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Title:Local properties of random link diagrams
Author(s):Obeidin, Malik
Director of Research:Dunfield, Nathan
Doctoral Committee Chair(s):Leininger, Christopher
Doctoral Committee Member(s):Hirani, Anil; Allen, Patrick
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):knot diagrams
link diagrams
knot theory
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.
Issue Date:2019-07-10
Rights Information:Copyright 2019 Malik Obeidin
Date Available in IDEALS:2019-11-26
Date Deposited:2019-08

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