Partially Ordering Unknotting Operations

TL;DR

This paper classifies extended $ST$-moves, a type of local move in knot theory, and explores their ability to unknot knots, establishing a framework for understanding their relative effectiveness.

Contribution

It introduces an equivalence classification of extended $ST$-moves and analyzes their role in unknotting, including the ability of certain moves to transform knots into trivial knots.

Findings

Extended $ST$-moves realize crossing change or $SH(2)$-move.

Most extended $ST$-moves can unknot knots with a single move.

A binary relation among moves is established and exemplified.

Abstract

In this paper, we introduce an equivalence relation on the set of local moves and classify local moves, called the extended $ST$-moves, up to the equivalence. Moreover, by inducing a binary relation on the set of equivalence classes of local moves, we show that an extended $ST$-move realizes the crossing change or the $SH(2)$-move. In addition, for any oriented knot and two extended $ST$-moves, we disscus the magnitude relation between the unknotting numbers of the knot via the moves, and show that there is an extended $ST$-move except $SH$-moves so that the knot can be transformed into the trivial knot by the single extended $ST$-move. Finally, we provide some examples of $ST$-moves with the binary relation.