nLab unknotting number

Idea

If you had a piece of string possibly tangled up, and could, at a crossing, pull one part of the string through the other, then, intuitively, repeating this enough times, the string would become unknotted. At the mathematical level, there is a corresponding notion of a crossing change on a diagram

Definition

A crossing change in a diagram exchanges an overpass and underpass at a crossing, as below:

(The central arrow should be a left-right arrow, but the arrowheads do not come out!)

Crossing changes will usually alter the isotopy type of the diagram.

Lemma

Let $D$ be a diagram with $c$ crossings, then changing at most $c/2$ crossings of $D$ produces a diagram of the unknot.

Definition

The unknotting number, $u(K)$, is the smallest number of crossing changes required to obtain the unknot from some diagram of the knot.

Of course, we know that $u(K)\leq c(K)/2$, but the natural difficulty of calculating $u(K)$ is made worse by the following result of Beiler (1984).

The unknotting number of a knot does not necessarily occur in a minimal diagram.

Beiler gave an example of a minimal diagram for a 10 crossing knot, which cannot be unknotted with fewer than 3 crossing changes, yet for which there is a 14 crossing diagram, which is isotopic to it, yet can be unknotted with just 2 crossing changes.