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Session I.4 - Computational Geometry and Topology

Wednesday, June 14, 16:30 ~ 17:00

Distribution of links and their volume in new random link model based on meanders

Anastasiia Tsvietkova

Rutgers University, Newark, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A link is an embedding of a union of circles in a 3-space, considered up to an ambient isotopy. It is among the main objects of study in low-dimensional geometry, topology, and knot theory. Random structures can be useful for proving statements about properties of a typical topological object. In this paper, we suggest a new random model for links based on meanders. We then prove that trivial links appear with vanishing probability in this model, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained on every step.

A random meander is obtained through matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of random links in this model, and, moreover, of the respective 3-manifolds that are link complements in 3-sphere. One of the strongest invariants for such manifolds is hyperbolic or simplicial volume. We give expected twist number of a link diagram and use it to bound expected hyperbolic and simplicial volume of random links. The tools from combinatorics that we use include Catalan and Narayana numbers, and Zeilberger's algorithm.

Joint work with Nicholas Owad (Hood College).

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