Counting surfaces in 3-manifolds

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.

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

2023

Author dc:creator

• Lee, Chaeryn

Contributors dc:contributor

• Dunfield, Nathan
• Hirani, Anil
• Albin, Pierre
• Samperton, Eric

• Topology
• Normal Surface Theory

Statement dc:rights

• Copyright 2023 Chaeryn Lee

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F121290