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:

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(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 DD be a diagram with cc crossings, then changing at most c/2c/2 crossings of DD produces a diagram of the unknot.

Definition

The unknotting number, u(K)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)c(K)/2u(K)\leq c(K)/2, but the natural difficulty of calculating u(K)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.

Revised on October 16, 2010 08:30:36 by Tim Porter (95.147.237.215)