Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?

TL;DR

This paper investigates the properties of Khovanov-Rozansky polynomials for torus knots, demonstrating that they exhibit evolution and recursion similar to classical knot polynomials, but with notable differences such as incompatibility with mirror symmetry.

Contribution

It extends the evolution and recursion properties known for classical knot polynomials to Khovanov-Rozansky polynomials, highlighting their persistence and differences.

Findings

KR polynomials satisfy evolution/recursion properties similar to classical polynomials

Some form of differential expansion is preserved in KR polynomials

Evolution in KR polynomials is incompatible with mirror symmetry

Abstract

$R$-coloured knot polynomials for $m$-strand torus knots $Torus_{[m,n]}$ are described by the Rosso-Jones formula, which is an example of evolution in $n$ with Lyapunov exponents, labelled by Young diagrams from $R^{\otimes m}$. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group $SL(N)$ only diagrams with no more than $N$ lines can contribute and the recursion degree is reduced. We claim that these properties (evolution/recursion and reduction) persist for Khovanov-Rozansky (KR) polynomials, obtained by additional factorization modulo $1+{\bf t}$, which is not yet adequately described in quantum field theory. Also preserved is some weakened version of differential expansion, which is responsible at least for a simple relation between {\it reduced} and {\it unreduced} Khovanov polynomials. However, in the KR case evolution is incompatible with the mirror symmetry under the change $n\longrightarrow -n$, what can signal about an ambiguity in the KR factorization even for torus knots. }