Hyperbolicity of Links in Thickened Surfaces

TL;DR

This paper extends Menasco's hyperbolicity results from links in $S^3$ to links in thickened surfaces $S imes I$, establishing conditions under which such links are hyperbolic.

Contribution

It introduces the concept of fully alternating links in $S imes I$ and proves their hyperbolicity, also providing a method to determine primeness in this setting.

Findings

Prime, fully alternating links in $S imes I$ are hyperbolic.

A fully alternating link is prime iff it is 'obviously prime'.

Hyperbolicity extends to prime links with fully alternating projections on essential surfaces.

Abstract

Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed thickened surfaces $S \times I$. We define a link to be fully alternating if it has an alternating projection from $S\times I$ to $S$ where the interior of every complementary region is an open disk. We show that a prime, fully alternating link in $S\times I$ is hyperbolic. Similar to Menasco, we also give an easy way to determine primeness in $S\times I$. A fully alternating link is prime in $S\times I$ if and only if it is "obviously prime". Furthermore, we extend our result to show that a prime link with fully alternating projection to an essential surface embedded in an orientable, hyperbolic 3-manifold has a hyperbolic complement.