Alternating links with totally geodesic checkerboard surfaces

TL;DR

This paper classifies certain alternating links with totally geodesic checkerboard surfaces, linking them to well-known polyhedral links and characterizing their geometric properties within hyperbolic geometry.

Contribution

It provides a complete classification of alternating links with two totally geodesic checkerboard surfaces and characterizes these links as right-angled completely realizable links.

Findings

Only three such links exist, corresponding to the octahedron, cuboctahedron, and icosidodecahedron.

Most hyperbolic weaving knots do not have both checkerboard surfaces totally geodesic.

The links are characterized as right-angled completely realizable links.

Abstract

We prove that alternating links with two totally geodesic checkerboard surfaces are three links with projection the 1-skeleton of the octahedron, the cuboctahedron and the icosidodecahedron. We also characterize these links as right-angled completely realisable links and show that all hyperbolic weaving knots with two exceptions have both checkerboard surfaces not totally geodesic.