Just how knotted is a knot? One measure of knottedness is the
**unknotting number **of a knot. It is the smallest number of **crossing changes** needed to obtain the unknot from some diagram of the knot. The
unknotting number is hard to calculate and remains unknown for
many knots.

A
**crossing change** looks like this:

The
**trefoil **has **crossing number 3** but can be unknotted by one change, so it has **unknotting number 1**

For the
**cinquefoil** we need two changes. Here is a first. Either crossing can then
be changed for the second.. So it has **unknotting number 2**, though **crossing number 5**.

It was shown as recently as 1983 that the simplest diagram of
a knot is not necessarily the easiest to unknot! This knot has
10 crossings. It cannot be drawn with fewer, so it has
**crossing number 10**. However to unknot this diagram we have to change 3 crossings.
Even so it does not have unknotting number 3 because ...

here is a different picture of the So the degree of unknottedness does not always arise in the simplest
picture of a knot. Question: Those who have difficulty with this should stare at the complicated
example of the unknot on the page

**same knot** with 14 crossings. We can change **2 crossings** here to obtain the unknot. Our knot therefore has **unknotting number 2**. **which 2 crossings?! ****When are two knots the same?**

**© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002**

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