Algorithmic problems in 3-manifold topology and geometry

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Malionek, Joseph D

Contributors dc:contributor

• Dunfield, Nathan M
• Hirani, Anil N
• Rasmussen, Jacob
• Rasmussen, Sarah

• Algorithms
• Hyperbolic Geometry
• 3-manifolds

Statement dc:rights

• Copyright 2024 Joseph Malionek

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/124322

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F124322