On the concordance genus of topologically slice knots

TL;DR

This paper establishes a new lower bound for the concordance genus of knots using knot Floer complexes and demonstrates the existence of topologically slice knots with minimal 4-ball genus but arbitrarily large concordance genus.

Contribution

It introduces a novel lower bound for the concordance genus based on knot Floer homology and constructs examples of topologically slice knots with specific genus properties.

Findings

Lower bound for concordance genus from knot Floer complex

Existence of topologically slice knots with 4-ball genus one and large concordance genus

Demonstrates the gap between 4-ball genus and concordance genus in certain knots

Abstract

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that there are topologically slice knots with 4-ball genus equal to one and arbitrarily large concordance genus.