On the number of unknot diagrams
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Let D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2 n diagrams. A folklore argument shows that at least one of these 2 n diagrams is unknot, from which it follows that every diagram has finite unknotting number. It is easy to see that this argument can be used to show that actually D has more than one unknot diagram, but it cannot yield more than 4n unknot diagrams. We improve this linear bound to a superpolynomial bound by showing that at least 2 3 n of the diagrams in D are unknot. We also show that either all the diagrams in D are unknot or there is a diagram in D that is a diagram of the trefoil knot. © 2019 Society for Industrial and Applied Mathematics.
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Knot diagrams; Knot shadows; Plane curves; Unknot diagrams Combinatorial mathematics; Mathematical techniques; Knot diagrams; Knot shadows; Linear bounds; Plane curves; Unknot diagrams; Graphic methods
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