A note on Gornik's perturbation of Khovanov-Rozansky homology

TL;DR

This paper demonstrates that Gornik's perturbation of Khovanov-Rozansky homology encodes a single integer invariant s_n(K), which acts as a homomorphism from the smooth knot concordance group to the integers, similar to Rasmussen's invariant.

Contribution

It establishes the equivalence between Gornik's homology and a single integer invariant s_n(K), and proves that s_n is a homomorphism from the smooth knot concordance group.

Findings

s_n(K) is an invariant derived from Gornik's homology.

s_n(K) is a homomorphism from the smooth knot concordance group.

The associated graded vector space contains the same information as s_n(K).

Abstract

We show that the information contained in the associated graded vector space to Gornik's version of Khovanov-Rozansky knot homology is equivalent to a single even integer s_n(K). Furthermore we show that s_n is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen's invariant coming from a perturbation of Khovanov homology.