{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/121290"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/121290","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Counting surfaces in 3-manifolds","abstract":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-08-01","abstract_html":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-08-01","abstract_has_math":false,"creators":["Lee, Chaeryn"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dunfield, Nathan","Hirani, Anil","Albin, Pierre","Samperton, Eric"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-08","date_published":"2023-08","updated_at":"2026-07-22T22:24:57Z","subjects":["Topology","Normal Surface Theory"],"languages":["en","eng"],"rights":["Copyright 2023 Chaeryn Lee"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/121290","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dunfield, Nathan","Hirani, Anil","Albin, Pierre","Samperton, Eric"]},{"key":"dc:creator","label":"Author","values":["Lee, Chaeryn"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-08","2023-07-07"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Topology","Normal Surface Theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2023 Chaeryn Lee"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/121290"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-08-01","The student, Chaeryn Lee, accepted the attached license on 2023-06-26 at 10:43.","The student, Chaeryn Lee, submitted this Dissertation for approval on 2023-06-26 at 10:53.","This Dissertation was approved for publication on 2023-07-07 at 15:44.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18975 on 2023-12-04 at 17:17:42","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."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Counting surfaces in 3-manifolds"]}]}],"canonical_facts":{"dc:contributor":["Dunfield, Nathan","Hirani, Anil","Albin, Pierre","Samperton, Eric"],"dc:creator":["Lee, Chaeryn"],"dc:date":["2023-08","2023-07-07"],"dc:description":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-08-01","The student, Chaeryn Lee, accepted the attached license on 2023-06-26 at 10:43.","The student, Chaeryn Lee, submitted this Dissertation for approval on 2023-06-26 at 10:53.","This Dissertation was approved for publication on 2023-07-07 at 15:44.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18975 on 2023-12-04 at 17:17:42","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."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/121290"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Chaeryn Lee"],"dc:subject":["Topology","Normal Surface Theory"],"dc:title":["Counting surfaces in 3-manifolds"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:57Z"}