Concordance homomorphisms from knot Floer homology

TL;DR

This paper introduces a new family of knot invariants derived from knot Floer homology that serve as homomorphisms from the smooth concordance group to integers, providing bounds on knot genera.

Contribution

It develops a modified construction of knot Floer homology to produce a one-parameter family of concordance homomorphisms with applications to knot genus bounds.

Findings

Provides bounds on the four-ball genus of knots.

Establishes homomorphisms from the concordance group to integers.

Demonstrates applications to knot theory problems.

Abstract

We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.