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

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Regular Article - Theoretical Physics


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 Rm. 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.


Chern-Simons Theories Integrable Field Theories Wilson ’t Hooft and Polyakov loops 


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© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia

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