A family of concordance homomorphisms from Khovanov homology

TL;DR

This paper introduces a new family of concordance homomorphisms derived from a modified Khovanov homology, providing tools to estimate slice genus and distinguish knots in the smooth concordance group.

Contribution

It constructs a 1-parameter family of concordance homomorphisms from a modified Khovanov homology, analogous to the Upsilon invariant, expanding the toolkit for knot concordance analysis.

Findings

Provides lower bounds on the slice genus of knots.

Proves certain pretzel knots are linearly independent in the concordance group.

Introduces a new invariant similar to the Upsilon from knot Floer homology.

Abstract

By considering a version of Khovanov homology incorporating both the Lee and $E(-1)$ differentials, we construct a $1$-parameter family of concordance homomorphisms similar to the Upsilon invariant from knot Floer homology. This invariant gives lower bounds on the slice genus and can be used to prove that certain infinite families of pretzel knots are linearly independent in the smooth concordance group.