Counting surfaces in 3-manifolds

Lee, Chaeryn

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

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

Handle URL

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

Copyright and License Information

Copyright 2023 Chaeryn Lee

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