Files
Abstract
We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices n. This model is an exciting new model which captures both the random geometry of space curve models of knotting as well as the ease of computing invariants from diagrams. This model of random knotting is similar to those studied by Diao et al., and Dunfield et al. We prove that unknot diagrams are asymptotically exponentially rare, an analogue of Sumners and Whittington's result for self-avoiding walks. Our proof uses the same idea: We first show that knot diagrams obey a pattern theorem and exhibit fractal structure. We use a rejection sampling method to present experimental data showing that these asymptotic results occur quickly, and compare parallels to other models of random knots. We finish by providing a number of extensions to the diagram model. The diagram model can be used to study embedded graph theory, open knot theory, virtual knot theory, and even random knots of fixed type. In this latter scenario, we prove a result still unproven for other models of random knotting. We additionally discuss an alternative method for randomly sampling diagrams via a Markov chain Monte Carlo method.