On the Number of Unknot Diagrams

Abstract

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. arXiv:1710.06470v1 [math.CO]

The knots that lie above all shadows

“…We make use of the following remark. By [4,Theorem 3], every shadow in which not every crossing is nugatory resolves into a trefoil knot. As we discussed above, a is a nugatory crossing of a shadow S if and only if a does not form part of any alternating pair of symbols in the Gauss code of S. Thus we have the following.…”

Structure and enumeration of K4-minor-free links and link-diagrams

“…In the last decades, there has also been interest in properties of random knots and links and their models, as well as random generation of them; see for instance [10,7,13,15] or [8,Chapter 25]. In parallel, various combinatorial and algorithmic questions of more deterministic nature have been addressed, for example in [1,9,22].…”

Structure and Enumeration of K 4 -minor-free links and link diagrams

“…In the last decades, there has also been interest in properties of random knots and links and their models, as well as random generation of them; see for instance [11,8,14,17] or [9,Chapter 25]. In parallel, various combinatorial and algorithmic questions of more deterministic nature have been addressed, for example in [1,10,23].…”