University of Illinois Urbana-Champaign

Local properties of random link diagrams

Obeidin, Malik

Content Files
OBEIDIN-DISSERTATION-2019.pdf

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https://hdl.handle.net/2142/105571

Description

Title
Local properties of random link diagrams
Author(s)
Obeidin, Malik
Issue Date
2019-07-10
Director of Research (if dissertation) or Advisor (if thesis)
Dunfield, Nathan
Doctoral Committee Chair(s)
Leininger, Christopher
Committee Member(s)
  • Hirani, Anil
  • Allen, Patrick
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2019-11-26T20:28:49Z
Keyword(s)
  • knot diagrams
  • link diagrams
  • knot theory
  • 3-manifolds
  • hyperbolic volume
  • random knot
  • satellite knot
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.
Graduation Semester
2019-08
Type of Resource
text
Permalink
http://hdl.handle.net/2142/105571
Copyright and License Information
Copyright 2019 Malik Obeidin

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