Volumes of Random Alternating Link Diagrams

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.