Unknotting number

In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number ${\displaystyle n}$, then there exists a diagram of the knot which can be changed to unknot by switching ${\displaystyle n}$ crossings. The unknotting number of a knot is always less than half of its crossing number.

Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the unknotting numbers for the first few knots:

Trefoil knot
unknotting number 1

Figure-eight knot
unknotting number 1

Cinquefoil knot
unknotting number 2

Three-twist knot
unknotting number 1

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