A note on the concordance $\mathbb{Z}$-genus

TL;DR

This paper demonstrates that the gap between the topological 4-genus of a knot and a specific decomposed minimal genus surface can be arbitrarily large, extending previous work on topologically slice knots and smooth concordance.

Contribution

It introduces a new measure of genus difference for knots and shows it can be arbitrarily large, expanding understanding of knot concordance and genus relationships.

Findings

The genus difference can be arbitrarily large.

Extends previous results on topologically slice knots.

Shows limitations of smooth concordance to trivial Alexander polynomial.

Abstract

We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden-Livingston-Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.