The genus filtration in the smooth concordance group

TL;DR

This paper introduces a genus-based filtration of the smooth concordance group and uses advanced invariants to show the resulting quotients are infinitely generated, with applications to specific knot families.

Contribution

It defines a new filtration of the smooth concordance group and demonstrates its properties using Heegaard Floer invariants, revealing infinite generation of quotient groups.

Findings

Quotient groups are infinitely generated.

Heegaard Floer invariants effectively distinguish filtration levels.

Applications to topologically slice knots demonstrate the filtration's utility.

Abstract

We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.