On transverse invariants from Khovanov homology

TL;DR

This paper introduces two refined invariants of transverse knots derived from Khovanov homology, enhancing the understanding of their invariance properties and relation to classical invariants.

Contribution

It presents two new refinements of Plamenevskaya's invariant, one in a deformed Khovanov complex and another as a cohomotopy element, with invariance under specific moves.

Findings

First refinement is invariant under negative flypes and SZ moves.

Both refinements are determined by classical invariants for small-crossing knots.

Plamenevskaya's original class is also shown to be invariant under these moves.

Abstract

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskaya's class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both refinements are determined by the classical invariants of K.