{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/105571"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/105571","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Local properties of random link diagrams","abstract":"We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random prime alternating diagrams, the expected hyperbolic volume is asymptotically linear in the number of crossings. If we do not restrict to prime alternating diagrams, and instead randomize the over/under strand at each crossing, it is known due to work of Chapman that the resulting diagrams are generically composite, as any tangle — including ones which, when inserted into a diagram, force a link to be composite — occurs many times in a large link diagram with high probability. Using enumerations of Bernardi and Fusy, we prove an asymptotic formula for probability that a tangle occurs in a specific location in a random (not necessarily prime) link diagram.","abstract_html":"We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random prime alternating diagrams, the expected hyperbolic volume is asymptotically linear in the number of crossings. If we do not restrict to prime alternating diagrams, and instead randomize the over/under strand at each crossing, it is known due to work of Chapman that the resulting diagrams are generically composite, as any tangle — including ones which, when inserted into a diagram, force a link to be composite — occurs many times in a large link diagram with high probability. Using enumerations of Bernardi and Fusy, we prove an asymptotic formula for probability that a tangle occurs in a specific location in a random (not necessarily prime) link diagram.","abstract_has_math":false,"creators":["Obeidin, Malik"],"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","Leininger, Christopher","Hirani, Anil","Allen, Patrick"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-11-26T20:28:49Z","date_published":"2019-11-26T20:28:49Z","updated_at":"2026-07-22T22:24:44Z","subjects":["knot diagrams","link diagrams","knot theory","3-manifolds","hyperbolic volume","random knot","satellite knot"],"languages":["en"],"rights":["Copyright 2019 Malik Obeidin"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/105571","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dunfield, Nathan","Leininger, Christopher","Hirani, Anil","Allen, Patrick"]},{"key":"dc:creator","label":"Author","values":["Obeidin, Malik"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-11-26T20:28:49Z","2019-07-10","2019-08"]},{"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":["knot diagrams","link diagrams","knot theory","3-manifolds","hyperbolic volume","random knot","satellite knot"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2019 Malik Obeidin"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/105571"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random prime alternating diagrams, the expected hyperbolic volume is asymptotically linear in the number of crossings. If we do not restrict to prime alternating diagrams, and instead randomize the over/under strand at each crossing, it is known due to work of Chapman that the resulting diagrams are generically composite, as any tangle — including ones which, when inserted into a diagram, force a link to be composite — occurs many times in a large link diagram with high probability. Using enumerations of Bernardi and Fusy, we prove an asymptotic formula for probability that a tangle occurs in a specific location in a random (not necessarily prime) link diagram.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Malik Obeidin, accepted the attached license on 2019-07-09 at 03:50.","The student, Malik Obeidin, submitted this Dissertation for approval on 2019-07-09 at 04:00.","This Dissertation was approved for publication on 2019-07-10 at 09:20.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13962 on 2019-11-26 at 12:48:17","Made available in DSpace on 2019-11-26T20:28:49Z (GMT). No. of bitstreams: 3 OBEIDIN-DISSERTATION-2019.pdf: 935632 bytes, checksum: e4005e8c8bd3cacee4ecca382aff2f7d (MD5) LICENSE.txt: 4210 bytes, checksum: f0f38dc402ef7814c98abbafaaa8def8 (MD5) PROQUEST_LICENSE.txt: 4556 bytes, checksum: ac7a848ddd80e3a677e81d6bf0b72ba6 (MD5) Previous issue date: 2019-07-10"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Local properties of random link diagrams"]}]}],"canonical_facts":{"dc:contributor":["Dunfield, Nathan","Leininger, Christopher","Hirani, Anil","Allen, Patrick"],"dc:creator":["Obeidin, Malik"],"dc:date":["2019-11-26T20:28:49Z","2019-07-10","2019-08"],"dc:description":["We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random prime alternating diagrams, the expected hyperbolic volume is asymptotically linear in the number of crossings. If we do not restrict to prime alternating diagrams, and instead randomize the over/under strand at each crossing, it is known due to work of Chapman that the resulting diagrams are generically composite, as any tangle — including ones which, when inserted into a diagram, force a link to be composite — occurs many times in a large link diagram with high probability. Using enumerations of Bernardi and Fusy, we prove an asymptotic formula for probability that a tangle occurs in a specific location in a random (not necessarily prime) link diagram.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Malik Obeidin, accepted the attached license on 2019-07-09 at 03:50.","The student, Malik Obeidin, submitted this Dissertation for approval on 2019-07-09 at 04:00.","This Dissertation was approved for publication on 2019-07-10 at 09:20.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13962 on 2019-11-26 at 12:48:17","Made available in DSpace on 2019-11-26T20:28:49Z (GMT). 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