Totally geodesic surfaces in twist knot complements

TL;DR

This paper constructs infinitely many non-commensurable hyperbolic 3-manifolds with exactly k totally geodesic surfaces, using twist knot complements, and addresses questions about their uniqueness and geometric properties.

Contribution

It provides explicit examples of hyperbolic 3-manifolds with a specified number of totally geodesic surfaces, solving open questions in the field.

Findings

Existence of infinitely many non-commensurable hyperbolic 3-manifolds with exactly k totally geodesic surfaces.

Construction of these manifolds from twist knot complements and their dihedral covers.

Demonstration that no twist knot complement with odd prime half twists is right-angled.

Abstract

In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher, Miller and Stover. The construction comes from a family of twist knot complements and their dihedral covers. The case $k=1$ arises from the uniqueness of an immersed totally geodesic thrice-punctured sphere, answering a question of Reid. Applying the proof techniques of the main result, we explicitly construct non-elementary maximal Fuchsian subgroups of infinite covolume within twist knot groups, and we also show that no twist knot complement with odd prime half twists is right-angled in the sense of Champanerkar, Kofman, and Purcell.