  A plane curve is a knot diagram in which each crossing is replaced by a
4-valent vertex, and so are dual to a subset of planar quadrangulations. The
aim of this paper is to introduce a new tool for sampling diagrams via sampling
of plane curves. At present the most efficient method for sampling diagrams is
rejection sampling, however that method is inefficient at even modest sizes. We
introduce Markov chains that sample from the space of plane curves using local
moves based on Reidemeister moves. By then mapping vertices on those curves to
crossings we produce random knot diagrams. Combining this chain with flat
histogram methods we achieve an efficient sampler of plane curves and knot
diagrams. By analysing data from this chain we are able to estimate the number
of knot diagrams of a given size and also compute knotting probabilities and so
investigate their asymptotic behaviour.
