  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 $n$, as first established in recent work with
Cantarella and Mastin. The knot diagram 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.
  We prove that unknot diagrams are asymptotically exponentially rare, an
analogue of Sumners and Whittington's landmark result for self-avoiding walks.
Our proof uses the same key idea: We first show that knot diagrams obey a
pattern theorem, which describes their fractal structure. We examine how
quickly this behavior occurs in practice. As a consequence, almost all diagrams
are asymmetric, simplifying sampling from this model. We conclude with
experimental data on knotting in this model. This model of random knotting is
similar to those studied by Diao et al., and Dunfield et al.
