Torsion in Khovanov homology of homologically thin knots

TL;DR

This paper proves that homologically thin links have no higher 2-power torsion in their Khovanov homology, leading to a complete determination of integer Khovanov homology for certain links by classical invariants.

Contribution

It establishes a new algebraic relation between Bockstein and Turner differentials, ruling out certain torsion in Khovanov homology of thin knots.

Findings

Homologically thin links lack 2^k-torsion for k>1 in Khovanov homology.

Integer Khovanov homology of non-split alternating links is determined by Jones polynomial and signature.

Proposes a conjecture on spectral sequence relations.

Abstract

We prove that every $\mathbb{Z}_2$H-thin link has no $2^k$-torsion for $k>1$ in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is based on establishing an algebraic relation between Bockstein and Turner differentials on Khovanov homology over $\mathbb{Z}_2$. We conjecture that a similar relation exists between the corresponding spectral sequences.