On the number of unknot diagrams

Carolina Medina , Jorge Luis Ramírez Alfonsín (1) , Gelasio Salazar (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 Department of Combinatorics and Optimization

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.

Domaines: Mathématiques [math]

Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar. On the number of unknot diagrams. SIAM Journal on Discrete Mathematics, 2019, 33 (1), pp.306-326. ⟨10.1137/17M115462X⟩. ⟨hal-02049077⟩