Patterns in odd Khovanov homology

TL;DR

This paper explores the properties of odd Khovanov homology, comparing it with the even version, and demonstrates its applications in bounding Legendrian link invariants and detecting specific knot types.

Contribution

It introduces new applications of odd Khovanov homology in bounding Thurston-Bennequin numbers and identifying quasi-alternating knots, expanding its utility in knot theory.

Findings

Provides an effective upper bound on Thurston-Bennequin numbers

Detects quasi-alternating knots using odd Khovanov homology

Discusses potential in identifying transversely non-simple knots

Abstract

We investigate properties of the odd Khovanov homology, compare and contrast them with those of the original (even) Khovanov homology, and discuss applications of the odd Khovanov homology to other areas of knot theory and low-dimensional topology. We show that it provides an effective upper bound on the Thurston-Bennequin number of Legendrian links and can be used to detect quasi-alternating knots. A potential application to detecting transversely non-simple knots is also mentioned.