Geometry of alternating links on surfaces

TL;DR

This paper generalizes the polyhedral decomposition of alternating links to links on surfaces in 3-manifolds, providing new insights into their hyperbolic geometry, diagrammatic properties, and volume bounds.

Contribution

It introduces a decomposition for surface-alternating links, extending Menasco's work, and analyzes their hyperbolic structures and geometric properties.

Findings

Decomposition into simpler pieces under mild conditions

Criteria for hyperbolicity of surface-alternating links

Bounds on volume and analysis of checkerboard surfaces

Abstract

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude exceptional Dehn fillings.