Geometry & Topology

Cusp volumes of alternating knots

Marc Lackenby and Jessica Purcell

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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.

Article information

Source
Geom. Topol., Volume 20, Number 4 (2016), 2053-2078.

Dates
Received: 26 November 2014
Revised: 19 August 2015
Accepted: 11 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859021

Digital Object Identifier
doi:10.2140/gt.2016.20.2053

Mathematical Reviews number (MathSciNet)
MR3548463

Zentralblatt MATH identifier
1378.57011

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds

Keywords
alternating knot cusp volume

Citation

Lackenby, Marc; Purcell, Jessica. Cusp volumes of alternating knots. Geom. Topol. 20 (2016), no. 4, 2053--2078. doi:10.2140/gt.2016.20.2053. https://projecteuclid.org/euclid.gt/1510859021


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