Geometry & Topology

Cusp volumes of alternating knots

Marc Lackenby and Jessica Purcell

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

alternating knot cusp volume


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

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