Gordian adjacency for torus knots

Abstract

A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using Levine-Tristram signatures as obstructions to Gordian adjacency. Finally, Gordian adjacency for torus knots is compared to the notion of adjacency for plane curve singularities.

21 pages, 8 figures. Minor corrections and implementation of referee's recommendations

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• On the values taken by slice torus invariants
• The four-genus of connected sums of torus knots
• Signature invariants related to the unknotting number