Unknotting numbers of diagrams of a given nontrivial knot are unbounded

TL;DR

The paper proves that for any nontrivial knot, diagrams with arbitrarily large unknotting numbers exist, and it characterizes when the crossing number equals twice the unknotting number.

Contribution

It demonstrates the unboundedness of unknotting numbers of diagrams for nontrivial knots and characterizes the cases of equality with crossing number.

Findings

Unknotting numbers of diagrams of nontrivial knots are unbounded.

Equality between crossing number and twice the unknotting number holds only for (2,p)-torus knots.

Provides a characterization of knots where the crossing number equals twice the unknotting number.

Abstract

We show that for any nontrivial knot $K$ and any natural number $n$ there is a diagram $D$ of $K$ such that the unknotting number of $D$ is greater than or equal to $n$. It is well known that twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that the equality holds only when $K$ is a $(2,p)$-torus knot.