A move on diagrams that generates S-equivalence of knots

TL;DR

This paper demonstrates that S-equivalence of knots, which relates to their Seifert matrices, can be generated by a specific move called the doubled-delta move, linking knot invariants to diagrammatic transformations.

Contribution

The paper establishes that S-equivalence is generated by doubled-delta moves, providing a new diagrammatic perspective on knot invariants and their transformations.

Findings

S-equivalence is generated by doubled-delta moves

Knots with trivial Alexander polynomial can be undone by doubled-delta moves

Provides a diagrammatic characterization of S-equivalence

Abstract

Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander polynomial if and only if it can be undone by doubled-delta moves.