Small knot complements, exceptional surgeries and hidden symmetries

Hoffman, Neil R

doi:10.2140/agt.2014.14.3227

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Small knot complements, exceptional surgeries and hidden symmetries

Neil R Hoffman

Algebraic & Geometric Topology 14 (2014) 3227–3258

DOI: 10.2140/agt.2014.14.3227

Abstract

This paper provides two obstructions to small knot complements in $S^{3}$ admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture of Boileau, Boyer, Cebanu and Walsh. We also provide a second obstruction to admitting hidden symmetries in the case where a small knot complement covers a manifold admitting some symmetry and at least two exceptional surgeries.

Keywords

knot complements, commensurability, hidden symmetries, exceptional surgeries, trace field

Mathematical Subject Classification 2010

Primary: 57M12, 57M25

Secondary: 57M10

References

Publication

Received: 15 December 2011

Revised: 9 December 2013

Accepted: 29 January 2014

Published: 15 January 2015

Authors

Neil R Hoffman
Department of Mathematics and Statistics University of Melbourne Parkville, Victoria 3010 Australia

Volume 14, issue 6 (2014)