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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 n {\displaystyle n} n, then there exists a diagram of the knot which can be changed to unknot by switching n {\displaystyle n} n 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:

• Trefoil knot
unknotting number 1

• Figure-eight knot
unknotting number 1

• Cinquefoil knot
unknotting number 2

• Three-twist knot
unknotting number 1

• Stevedore knot
unknotting number 1

• 6₂ knot
unknotting number 1

• 6₃ knot
unknotting number 1

• 7₁ knot
unknotting number 3

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 (p,q)-torus knot is equal to (p-1)(q-1)/2.[3]
The unknotting numbers of prime knots with nine or fewer crossings have all been determined.[4] (The unknotting number of the 1011 prime knot is unknown.)

Other numerical knot invariants

Crossing number
Bridge number
Stick number

Unknotting problem

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. " $${\displaystyle {\frac {1}{2}}(p-1)(q-1)}$$".
Weisstein, Eric W. "Unknotting Number". MathWorld.