Null-homologous unknottings

TL;DR

This paper proves that any genus g knot can be unknotted with 2g null-homologous twists, establishing a precise relationship between knot genus and the minimal number of such twists needed for unknottings.

Contribution

It introduces a new bound on the number of null-homologous twists required to unknot a genus g knot, and shows that this bound is sharp for some knots.

Findings

Any genus g knot can be unknotted with 2g null-homologous twists.

There exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.

Abstract

Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.