Towards tangle calculus for Khovanov polynomials

TL;DR

This paper explores the application of tangle calculus and evolution to Khovanov polynomials, revealing that jumps are less frequent than expected and enabling the definition of a jump-free component of colored Khovanov polynomials.

Contribution

It introduces a new approach to analyze Khovanov polynomials using tangle calculus, identifying a separation between jumping and smooth parts via a new invariant called "Thickness."

Findings

Jumps in evolution are less common than previously thought.

Main contributions in torus and twist satellites do not jump.

A new invariant "Thickness" helps distinguish jumping and smooth parts.

Abstract

We provide new evidence that the tangle calculus and "evolution" are applicable to the Khovanov polynomials for families of long braids inside the knot diagram. We show that jumps in evolution, peculiar for superpolynomials, are much less abundant than it was originally expected. Namely, for torus and twist satellites of a fixed companion knot, the main (most complicated) contribution does not jump, all jumps are concentrated in the torus and twist part correspondingly, where these jumps are necessary to make the Khovanov polynomial positive. Among other things, this opens a way to define a jump-free part of the colored Khovanov polynomials, which differs from the naive colored polynomial just "infinitesimally". The separation between jumping and smooth parts involves a combination of Rasmussen index and a new knot invariant, which we call "Thickness".