Cusp volumes of alternating knots

Lackenby, Marc; Purcell, Jessica

doi:10.2140/gt.2016.20.2053

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Cusp volumes of alternating knots

Marc Lackenby and Jessica S Purcell

Geometry & Topology 20 (2016) 2053–2078

DOI: 10.2140/gt.2016.20.2053

Abstract

We show that the cusp volume of a hyperbolic alternating knot can be bounded above and below in terms of the twist number of an alternating diagram of the knot. This leads to diagrammatic estimates on lengths of slopes, and has some applications to Dehn surgery. Another consequence is that there is a universal lower bound on the cusp density of hyperbolic alternating knots.

Keywords

alternating, knot, cusp, volume

Mathematical Subject Classification 2010

Primary: 57M25, 57M50

References

Publication

Received: 26 November 2014

Revised: 19 August 2015

Accepted: 11 October 2015

Published: 15 September 2016

Proposed: Ian Agol

Seconded: Martin Bridson, Walter Neumann

Authors

Marc Lackenby
Mathematical Institute University of Oxford Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG United Kingdom

Jessica S Purcell
School of Mathematical Sciences Monash University 9 Rainforest Walk Victoria 3800 Australia

Volume 20, issue 4 (2016)