On the number of unknot diagrams

TL;DR

This paper proves that for any knot diagram with n crossings, the set of diagrams obtained by crossing exchanges contains at least 2^{n^{1/3}} unknot diagrams, significantly improving previous bounds.

Contribution

It establishes a superpolynomial lower bound on the number of unknot diagrams obtainable from a given diagram via crossing exchanges, advancing understanding of unknotting complexity.

Findings

At least 2^{n^{1/3}} unknot diagrams exist in the set from crossing exchanges.

Either all diagrams are unknot or one is a trefoil diagram.

Improves previous linear bounds to superpolynomial bounds.

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 $2^n$ 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 ${\mathcal D}$ 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 $2^{\sqrt[3]{n}}$ 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.