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 , then there exists a diagram of the knot which can be changed to unknot by switching crossings.^{[1]} The unknotting number of a knot is always less than half of its crossing number.^{[2]}

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:

In general, it is relatively difficult to determine the unknotting number of a given knot. Known cases include:

- The unknotting number of a nontrivial twist knot is always equal to one.
- The unknotting number of a -torus knot is equal to .
^{[3]} - The unknotting numbers of prime knots with nine or fewer crossings have all been determined.
^{[4]}(The unknotting number of the 10_{11}prime knot is unknown.)

## Other numerical knot invariants

## See also

## References

**^**Adams, Colin Conrad (2004).*The knot book: an elementary introduction to the mathematical theory of knots*. Providence, Rhode Island: American Mathematical Society. p. 56. ISBN 0-8218-3678-1.**^**Taniyama, Kouki (2009), "Unknotting numbers of diagrams of a given nontrivial knot are unbounded",*Journal of Knot Theory and its Ramifications*,**18**(8): 1049–1063, arXiv:0805.3174 , doi:10.1142/S0218216509007361, MR 2554334.**^**"Torus Knot",*Mathworld.Wolfram.com*. "".**^**Weisstein, Eric W. "Unknotting Number".*MathWorld*.