Local properties of random link diagrams

Obeidin, Malik

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

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

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

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

Copyright and License Information

Copyright 2019 Malik Obeidin

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