Colored HOMFLY polynomials of knots presented as double fat diagrams

TL;DR

This paper introduces a new method for computing colored HOMFLY polynomials of knots using double fat diagrams, leveraging fusion and braiding matrices, and verifies the approach with various knot examples.

Contribution

It proposes a conjecture for colored HOMFLY polynomials of knots represented as double fat graphs, incorporating non-rectangular representations and fusion matrices, verified through extensive comparisons.

Findings

Conjectured form effectively computes [21]-colored HOMFLY polynomials.

Distinguishes certain pretzel mutants, fails for others.

Difference between mutant polynomials follows a general A-dependent pattern.

Abstract

Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.