Local properties of random link diagrams

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

Obeidin, Malik

关键词: knot diagrams;    link diagrams;    knot theory;    3-manifolds;    hyperbolic volume;    random knot;    satellite knot;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/105571/OBEIDIN-DISSERTATION-2019.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.