Recurrence Relations and Asymptotics of Colored Jones Polynomials

We consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-difference equations for colored Jones polynomials. These sequences of polynomials are invariants for the knots and their asymptotics plays an important role in the famous volume conjecture for the complement of the knot to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document}d sphere. We give an introduction to the theory of hyperbolic volume of the knots complements and study the asymptotics of the solutions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-recurrence relations of high order.


INTRODUCTION
At first we give some definitions concerning knots from the paper [52] and the books [1,9,49]. A knot in 3-sphere S 3 is a continuous embedding of a topological circle in 3-dimensional space considered up to continuous deformation (isotopy). The simplest knot of all is unknotted circle, which we call the unknot or the trivial knot. The next simplest non-trivial knot is called a trefoil knot. A link is a collection of knots which do not intersect, but which may be linked together. An oriented link L is a compact 1-dimensional oriented smooth submanifold of R 3 ⊂ S 3 . A framed oriented link L is a link equipped with a smooth normal vector field V , which is a function V : L → R 3 , such that V (x) is not in the tangent space T x L for every x ∈ L.
A hyperbolic knot K in S 3 is defined to be a knot such that the complement S 3 \ K is hyperbolic 3-manifold. Note that S 3 \ K is a finite volume but noncompact hyperbolic 3-manifold (see [1]). A torus knot is a knot which can be placed on an ordinary torus in S 3 . With any nontrivial knot K there is associated a whole collection of other knots, known as satellites of K; these are knots which are obtained by a nontrivial embedding of a circle in a small solid torus neighborhood of K. Here, "nontrivial" means that the embedding is not isotopic to K itself and is not contained within a ball inside the solid torus. A knot is a satellite knot if it is a satellite of a nontrivial knot, see also [40,Definition 1.2].
The famous Thurston's theorem [52,Section 2.5] states that every knot in S 3 is either a torus knot or a satellite knot or a hyperbolic knot, see also [51].
In algebraic topology, a way of studying knots is to associate some invariants, which are useful to separate knots. In the given paper, we consider one of such invariants. In 1985, Jones introduced his celebrated polynomial invariant, the Jones polynomial J K (q) associated to a knot K [26,27]. At first, this polynomial was defined by using the operator algebra, namely, the Yang-Baxter operator and the R-matrix. Shortly after, another version of this invariant was given by using the Kauffman bracket [31]. Colored versions of these invariants were defined, via quantum groups [47]. For details on the definitions of the colored Jones polynomial, see, e.g., [40,Chapter 2]. Definition 1 [13]. (i) The colored Jones function J K : N → Z[q ±1 ] of a knot K in 3-space is a sequence of Laurent polynomials whose nth term J K (n) is the Jones polynomial of a knot K with the n dimensional irreducible representation sl 2 . We normalize the Jones function so that the value for the trivial knot (unknot) is one, J unknot (n) = 1 for all n.
(ii) We denote by J K : N → Z[q ±1/2 ] the colored Jones function with normalization by J unknot (n) = [n] := q n/2 − q −n/2 Hence, J K (n) = [n]J K (n). Below we use also notation: J K ≡ J K (n) ≡ J K (n, q). For any knot, J K (1) = 1 and J K (2) coincides with the Jones polynomial of a knot K.
For a detailed discussion on the polynomial invariants of knots that come from quantum groups, see, e.g., [58,59].
A new approach to the study of the colored Jones function was proposed by Garoufalidis and Lˆe [12] by using a q-difference equation.
where b j (u, v) are smooth functions and f (n, q) = f (n)(q). The characteristic polynomial (see [14, Theorem 1 [12,Theorem 1]. The colored Jones function of any knot satisfies a q-difference equation. Recall that a sequence f : N → Q[q] is q-holonomic if it satisfies a q-linear difference equation, i.e., there exist a number d ∈ N and polynomials b j (u, v) ∈ Q[u, v] for j = 0, 1, . . . , d with b d = 0 such that for all n ∈ N the sequence f satisfies (2). Thus, Theorem 1 states that the colored Jones function is q-holonomic.
Holonomic functions and systems were first introduced and studied by Bernstein [7,8]. The algorithmic significance of q-holonomic sequence was first recognized by Zeilberger [62]. To prove Theorem 1, Garoufalidis and Lˆe showed (see [12,Proposition 3.7]) that the colored Jones function can be written as a multisum of a proper q-hypergeometric function F (n, k 1 , . . . , k r ) (see [12,Definition 2.3] or [61, p. 589]). In (4), for a fixed positive n, only finitely many terms are nonzero. In turn, Zeilberger [62] showed how to check to holonomicity and proved that a class of proper-hypergeometric functions is holonomic. Moreover, finite (multi-dimensional) sums of proper q-hypergeometric terms are q-holonomic in the remaining free variables. This is the Fundamental Theorem of Wilf and Zeilberger [61]. Hence, the terms F in (4) are q-holonomic in all r + 1 variables [12]. Recall that a discrete function f ∈ Z r → Q(q) is q- Here the operators E i , i = 1, . . . , r, act on the discrete functions by E i f (n 1 , . . . , n r ) = f (n 1 , . . . , n i−1 , n i + 1, n i+1 , . . . , n r ).
The key role in the proof in [14] plays the translation of asymptotics of solutions of q-difference equations in the terms of asymptotics of solutions of ε-difference equations [14,Section 2]. In turn, the ε-difference equations of the form l j=0 a j (kε, ε)y k+j = 0, k ∈ Z (l fixed), were considered by O. Costin and R. Costin [10]. Using the WKB method, they studied the asymptotics of solutions as ε → 0.
2. VOLUME CONJECTURE In this section, we describe some conjectures concerning the colored Jones function.

Hyperbolic Volume Conjecture
Kashaev [28,29] defined a complex-valued knot invariant K n of a knot K for n = 2, 3, ... by using the quantum dilogarithm. He proposed a conjecture that his invariant K n would grow exponentially w.r.t. n and that its growth rate would give the volume of the complement of a hyperbolic knot: where vol(S 3 \ K) is the volume of a complete hyperbolic metric in the knot complement S 3 \ K. Later, H. Murakami and J. Murakami [39,Theorem 4.9] proved that Kashaev's invariant with parameter n coincides with n dimensional colored Jones polynomial evaluated at the n-th root of unity, The value K n is called the Kashaev invariant of a knot K at q = exp{2πi/n}. For example, see [29], unknot n = 1, where the q-factorial (q) m , m ∈ N, is defined by and q denotes the complex conjugation of q.
The Hyperbolic Volume Conjecture (HVC, in short) states as follows.
(HVC): Let K be a hyperbolic knot in S 3 . Then The (HVC) was generalized by H. Murakami and J. Murakami [39] to the any knot as follows. Volume Conjecture: Let K be any knot in S 3 . Then where v 3 is the volume of the ideal regular hyperbolic tetrahedron in 3-dimensional hyperbolic space H 3 and ||M || is the Gromov norm [20] (or simplicial volume). In particular if the knot K is hyperbolic, then the r.h.s. of (9) coincides with the volume of the knot complement vol(S 3 \ K).
Remarks. (i) The simplicial volume of K, ||S 3 \ K||, is equal to the sum of the hyperbolic volumes of the hyperbolic pieces of the torus decomposition of S 3 \ K divided by v 3 .
(iii) For the first time, the regular hyperbolic 3-simplex was considered by Gieseking in 1912 (see [19]). Consider a regular tetrahedron in Euclidean space, inscribed in the unit sphere, so that its vertices are on the sphere. Now interpret this tetrahedron to lie in the projective model for hyperbolic space, so that it determines an ideal hyperbolic simplex. The dihedral angles of the hyperbolic simplex are π/3. The volume of regular ideal 3-simplex is equal to v 3 = 3Λ(π/3) = 1.01494 . . . (see [38]), where Λ(z) is the Lobachevskii function, Note that the Lobachevskii function can be expressed by the imaginary part of the Euler dilogarithm Li 2 (z) [29]. For z ∈ C such that |z| ≤ 1, the Euler dilogarithm is defined by For z ∈ C such that | arg(z)| < π, the analytic continuation of the dilogarithm function is given by Then, for θ ∈ R, the Lobachevskii function is defined by the rule (iv) In 1975, R. Riley [48] showed that the figure-eight knot complement has a hyperbolic structure. Thus, the figure-eight knot (the 4 1 knot in Rolfsen' notation [49]) is a hyperbolic knot. In 1978, Thurston proved that the figure-eight complement can be triangulated by two copies of a maximal 3simplex. Hence, vol(S 3 \ 4 1 ) = 2v 3 = 6Λ(π/3) = 2.02988 . . . This formula was obtained by Milnor [38]. Similarly, the complement of the Whitehead link was constructed from a regular ideal octahedron which in turn, is formed by gluing two copies of the infinite cone on a regular planar quadrilateral. Thus, its volume equals 8Λ(π/4) = 3.66386 . . .. The complement of the Borromean rings has volume 16Λ(π/4) = 7.32772 . . ., since it is obtained by gluing two ideal octahedra together, see Example in [51,Sect. 7.2].
(v) There are examples of knots all with the same volume. For example, the 5 2 knot and a 12-crossing knot 12n_242 have the same volume, vol(S 3 \ 5 2 ) = vol(S 3 \ 12n_242) = 2, 82812 . . . (cf. [36]). There are other invariants which can be used to distinguish between manifolds of the same volume, for example, the maximal cusp volume [1,Sections 3,4].
To numerically confirm the volume conjecture, the authors use the explicit multisum formulas for the Kashaev invariant (cf. (7)). These formulas allow to reduce the finding the limit in the l.h.s. of (9) to the study of the asymptotics of special functions such that the Lobachevskii function Λ(z), the Euler dilogarithm Li 2 (z), with applying the saddle point method. Namely, the first step in this approach is to write the Kashaev invariant K N in the form where p is the number of summations appearing in K N . The function V K (z), z ∈ C p , is called a potential. To find this function, one uses the following asymptotic behavior The second step is to find a saddle point z 0 which is a solution to ∂ [24, p. 333]).

AJ Conjecture
The next AJ conjecture (made by Garoufalidis [13, p. 297]) relates the colored Jones polynomial and the A-polynomial. (AJ are the initials of the A-polynomial and the colored Jones polynomial J). The Apolynomial of a knot was introduced in [11]. It is a two-variable polynomial, usually written in the terms of the meridian and longitude variables M and L. The A-polynomial, A K (M, L), parameterizes the affine variety of SL(2, C) representations of the knot complement, viewed from the boundary torus [11]. The A-polynomial is defined up to multiplication by a rational function of M and a power of L [13].
Since the colored Jones polynomial is defined by a multidimensional sum of a proper q-hypergeometric term, numerous algorithms can produce a linear recurrence with polynomial coefficients; see, e.g., the book of Petkovˇsek, Wilf and Zeilberger [46]. Different algorithms in general produce different recurrences, which may not be of minimal order. The non-commutative A-polynomial of the twist knots K p was computed with a certificate by Garoufalidis and Sun [18] for p = −15; ...; 15. This polynomial was calculated for most knots up to 12 crossings, see [11,36]. To state the AJ conjecture we write q-difference equations in the operator form The operators E, Q, and q act on a discrete function f : Then, we put q = t 4 , (Lf )(n) := f (n + 1), (Mf )(n) := t 2n f (n). Then J K (n) satisfies the following equation The AJ conjecture (cf. [13]): up to a multiplication by a polynomial in v. For example, for K = 3 1 , [11] computed the A-polynomial of the 3 1 and 4 1 knots (see also [13, p. 300]) where the −3 1 knot is the mirror image of the 3 1 knot. The A-polynomial of the 5 2 knot is given by (cf. [36,11]) Correspondingly (we change L into L −1 and omit the factor L −3 ), The AJ conjecture was confirmed for the knots 3 1 , 4 1 , 7 4 (by Garoufalidis [13] and Koutschan-Garoufalidis [17]); for torus knots (by Hikami [25], Tran [54]), for some classes of two-bridge knots, including all twist knots, and pretzel knots (by Le [33], Le-Tran [34], Le-Zhang [35]).
Remark. There are new conjectures about the colored Jones polynomial, for details, see, e.g., [53].
3. THE COLORED JONES FUNCTION Habiro [23] obtained the following expansion of the colored Jones function, known as the cyclotomic expansion. He showed that for every zero-framed knot K, there exists a cyclotomic function C K : where C K (0) := 1, the cyclotomic kernel s(n, k) is a proper q-hypergeometric term given by (see e.g., [13, p. Here {n} := q n/2 − q −n/2 . Masbaum [37] obtained a formula for the colored Jones polynomial of twist knots generalizing formula of Habiro and Lˆe [32] for the 3 1 and 4 1 knots. Note that Garoufalidis and Lˆe [12] proved that the function C K of every knot K is q-holonomic. For the unknot, the function C unknot (k) = δ k0 , where δ k0 denotes the Kronecker delta. Below we give examples of the function C K and the colored Jones function J K for the 3 1 , 4 1 and 5 2 knots. The unknot and the first four knots are shown in Fig. 1. The pictures for the next knots see in [49].

q-DIFFERENCE EQUATIONS
The recurrence relations for the colored Jones function can be found from multisum-formulas for the colored Jones function of a form (16) by computer calculations. For details, see [13,Chapter 3.2]. There are various programs that can compute the recursion relations for multisums, see, e.g., [18,45,46] and the references therein. Below we give q-difference equations in the case of the 3 1 and 4 1 knots. For the 5 2 knot, the q-difference equation is given in Section 5.

q-Difference
which is converted into a third-order homogeneous recurrence relation of the form (after change n → n + 3) where the coefficients b j ≡ b j (q n , q), j = 0, 1, 2, 3, are and Remark. Note that the coefficients b j in (24) and (27) have the factor (q n+j − 1) for all j. This fact is valid for any knot and follows from (16)  The initial conditions for (25) are J 4 1 (0) = 0, J 4 1 (1) = 1, In this case, the characteristic polynomial is Hence, for the 4 1 knot, we have 3 eigenvalues, Thus, the regularity condition does not hold.
Proof. At first, we find the corresponding to (28) fourth-order homogeneous q-difference equation. Indeed, the solution of (28) satisfies the following equation where the coefficients b j := b j (q n , q) are Secondly, the system (31) can be rewritten in the operator form where the operators q, Q and E are defined in (12). Further, let us put (cf. (13)) Using (32), we obtain (see AJ conjecture in Section 2.2) (15). Using A j defined in (29) we writẽ A j (q n , q) := (q n+j − 1)A j (q n , q), j = 0, 1, 2, 3.
Finally, the initial conditions for (28) are (cf. (21)) f n | n=k = (q n − 1)J −5 2 (n, q)| n=k , k = 1, 2, 3. P Remark. We note that the regularity condition does not hold for the q-recurrences (31)- (32). Indeed, put q n = v and q = 1 in the coefficients b j (q n , q) and obtain The characteristic polynomial for (31) is where It can be easily checked (details see below) that there exist points v ∈ S 1 such that Dsc λ P (v, λ) = 0.
The q-recurrence relations (28)- (30) have been presented to us by Stavros Garoufalidis 1) . Being motivated by the Volume Conjecture to develop technique for asymptotic analysis of the solutions of q-difference equations, we have started in preprint [5] to investigate the asymptotics of the polynomialṡ J n (q) := f n (q) to problem (28)- (30) in a double scale as Here we report on some first steps in this direction.
5.1. Homogeneous Problem. New Variables To find the asymptotics of a sequence of polynomials defined by inhomogeneous recurrences, we first construct an asymptotic expansion basis of solutions to the homogeneous problem where the elements of the transition matrix are given in (29). Instead of the basic variables q and n, we will consider the following parameters. We will distinguish between large N and current n: Next, recall the previously introduced variable z := q n . Finally, define a parameter H such that As a result, instead of the variables n, q and q n , the basic variables will be (n, q, q n ) −→ (tN, H/N, z).
In view of (8) we have H = 2πi. The small parameter for the power expansions will be (1/N ).

Spectral
Curve of the Homogeneous Problem. Branch Points Substituting into the characteristic equation det(A n − Iλ) = 0 the limit values q n =: z, q =: 1, we obtain the equation for the limiting spectral curve, where a 1 and a 2 are defined in (36). The genus of the algebraic function λ(z) is 2. It has eight branch points, two of them lie on the unit circle centered at origin their numerical values are equal to z 1,2 = −0.7422176660 ± i0.6701588892. Note also that although the points z = ±1 are not branch points, however at these points all three branches of the function λ(z) = {λ j (z)} 2 j=0 coincide: λ 0 (±1) = λ 1 (±1) = λ 2 (±1).

Road Map for the Asymptotics
Recall that our goal is to find the asymptotics (8) for the q-holonomic system {J n (q)} defined by recurrence relations (28)- (29) and the initial conditions (30). That is, if we consider the polynomialJ n (q) =:Q n (q, q n ) as a function of two variables q = e 2πi N and z := q n , then we start with the functions {Q(q, q n )} 3 n=1 taken at the points q, q 2 , q 3 , and calculating the functions {Q(q, q n )} N n=4 by (28)- (29) at the consecutive roots of unity z := q n , moving with respect to z along the unit circle from z := q to z := q N = e 2πi , we come toQ(q, q N ). To determine lim N →∞Q n (q, q N ) we first find the asymptotics of the basis of solutions to the homogeneous problem (38) for z K for compact sets K covering {|z| = 1}.
Let us note a circumstance that complicates the task. The unit circle does not lie entirely within the zone where the eigenvalues are separated: our path along {e iφ } 2πi φ=0 starts and ends in the zone where the eigenvalues are close to each other (the neighborhood of the point 1), and also along the path such zones appear in the neighborhoods of the points z 1 , −1, z 2 , see (41) and (42). Therefore, after finding this basis in the zone of separated eigenvalues, in order to find a global (along the entire circle) asymptotic basis of solutions of the homogeneous problem, we have to find bases in four local zones close to eigenvalues (in the neighborhood of the points 1, z 1 , −1, z 2 ) and the transition matrix for the basis in the zone of separated eigenvalues at the entrance and exit from local zones, where eigenvalues are close.
Finally, having at our disposal an asymptotic expansion of the general homogeneous solution along the whole unit circle, by the method of variation of constants, we find the general solution of the inhomogeneous problem (28), (29), which we match with the initial data (30). The special solution obtained in this way for the inhomogeneous problem with the help of the asymptotic basis of the homogeneous problem, we can continue from the initial zone (the neighborhood of the point z = 1) along the whole circle to the finite zone which is the neighborhood of the point z = e 2πi .
In the present paper, we construct the spectral curve parametrization which is needed for realisation of this plan.

Spectral Curve Parametrization
Theorem 2. Let s ∈ C. Then for the points of the curve λ(z), see (40) where Moreover, for the branches of λ(z) we have Remark. If for some s =: s 0 by (43) we have the value of the branch λ 0 = L(z(s 0 ), s 0 ), then to find other branches λ 1,2 by (45) we have to solve the cubic equation R 3 (s 0 ) = R 3 (s) to find s = s 1 , s 2 . In practice, however, we reduce the problem to a quadratic equation as follows. In the equation in (40), we replace z with its parametrization by variable s (the first line in (43)). We divide the resulting expression (depending on λ and s) by (λ − L(z, s)), while in the expression L(z, s) (the second line in (43)) parametrization is also substituted. Symbolic calculations allow us to implement this division procedure. The result is a quadratic equation for another branches λ 1 and λ 2 .
The inverse Zhukovsky transform can be written as Then Note that the parametrization (43) proved in the theorem is more compact and convenient compared to (48) (in particular, to calculate λ we use the already calculated variable z).

OPEN ACCESS
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