Local properties of random link diagrams

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.

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

2019

Author dc:creator

• Obeidin, Malik

Contributors dc:contributor

• Dunfield, Nathan
• Leininger, Christopher
• Hirani, Anil
• Allen, Patrick

• knot diagrams
• link diagrams
• knot theory
• 3-manifolds
• hyperbolic volume
• random knot
• satellite knot

Statement dc:rights

• Copyright 2019 Malik Obeidin

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/105571