Genus non-increasing totally positive unknotting number

TL;DR

This paper introduces a new knot invariant called the genus non-increasing totally positive unknotting number, which measures the minimal positive-to-negative crossing changes needed to unknot a knot without increasing genus, and shows it can be arbitrarily large for genus one knots.

Contribution

It defines a novel unknotting number constrained by genus and positivity, and demonstrates its unboundedness for genus one knots.

Findings

The invariant can be arbitrarily large for genus one knots.

Crossing changes are restricted to positive-to-negative without increasing genus.

The invariant provides new insights into knot unknotting complexity.

Abstract

The genus non-increasing totally positive unknotting number is the minimum number of crossing changes that transform a knot into the unknot, such that all the crossing changes are positive-to-negative crossing changes that do not increase the genus. We show that the genus non-increasing totally positive unknotting number can be arbitrary large for genus one knots.