = (A 1 , A 2 , A 3 ), such that A 1 is an eigenvector of λ 3 , A 2 is an eigenvector of λ 2 , and A 3 is an eigenvector of λ 1 .Let B be as in (5.10).Consider M = BA −1 ∈ G.Then, γ = M γ is a standard configuration of a critical curve with quantum numbers q 1 and q 3 .Remark 6.5.It is clear that, being numerical approximations, the parametrizations obtained with this procedure are only approximately periodic.
where I 3 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

Definition 2 . 16 .
Let [z] ∈ CP 2 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]

Figure 1 .
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.

Figure 2 .
Figure 2. On the left: the phase curve of c ∈ A. On the right: the phase curve of c ∈ B.

Figure 3 .
Figure 3. On the left: the phase curve of c ∈ C, c 1 ̸ = 0. On the center: the phase curve of c ∈ C, c 1 = 0 and c 2 ̸ = 0. On the right: the phase curve of c = (0, 0).

4. 3 . 2
The twist of a critical curve of type B ′ Let e 1 < e 2 < e 3 be the simple real roots of P c .The highest root e 3 is positive.The lower roots e 1 and e 2 are either both negative or both positive and P c is positive on (e 1 , e 2 ).Let ω c > 0 be the complete hyperelliptic integral of the first kind ω c = sign(e 1 )

Figure 4 .
Figure 4. On the left: the graph of the function h c , c = (1/2, −4.8) ∈ A. The red line is the horizontal asymptote y = e 1 .On the right: the graph of the twist.The red lines are the vertical asymptotes x = ±ω c .

Figure 5 .
Figure 5. On the left: the graph of the function h c 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 2ω c .The lowest roots e 1 and e 2 are negative.

Figure 7 .
Figure 7. On the left: the connected component B − 1 (dark brown) of B 1 .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 B 1 .

Figure 9 .
Figure 9. On the left: the graph of the function f 11 .On the right: the graph of the function f 12 .

Figure 10 .
Figure 10.On the left: the graph of the function f 31 .On the right: the graph of the function f 32 .

Figure 11 .
Figure 11.The plot of the map P.

Figure 12 .
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 δ q .

Figure 13 .
Figure 13.The Heisenberg projection of a standard configuration of a critical curve with quantum numbers q 1 = −2/15 and q 2 = −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 15 .
Figure 15.On the left: the Heisenberg projection of a standard configuration of a critical curve of type B ′ −1 , with quantum numbers q 1 = 5/49, q 2 = −4/7.On the right: the Heisenberg projection of a standard configuration of a critical curve of type B ′ − 1 , with quantum numbers q 1 = −7/36, q 2 = −23/54.
This proves the claim.It also shows that P c 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, P c 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

6
Experimental evidence of the existence of infinite countably many closed critical curves of type B ′ 1 and examplesThis 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 B 1 (cf.Remark 4.14).The same reasoning applies, as well, if the modulus belongs to the other connected component B + 1 of B 1 .We parametrize B − 1 by the map ψ − : K − → B − 1 , defined in (4.6), where K − is the rectangle Ĵ−