Random meander model for links

TL;DR

This paper introduces a new probabilistic model for links based on meander diagrams, analyzing their properties and geometric features using combinatorics, with implications for understanding random links and their 3-manifold complements.

Contribution

It proposes a novel random link model using meander diagrams, connecting combinatorics with knot theory and 3-manifold geometry, and derives properties like probability bounds and expected volumes.

Findings

Trivial links occur with vanishing probability.

No link is obtained with probability 1.

Lower bound for non-isotopic knots with fixed crossings.

Abstract

We suggest a new random model for links based on meander diagrams and graphs. 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 for a fixed number of crossings. A random meander diagram 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. We use this for exploring geometric properties of a link complement. Specifically, 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.