Abstract
We remind the method to calculate colored Jones polynomials for the plat representations of knot diagrams from the knowledge of modular transformation (monodromies) of Virasoro conformal blocks with insertions of degenerate fields. As an illustration we use a rich family of pretzel knots, lying on a surface of arbitrary genus g, which was recently analyzed by the evolution method. Further generalizations can be to generic Virasoro modular transformations, provided by integral kernels, which can lead to the Hikami invariants.
Highlights
Knot polynomials are Wilson loop averages in 3d Chern-Simons theory [1, 2], HRL(q, G) = Tr RP exp A L exp 2πi Tr Adj M AdA + 2 A3 3 (1)where A is the G-connection on a 3d manifold M, to which a line L belongs
In the present paper we describe the results of these calculations and explain the steps that lead to the proposed formula in the case of Jones polynomials
The pretzel knot on genus g surface is labeled by g + 1 integers n1, . . . , ng+1
Summary
Today there are two effective ways to calculate “ordinary” knot polynomials: by the Reshetikhin-Turaev (RT) group theory method [4, 5, 6, 7], using explicit quantum R-matrices in various representations of various groups (well-known skein relations are a simple particular case of this), and by Khovanov’s hypercube method [8] in the modified version of [9], applicable to algebra G = SU (N ) with arbitrary N Both approaches are partly related, first, via peculiar Kauffman’s R-matrix for N = 2 [10] and, more generally, in [11]. This is a interesting question because the pretzel knot family contains a vast set of mutant knots, which are undistinguishable at the level of (anti)symmetric representations, and, in particular, by any colored Jones polynomials
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