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

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. }