On the number of unknot diagrams

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.