Gordian adjacency for torus knots

Feller, Peter

doi:10.2140/agt.2014.14.769

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Gordian adjacency for torus knots

Peter Feller

Algebraic & Geometric Topology 14 (2014) 769–793

DOI: 10.2140/agt.2014.14.769

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.

Keywords

Gordian distance, unknotting number, torus knots, plane curve singularities, adjacency

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 14B07

References

Publication

Received: 21 March 2013

Revised: 21 August 2013

Accepted: 22 August 2013

Published: 30 January 2014

Authors

Peter Feller
Universität Bern Sidlerstrasse 5 CH-3012 Bern Switzerland

Volume 14, issue 2 (2014)