Feature Column from the AMS

The Unknotting Number of a Knot

The unknotting number of a knot is defined to be the minimum number ofcrossings that must be switched in order to unknot it. (A knot isconsidered to be unknotted when it can be deformed into a circle.)Despite the simplicity of this definition, the unknotting number of a knot isa relatively intractable invariant of it.

Here is a diagram of a knot with 10 crossings (the minimum possible for thisknot). In this diagram, 3 crossings must be switched in order to unknot it.

Here is a diagram of the same knot with 14 crossings. In this diagram, only 2crossings must be switched in order to unknot it.

- Steven Weintraub

For more information, see the entry Exhibition: Mathematics and Knotson the What's New in Mathematics home page.