Algorithmic problems in 3-manifold topology and geometry
Malionek, Joseph D
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https://hdl.handle.net/2142/124322
Description
Title
Algorithmic problems in 3-manifold topology and geometry
Author(s)
Malionek, Joseph D
Issue Date
2024-04-18
Director of Research (if dissertation) or Advisor (if thesis)
Dunfield, Nathan M
Doctoral Committee Chair(s)
Hirani, Anil N
Committee Member(s)
Rasmussen, Jacob
Rasmussen, Sarah
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
algorithms
hyperbolic geometry
3-manifolds
Abstract
This is a study of two algorithmic problems in low-dimensional topology in geometry. The first chapter presents some results on twisted Alexander polynomials. I show that for torsion polynomials twisted by 2-dimensional representations, the coefficients are determined solely by the traces of the matrices in the representation. The second chapter details joint work with Brannon Basilio and Chaeryn Lee concerning totally geodesic surfaces. We produce an algorithm which determines when a hyperbolic 3-manifold contains a totally geodesic surface. For both of these projects, practical algorithms were implemented and extensive examples were calculated.
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