Unknotting number
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:
- 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 -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.)
Other numerical knot invariants
- Crossing number
- Bridge number
- Linking number
- Stick number
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.
- ^ Wendt, Hilmar (December 1937). "Die gordische Auflösung von Knoten". Mathematische Zeitschrift. 42 (1): 680–696. doi:10.1007/BF01160103.
- ^ "Torus Knot", Mathworld.Wolfram.com. "".
- ^ Weisstein, Eric W. "Unknotting Number". MathWorld.
External links
- "Three_Dimensional_Invariants#Unknotting_Number", The Knot Atlas.
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