Unknotting number

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

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, then there exists a diagram of the knot which can be changed to unknot by switching 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:

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.
  • The unknotting numbers of prime knots with nine or fewer crossings have all been determined.[3] (The unknotting number of the 1011 prime knot is unknown.)

Other numerical knot invariants[edit]

See also[edit]


  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, doi:10.1142/S0218216509007361, MR 2554334 .
  3. ^ Weisstein, Eric W., "Unknotting Number", MathWorld.

External links[edit]