The Reidemeister graph is a complete knot invariant

TL;DR

This paper introduces Reidemeister graphs associated with knots, proves their effectiveness as complete invariants up to mirroring, and explores their properties and relations to other knot graphs.

Contribution

It defines two Reidemeister graphs, analyzes their properties, and proves one as a complete knot invariant up to mirroring, advancing knot classification methods.

Findings

Reidemeister graphs are locally finite and have specific global properties.

One Reidemeister graph type is a complete knot invariant up to mirroring.

A new object relating Reidemeister and Gordian graphs is introduced and analyzed.

Abstract

We describe two locally finite graphs naturally associated to each knot type K, called Reidemeister graphs. We determine several local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly, we introduce another object, relating the Reidemeister and Gordian graphs, and determine some of its properties.