On the complexity of 2-bridge link complements

TL;DR

This paper investigates the minimal triangulations of 2-bridge link complements, providing explicit constructions and improved volume-based complexity bounds for certain classes of these links.

Contribution

It offers a new necessary condition for minimality and constructs explicit angle structures to better estimate the complexity of 2-bridge link complements.

Findings

Established a necessary condition for minimal Sakuma-Weeks triangulations.

Constructed explicit angle structures for specific 2-bridge links.

Provided improved lower bounds on link complement complexity via volume estimates.

Abstract

We reprove a necessary condition for the Sakuma-Weeks triangulation of a 2-bridge link complement to be minimal in terms of the mapping class describing its alternating 4-string braid construction. For the 2-bridge links satisfying this condition we construct explicit angle structures on the Sakuma-Weeks triangulations and compute both multiplicative and additive lower bounds on the complexity of the link complements via volume estimates. These lower bounds are an improvement on existing volume estimates for the 2-bridge links examined.