Colored knot polynomials for Pretzel knots and links of arbitrary genus

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformation of toric conformal block.

It depends on g + 1 integers n 0 , . . . , n g , the algebraic lengths of the constituent 2-strand braids. Orientation of lines does not matter, when one considers the Jones polynomials (not HOMFLY!). Also, these polynomials are defined only for the symmetric representations [r] and, hence, do not change under arbitrary permutations of parameters n i (though the knot/link itself has at best the cyclic symmetry n i −→ n i+1 , and even this is true only for particular orientations). A particular manifestation of this enhanced symmetry has been recently noted in [8].
The answer for the colored Jones polynomials for this entire family can be written in full generality, and is wonderfully simple and is made out of the Racah matrix [9] of SU q (2) in representation [r] (spin r/2). Orthogonality of S implies that Quantum numbers in these formulas are defined as [n] = q n −q −n q−q −1 .
The first of the r + 1 polynomials is just the Jones polynomial for the 2-strand torus knot/link T [2, n]: The origin of its orthonormal "satellites" P (n) k (q|r) with k = 1, . . . , r and of entire rotation from the monomial basis {λ n m } is yet unknown.
Naturally, a straightforward generalization exists to the HOMFLY polynomials: which looks like a modular transformation of toric conformal block, summed over intermediate states X in the loop: ?
Here dim X and A are the universal A-dependent dimension of representation X (which is equal at A = q N to the quantum dimension of representation X of SU q (N )) and rescaled Racah matrix respectively, interpolating between those for SU q (N ) at A = q N . This formula is indeed true [10], at least when all the representations are symmetric or their conjugate: R i = [r], [r]. The answer for antisymmetric representations then follows from the general transposition rule [11,12] . It should possess further continuation a la [13] to superpolynomials, thus providing a β-deformation [14] of the universal Racah matrix.
Eqs. (1) and (7) result from a tedious calculation in [10,15] with the help of evolution [11,16], modernized Reshetikhin-Turaev [3,5] and modular matrix [2,4,6] methods. Of course, a simple conceptual derivation should exist for such a simple and general formula, but it still remains to be found. The value of (1) is independent of the derivation details, and it is high, because this only formula (together with its lifting to the HOMFLY polynomials in [10]) contains almost all what is currently known about explicit colored knot polynomials beyond torus links: in particular, all the twist and 2-bridge knots are small subsets in the Pretzel family (however, among torus knots with more than two strands, only [3,4] and [3,5] belong to it).
Examples: We give them here mostly for the Jones case, for an exhaustive description of the symmetric HOMFLY polynomials for all Pretzel links see [10]. First of all, we list the first few S and A matrices for the lowest representations of SU q (2): In general, from (3) and [9] one has: Given these matrices, eq.(1) provides absolutely explicit expressions for all genus-g knots/links in the corresponding representations [15]: J (n0,...,ng) 2 . . .
Similarly, the fundamental HOMFLY from [10] is where χ m and ∆ k are quantum dimensions of the representations appearing in the products [r]⊗[r] and [r]⊗[r] respectively (i.e. restrictions of the Schur functions to the topological locus, described by the hook formulas). One more new parameter is the number 2g || of parallel braids (g || = g ↑↑ = g ↓↓ ), the remaining g ↑↓ = g + 1 − 2g || are antiparallel.
As to practical applications, formula (10)  . . , n) [2] = 1 [3] (upper/lower signs are for odd/even n) can look a little too long as compared to the usual for the two-strand knots/links [2, n + 1] = (n + 1) 1 (when a = 1, b = 1) or for the twist knots (n + 2) 2 or (|n| + 1) 1 (when a = 2, b = 1, n odd): a lot of cancelations happen in these special cases. However, in generic situation such cancelations do not take place, and just the same expression describes the colored Jones polynomials for, say, the Pretzel (1,1,1,1,1,1,1,1,19,19,19,19,19,19,19) with a = 8, b = 7, n = 19, which is about one page long. To avoid possible confusion, we note that the Jones polynomials are unreduced and appear in this formalism in the vertical framing, the one consistent with the orientation independence. Conversion to the topological framing inserts the factor q −8 for each vertex in the parallel braids, this means a total q −8(n+1) for (n + 1) 1 , while for the twist knots the total factor is either q 16 or just 1, if tw k is represented as (−1, −1, 2k) or as (1, 1, 2k − 1).