An annular refinement of the transverse element in Khovanov homology

TL;DR

This paper introduces a refined invariant in Khovanov homology using annular grading, which helps distinguish certain braids, obstruct negative destabilization, and analyze spectral sequences, revealing new structural insights.

Contribution

It constructs a new braid conjugacy class invariant $ppa$ via annular grading, providing tools to distinguish braids and analyze spectral sequences in Khovanov homology.

Findings

ppa distinguishes some braids with identical classical invariants.

ppa provides an obstruction to negative destabilization.

The spectral sequence from annular to ordinary Khovanov homology may not collapse immediately.

Abstract

We construct a braid conjugacy class invariant $\kappa$ by refining Plamenevskaya's transverse element $\psi$ in Khovanov homology via the annular grading. While $\kappa$ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using $\kappa$ we construct an obstruction to negative destabilization (stronger than $\psi$) and a solution to the word problem in braid groups. Also, $\kappa$ is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete examples in which this spectral sequence does not collapse immediately. In addition, we study these constructions in reduced Khovanov homology and illustrate that the two reduced versions are fundamentally different with respect to the annular filtration.