Translator Disclaimer
2022 Cusp volumes of alternating knots on surfaces
Brandon Bavier
Algebr. Geom. Topol. 22(5): 2467-2532 (2022). 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.

Citation

Download Citation

Brandon Bavier. "Cusp volumes of alternating knots on surfaces." Algebr. Geom. Topol. 22 (5) 2467 - 2532, 2022. https://doi.org/10.2140/agt.2022.22.2467

Information

Received: 24 November 2020; Revised: 11 March 2021; Accepted: 18 April 2021; Published: 2022
First available in Project Euclid: 10 November 2022

Digital Object Identifier: 10.2140/agt.2022.22.2467

Subjects:
Primary: 57K10 , 57K31 , 57K32

Keywords: alternating links , cusp , essential surfaces , hyperbolic links , Volume

Rights: Copyright © 2022 Mathematical Sciences Publishers

JOURNAL ARTICLE
66 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.22 • No. 5 • 2022
MSP
Back to Top