Knot complements decomposing into prisms

TL;DR

This paper identifies four hyperbolic knot complements in three-dimensional space that cover prism orbifolds, revealing their hidden symmetries and the presence of totally geodesic surfaces.

Contribution

It introduces specific examples of knot complements covering prism orbifolds with unique geometric properties, including hidden symmetries and embedded geodesic surfaces.

Findings

Four hyperbolic knot complements cover prism orbifolds.

These orbifolds contain totally geodesic hyperbolic triangle sub-orbifolds.

Knot complements exhibit hidden symmetries and contain embedded geodesic surfaces.

Abstract

We describe four hyperbolic knot complements in $\mathbb{S}^3$, each of which covers a prism orbifold: the quotient of $\mathbb{H}^3$ by the action of a discrete group generated by reflections in the faces of a polyhedron that has the combinatorial type of a triangular prism. The prism orbifolds are rigid-cusped and contain compact, totally geodesic hyperbolic triangle sub-orbifolds; as a result, the knot complements covering them have hidden symmetries and contain closed, embedded, totally geodesic surfaces.