Vol. 319, No. 1, 2022

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Totally geodesic surfaces in twist knot complements

Khanh Le and Rebekah Palmer

Vol. 319 (2022), No. 1, 153–179

We give explicit examples of infinitely many noncommensurable (nonarithmetic) 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 nonelementary 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.

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totally geodesic surface, twist knots
Mathematical Subject Classification
Primary: 57K10, 57K32
Supplementary material

Magma and SageMath codes used to verify Theorem 1.3 and to produce Table 1

Received: 1 June 2021
Revised: 22 March 2022
Accepted: 26 March 2022
Published: 28 August 2022
Khanh Le
Department of Mathematics
Temple University
Philadelphia, PA
United States
Rebekah Palmer
Department of Mathematics
Temple University
Philadelphia, PA
United States