Right-angled links in thickened surfaces

TL;DR

This paper extends the study of alternating links to higher genus surfaces, defining a new class called RGCR links, and classifies them using geometric and diagrammatic properties, including bounds and explicit examples.

Contribution

It introduces the concept of RGCR links on higher genus surfaces, characterizes them via geometric and diagrammatic conditions, and classifies and bounds these links.

Findings

RGCR links are equivalent to having two totally geodesic checkerboard surfaces.

The paper provides a classification of RGCR links based on their checkerboard polygons.

It establishes bounds on the number of RGCR links for a given genus surface.

Abstract

Traditionally, alternating links are studied with alternating diagrams on $S^2$ in $S^3$. In this paper, we consider links which are alternating on higher genus surfaces $S_g$ in $S_g \times I$. We define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property is equivalent to the link having two totally geodesic checkerboard surfaces, equivalent to each checkerboard surface consisting of one type of polygon, and equivalent to a set of restrictions on the link's alternating projection diagram. We then use these diagram restrictions to classify RGCR links according to the polygons in their checkerboard surfaces, provide a bound on the number of RGCR links for a given surface of genus $g$, and find an RGCR knot. Along the way, we answer a question posed by Champanerkar, Kofman, and Purcell about links with alternating projections on the torus.