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] This invariant was first defined by Hilmar Wendt in 1936.[3]

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 .[4]
  • The unknotting numbers of prime knots with nine or fewer crossings have all been determined.[5] (The unknotting number of the 1011 prime knot is unknown.)
  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. ^ Wendt, Hilmar (December 1937). "Die gordische Auflösung von Knoten". Mathematische Zeitschrift. 42 (1): 680–696. doi:10.1007/BF01160103.
  4. ^ "Torus Knot", Mathworld.Wolfram.com. "".
  5. ^ Weisstein, Eric W. "Unknotting Number". MathWorld.

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