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⊗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 + 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 reduced and unreduced Khovanov polynomials. However, in the KR case evolution is incompatible with the mirror symmetry under the change n −→ −n, what can signal about an ambiguity in the KR factorization even for torus knots.
Highlights
The first task in the theory of knot polynomials is to describe their dependence on any of the integer-valued parameters n
R-coloured knot polynomials for m-strand torus knots T orus[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⊗m
In the KR case evolution is incompatible with the mirror symmetry under the change n −→ −n, what can signal about an ambiguity in the KR factorization even for torus knots
Summary
For m-strand torus knot HOMFLY polynomial is given by the Rosso-Jones evolution formula [56, 62,63,64,65,66]. Where and the degree of the difference equation is naively the number of Young diagrams Q ∈ R2m This fact is independent on the actual value of the coefficients CQ, i.e. from the point of view of the equation, they are the free parameters, parametrising its solution. In this sense the equation does not provide too much information about the HOMFLY polynomials, still it reflects an important property — evolution rule — of the torus family with particular number m of strands. The transformation n ↔ −n acts badly on the evolution formulas for KR polynomials, and this way to extend the set of available initial conditions does not seem to work
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