Factorization of colored knot polynomials at roots of unity

TL;DR

This paper uncovers new recursive and duality properties of colored knot polynomials at roots of unity, revealing universal relations that could advance the understanding of knot invariants.

Contribution

It introduces a recursion identity for HOMFLY polynomials at roots of unity and proposes a conjecture extending this to arbitrary representations, supported by checks on torus knots.

Findings

HOMFLY polynomials satisfy a recursion at roots of unity

Kashaev polynomial equals a special polynomial with A replaced by A^|R| for single-hook R

Universal relations constrain the coefficients of differential expansions

Abstract

From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a generalization of the property H_r = (H_1)^r for special polynomials at q=1. We conjecture a natural generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from H_R at q=exp(i\pi/|R|), turns equal to the special polynomial with A substituted by A^|R|, provided R is a single-hook representations (e.g. arbitrary symmetric) -- what provides a q-A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots -- existence of such universal relations means that these variables are still not unconstrained.