Evolution for Khovanov polynomials for figure-eight-like family of knots

Authors: Petr Dunin-Barkowski, Aleksandr Popolitov, Svetlana Popolitova

Preprint number: UUITP-60/18

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots – a two-parametric family of knots which "grows" from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones (t → -1) limit.