Unknotting numbers of diagrams of a given nontrivial knot are unbounded

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.

13 pages, 14 figures. Theorem 1.4 and Theorem 1.5 added