Khovanov homology: torsion and thickness

TL;DR

This paper advances understanding of Khovanov homology by addressing torsion conjectures, establishing size bounds for almost alternating links, and relating homologies of connected sums through exact sequences.

Contribution

It partially resolves Shumakovitch's torsion conjecture, provides size restrictions for almost alternating links, and links the homology of connected sums with individual link diagrams.

Findings

Partial proof of Shumakovitch's torsion conjecture for prime, non-split links

Size restrictions on Khovanov homology for almost alternating links

Existence of a long exact sequence connecting reduced and unreduced Khovanov homology

Abstract

We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homology of the diagram via a short exact sequence of homology which splits. Finally we show that our results can be adapted to reduced Khovanov homology and we show that there is a long exact sequence connecting reduced Khovanov homology with unreduced homology.