Asian Journal of Mathematics

Volume 22 (2018)

Number 6

Determining isotopy classes of crossing arcs in alternating links

Pages: 1005 – 1024



Anastasiia Tsvietkova (Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey, U.S.A.)


Given a reduced alternating diagram for a link, we obtain conditions that guarantee that the link complement has a complete hyperbolic structure, crossing arcs are the edges of an ideal geodesic triangulation, and every crossing arc is isotopic to a simple geodesic. The latter was conjectured by Sakuma and Weeks in 1995. We provide new infinite families of closed braids for which our conditions hold.


alternating link, link complement, hyperbolic structure, geodesic

2010 Mathematics Subject Classification

57M25, 57M50

Received 27 June 2016

Accepted 26 January 2017

Published 6 February 2019