The Reidemeister graph is a complete knot invariant

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.