Gordian adjacency for torus knots

摘要

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 × S~1 ×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.