Asymptotic laws for knot diagrams

Harrison Chapman

doi:10.46298/dmtcs.6329

Abstract

We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and sampling them with the counting measure on from sets of a fixed number of vertices n. We prove that random rooted knot diagrams are highly composite and hence almost surely knotted (this is the analogue of the Frisch-Wasserman-Delbruck conjecture) and extend this to unrooted knot diagrams by showing that almost all knot diagrams are asymmetric. The model is similar to one of Dunfield, et al.

Highlights

IntroductionA planar map (or map) is a graph embedded in the sphere S2 up to continuous deformation
A planar map is a graph embedded in the sphere S2 up to continuous deformation
The tangle diagram P can be appropriately attached to knot diagrams in K if there exists a tangle diagram Q containing P which can be attached to each n-crossing diagram K in H in such a way that, 1. for some fixed, positive integer k, at least n/k possible non-conflicting places of attachment exist, 2. only diagrams in K are produced, 3. for any diagram so produced we can identify the copies of Q that have been added and they are all pairwise vertex-disjoint, and

Summary

Introduction

A planar map (or map) is a graph embedded in the sphere S2 up to continuous deformation. It is 4-valent if the underlying graph is (i.e. all vertices are of degree 4); we will call 4-valent maps link shadows. Vertices of link shadows will be called crossings. A link component of a link shadow is an equivalence class of edges meeting opposite across crossings. If a link shadow consists of precisely one link component, it is called a knot shadow. We note that knot shadows are interesting in their own right; they are precisely curve immersions on the sphere studied by Gauss, Arnol’d (1995), and others

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Conclusion