Gordian adjacency for torus knots

Abstract

A knot $K_1$ is called Gordian adjacent to a knot $K_2$ if there exists an unknotting sequence for $K_2$ containing $K_1$. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus $S^1\times S^1\times ℝ$. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using Levine–Tristram signatures as obstructions to Gordian adjacency. Our study of Gordian adjacency is motivated by the concept of adjacency for plane curve singularities. In the last section we compare these two notions of adjacency.