Unknotting via null-homologous twists and multitwists

Samantha Allen, Kenan İnce, Seungwon Kim, Benjamin Matthias Ruppik and Hannah Turner

Vol. 330 (2024), No. 1, 25–41

DOI: 10.2140/pjm.2024.330.25

Abstract

The untwisting number of a knot $K$ is the minimum number of null-homologous twists required to convert $K$ to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus $1$.

This collaborative work was based on the proposed problem and prior work of our senior member, Kenan İnce. After several years of work and the writing and submission of this paper, Kenan passed away unexpectedly. They were an exceptional human, mathematician, and advocate. We dedicate this paper to them.

Keywords

4-manifolds, surgery diagram, unknotting operation, untwisting number

Mathematical Subject Classification

Primary: 57K10, 57K40

Milestones

Received: 21 July 2023

Revised: 16 April 2024

Accepted: 27 April 2024

Published: 22 July 2024