Amphichiral knots with large 4-genus

TL;DR

This paper constructs infinitely many strongly negative amphichiral knots with large 4-genus, showing they cannot bound low-genus surfaces in the 4-ball, and addresses a question about the 4-dimensional clasp number.

Contribution

It provides explicit examples of amphichiral knots with arbitrarily large 4-genus, expanding understanding of knot concordance and 4-dimensional topology.

Findings

Existence of infinitely many amphichiral knots with large 4-genus

These knots are rationally slice and 2-torsion in the concordance group

Answer to the question on the 4-dimensional clasp number

Abstract

For each $g>0$ we give infinitely many knots that are strongly negative amphichiral, hence rationally slice and representing 2-torsion in the smooth concordance group, yet which do not bound any locally flatly embedded surface in the 4-ball with genus less than or equal to $g$. Our examples also allow us to answer a question about the 4-dimensional clasp number of knots.