Colored knot polynomials for Pretzel knots and links of arbitrary genus

TL;DR

This paper proposes a simple, explicit formula for colored Jones and HOMFLY polynomials of Pretzel knots and links, connecting them to Racah matrices and modular transformations in conformal field theory.

Contribution

It introduces a conjectured explicit expression for these polynomials for a broad family of Pretzel knots and links, linking knot invariants to quantum group Racah matrices.

Findings

Explicit formulas for colored Jones and HOMFLY polynomials for Pretzel knots.

Connection between knot polynomials and Racah matrices of U_q(SU_N).

Relation to modular transformations of toric conformal blocks.

Abstract

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.