On a move reducing the genus of a knot diagram

TL;DR

This paper introduces a new operation on knot diagrams that preserves the knot type and does not increase genus, with conditions for decreasing genus, supported by relations between virtual and knotoid diagram genera, and is computationally implementable.

Contribution

It presents a novel genus-reducing operation for knot diagrams, linking virtual and knotoid diagram genera, with a simple Gauss code interpretation.

Findings

Operation does not increase genus or change knot type.

Condition identified for guaranteed genus reduction.

Operation is easily implementable via Gauss codes.

Abstract

For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof involves the study of a relation between the genus of a virtual knot diagram and the genus of a knotoid diagram, the former of which has been introduced by Stoimenow, Tchernov and Vdovina, and the latter by Turaev recently. Our operation has a simple interpretation in terms of Gauss codes and hence can easily be computer-implemented.