TL;DR
This paper investigates Gordian adjacency among torus knots, providing conditions, complete classifications for certain indices, and comparing it to plane curve singularity adjacency, advancing understanding of knot transformations.
Contribution
It offers a new sufficient condition for Gordian adjacency of torus knots and fully characterizes adjacency for index 2 and 3 using Levine-Tristram signatures.
Findings
Complete description of Gordian adjacency for index 2 and 3 torus knots.
A new sufficient condition for Gordian adjacency via knots in the thickened torus.
Comparison between Gordian adjacency and adjacency for plane curve singularities.
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.
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