Cusp volumes of alternating knots on surfaces

Bavier, Brandon

doi:10.2140/agt.2022.22.2467

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Cusp volumes of alternating knots on surfaces

Brandon Bavier

Algebraic & Geometric Topology 22 (2022) 2467–2532

DOI: 10.2140/agt.2022.22.2467

Abstract

We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed $3$ –manifolds. We show that, under mild hypothesis, their cusp area admits two-sided bounds in terms of the twist number of the alternating projection and the genus of the projection surface. As a result, we derive diagrammatic estimates of slope lengths and give applications to Dehn surgery. These generalize results of Lackenby and Purcell about alternating knots in the $3$ –sphere.

Using a result of Kalfagianni and Purcell, we point out that alternating knots on surfaces of higher genus can have arbitrarily small cusp density, in contrast to alternating knots on spheres whose cusp densities are bounded away from zero due to Lackenby and Purcell.

Keywords

hyperbolic links, alternating links, cusp, volume, essential surfaces

Mathematical Subject Classification

Primary: 57K10, 57K31, 57K32

References

Publication

Received: 24 November 2020

Revised: 11 March 2021

Accepted: 18 April 2021

Published: 25 October 2022

Authors

Brandon Bavier
Department of Mathematics Michigan State University East Lansing, MI United States

Volume 22, issue 5 (2022)