Determining isotopy classes of crossing arcs in alternating links

TL;DR

This paper establishes conditions under which crossing arcs in reduced alternating link diagrams are isotopic to simple geodesics, confirming a conjecture and providing infinite examples.

Contribution

It introduces new criteria for the hyperbolic structure and geodesic isotopy of crossing arcs in alternating links, confirming a longstanding conjecture.

Findings

Conditions guarantee hyperbolic structure of link complements

Crossing arcs are isotopic to simple geodesics under these conditions

Infinite families of braids satisfying the conditions are provided

Abstract

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 infinite families of closed braids for which our conditions hold.