A Markov Chain Sampler for Plane Curves
| Autor: | Harrison Chapman, Andrew Rechnitzer |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Markov chain
Plane curve General Mathematics 010102 general mathematics Geometric Topology (math.GT) 0102 computer and information sciences Computer Science::Computational Geometry 01 natural sciences Mathematics::Geometric Topology Combinatorics Mathematics - Geometric Topology Planar Computer Science::Discrete Mathematics 010201 computation theory & mathematics FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Mathematics Knot (mathematics) |
| DOI: | 10.48550/arxiv.1804.03311 |
| Popis: | 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. Comment: 41 pages, 30 figures |
| Databáze: | OpenAIRE |
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