On the number of unknot diagrams

2017/10/17 by Medina, Carolina, Ramírez-Alfonsín, Jorge, Salazar, Gelasio
#57M25 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.06470

Abstract

Let D be a knot diagram, and let \mathcal D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then \mathcal D consists of 2ⁿ diagrams. A folklore argument shows that at least one of these 2ⁿ 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 \mathcal 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 \mathcal D are unknot. We also show that either all the diagrams in \mathcal D are unknot, or there is a diagram in \mathcal D that is a diagram of the trefoil knot.

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