Winding number $m$ and $-m$ patterns acting on concordance

TL;DR

This paper proves that for any positive winding number pattern and its negative counterpart, there exist knots with arbitrarily large minimal genus cobordisms between their images, answering a question in knot concordance theory.

Contribution

It establishes the existence of knots with arbitrarily large cobordism genus between patterns of opposite winding numbers, generalizing previous results.

Findings

Existence of knots with arbitrarily large cobordism genus between $P(K)$ and $Q(K)$

Answers a question posed by Cochran-Harvey

Generalizes a result of Kim-Livingston

Abstract

We prove that for any winding number $m>0$ pattern $P$ and winding number $-m$ pattern $Q$, there exist knots $K$ such that the minimal genus of a cobordism between $P(K)$ and $Q(K)$ is arbitrarily large. This answers a question posed by Cochran-Harvey [CH17] and generalizes a result of Kim-Livingston [KL05].