Khovanov homology and strong inversions

TL;DR

This paper explores the relationship between strong inversions on knots and four-ended tangles, computing their Khovanov homology for knots up to 9 crossings, and provides counterexamples and refinements to existing conjectures.

Contribution

It computes Khovanov homology for all strong inversions on knots up to 9 crossings and offers counterexamples and refinements to previous conjectures.

Findings

Counterexample to Conjecture 29 from arXiv:1311.1085

Refinement and additional evidence for Conjecture 28 from arXiv:1311.1085

Computed reduced Khovanov homology for all relevant tangles up to 9 crossings

Abstract

There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author. In particular, we provide a counterexample to [Conjecture 29, arXiv:1311.1085] as well as a refinement of and additional evidence for [Conjecture 28, arXiv:1311.1085].