Volume 20, issue 4 (2016)

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Cusp volumes of alternating knots

Marc Lackenby and Jessica S Purcell

Geometry & Topology 20 (2016) 2053–2078
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.

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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