An odd analog of Plamenevskaya's invariant of transverse knots

TL;DR

This paper introduces an analog of Plamenevskaya's invariant within odd Khovanov homology, establishing it as a transverse link invariant with comparable properties, and demonstrates its computation on various knots.

Contribution

It defines a new invariant in odd Khovanov homology that parallels Plamenevskaya's invariant and proves its invariance and computability for transverse links.

Findings

The invariant is preserved under transverse isotopy.

It can be identified with an invariant in reduced odd Khovanov homology.

Computations show the invariant distinguishes certain transverse knot pairs.

Abstract

Plamenevskaya defined an invariant of transverse links as a distinguished class in the even Khovanov homology of a link. We define an analog of Plamenevskaya's invariant in the odd Khovanov homology of Ozsv\'ath, Rasmussen, and Szab\'o. We show that the analog is also an invariant of transverse links and has similar properties to Plamenevskaya's invariant. We also show that the analog invariant can be identified with an equivalent invariant in the reduced odd Khovanov homology. We demonstrate computations of the invariant on various transverse knot pairs with the same topological knot type and self-linking number.