We give explicit examples of infinitely many noncommensurable (nonarithmetic) hyperbolic
-manifolds
admitting exactly
totally geodesic surfaces for any positive integer
,
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
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.
