Volumes of Random Alternating Link Diagrams

TL;DR

This paper introduces a model for random link diagrams based on 4-valent maps, analyzing their hyperbolic volume and structural properties, with empirical validation through computer experiments.

Contribution

It provides a new probabilistic model for random links and explores the relationship between diagram complexity and hyperbolic volume, including computational results.

Findings

Expected hyperbolic volume is asymptotically linear in crossings for alternating diagrams.

Probability estimates for arbitrary tangles in nonalternating diagrams.

Random link diagrams tend to be highly composite.

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 alternating diagrams, the expected hyperbolic volume is asymptotically linear in the number of crossings. For nonalternating diagrams, we compute the probability of finding a given, arbitrary tangle around a given crossing, and show that a random link diagram will be highly composite. Additionally, we present some results of computer experiments obtained from implementing the model in the program SnapPy.