Khovanov homology and rational unknotting

TL;DR

This paper introduces a new knot invariant derived from Khovanov homology, providing bounds for rational unknotting and demonstrating its effectiveness across various knots.

Contribution

It defines a novel invariant from universal Khovanov homology that bounds the rational unknotting number and constructs knots with arbitrary invariant values.

Findings

$ ext{lambda}$ is a lower bound for the rational unknotting number

Existence of knots with arbitrary $ ext{lambda}$ values

Computed Bar-Natan complexes for rational tangles

Abstract

Building on work by Alishahi-Dowlin, we extract a new knot invariant $\lambda \ge 0$ from universal Khovanov homology. While $\lambda$ is a lower bound for the unknotting number, in fact more is true: $\lambda$ is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that for all $n \ge 0$, there exists a knot K with $\lambda(K) = n$. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.