University of Illinois Urbana-Champaign

Counting surfaces in 3-manifolds

Lee, Chaeryn

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https://hdl.handle.net/2142/121290

Description

Title
Counting surfaces in 3-manifolds
Author(s)
Lee, Chaeryn
Issue Date
2023-07-07
Director of Research (if dissertation) or Advisor (if thesis)
Dunfield, Nathan
Doctoral Committee Chair(s)
Hirani, Anil
Committee Member(s)
  • Albin, Pierre
  • Samperton, Eric
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • topology
  • normal surface theory
Abstract
In this paper we are interested in counting the number of isotopy classes of essential surfaces in 3-manifolds. We first look at a specific manifold, the exterior B of the knot K13n586 and count the number of isotopy classes of closed, connected, orientable, essential surfaces. The main result is that the count of surfaces by genus is equal to the Euler totient function. The main argument is to show when normal surfaces in B are connected by counting their number of components. We implement tools from Agol, Hass and Thurston to convert the problem of counting components of surfaces into counting the number of orbits in a set of integers under a collection of bijections defined on its subsets. Results from Dunfield, Garoufalidis and Rubinstein show that the count of isotopy classes of closed, orientable, essential surfaces by Euler characteristic admits quasi-polynomial behaviour. This holds for 3-manifolds that do not contain nonorientable essential surfaces. We attempt to see if this property extends to manifolds that do contain nonorientable surfaces by performing computations on a database of manifolds provided by SnapPy and Twister.
Graduation Semester
2023-08
Type of Resource
Thesis
Copyright and License Information
Copyright 2023 Chaeryn Lee

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