Unknotting number

Trefoil knot without 3-fold symmetry being unknotted by one crossing switch.
Whitehead link being unknotted by undoing one crossing

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:

Other numerical knot invariants[]

See also[]


  1. ^ 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.
  2. ^ 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.
  3. ^ "Torus Knot", Mathworld.Wolfram.com. "".
  4. ^ Weisstein, Eric W. "Unknotting Number". MathWorld.

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