Knot invariants from Virasoro related representation and pretzel knots

TL;DR

This paper presents a method to compute colored Jones polynomials for pretzel knots using Virasoro conformal blocks and modular transformations, extending to potential Hikami invariants through integral kernels.

Contribution

It introduces a novel approach linking Virasoro conformal blocks with knot invariants, specifically for pretzel knots, and discusses possible generalizations to Hikami invariants.

Findings

Computed colored Jones polynomials for pretzel knots using Virasoro conformal blocks.

Demonstrated the method on a family of pretzel knots on surfaces of arbitrary genus.

Outlined potential extensions to Hikami invariants via integral kernels.

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.