Enumerating the Covers of Prism Orbifolds

Abstract: A major goal in low-dimensional topology is to classify all 3-manifolds. Thurston’s “Geometrization conjecture”, proved by Grigori Perelman, is a key result towards fulfilling that purpose. This conjecture asserts that every compact 3-manifold can be cut into “geometric” pieces in a canonical way. Among such pieces the most complicated ones are hyperbolic 3-manifolds. A rich collection of hyperbolic 3- manifolds are knot and link complements. Hence, the study of hyperbolic links and knots (those who admit a hyperbolic structure) is essential to the theory of hyperbolic 3-manifolds, and consequently 3-manifolds. There is a conjecture stated by Neumann and Reid, which says the only hyperbolic knots that admit a hidden symmetry are the figure-8 knot and the two dodecahedral knots. They showed that a hyperbolic knot has hidden symmetries if and only if it covers a rigid-cusped orbifold. In my project, I am trying to test for counterexamples that can be generated from a collection of orbifold covers that we will build from prism orbifolds that are rigid-cusped.

Friday, November 17, 2023 - 14:00 to 15:00

Thackeray 427

Speaker Information
Arshia Gharagozlou
Graduate Student
University of Pittsburgh