English

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

High Energy Physics - Theory 2018-05-29 v1 Mathematical Physics Geometric Topology math.MP

Abstract

RR-coloured knot polynomials for mm-strand torus knots Torus[m,n]Torus_{[m,n]} are described by the Rosso-Jones formula, which is an example of evolution in nn with Lyapunov exponents, labelled by Young diagrams from RmR^{\otimes m}. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group SL(N)SL(N) only diagrams with no more than NN 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+t1+{\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 nnn\longrightarrow -n, what can signal about an ambiguity in the KR factorization even for torus knots. }

Keywords

Cite

@article{arxiv.1802.09383,
  title  = {Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?},
  author = {A. Anokhina and A. Morozov},
  journal= {arXiv preprint arXiv:1802.09383},
  year   = {2018}
}

Comments

23 pp