Witten zeta function: Difference between revisions
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[[Image:Unknotting trefoil.png|thumb|Trefoil knot without 3-fold symmetry being unknotted by one crossing switch.]] | |||
In the [[mathematics|mathematical]] area of [[knot theory]], the '''unknotting number''' of a [[knot (mathematics)|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 <math>n</math>, then there exists a [[knot diagram|diagram]] of the knot which can be changed to [[unknot]] by switching <math>n</math> crossings.<ref>{{cite book |author=Adams, Colin Conrad |title=The knot book: an elementary introduction to the mathematical theory of knots |publisher=American Mathematical Society |location=Providence, Rhode Island |year=2004 |pages=56 |isbn=0-8218-3678-1}}</ref> The unknotting number of a knot is always less than half of its [[crossing number (knot theory)|crossing number]].<ref>{{citation | |||
| last = Taniyama | first = Kouki | |||
| doi = 10.1142/S0218216509007361 | |||
| issue = 8 | |||
| journal = Journal of Knot Theory and its Ramifications | |||
| mr = 2554334 | |||
| pages = 1049–1063 | |||
| title = Unknotting numbers of diagrams of a given nontrivial knot are unbounded | |||
| volume = 18 | |||
| year = 2009}}.</ref> | |||
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: | |||
<gallery> | |||
Image:Blue Trefoil Knot.png|[[Trefoil knot]] <br> unknotting number 1 | |||
Image:Blue Figure-Eight Knot.png|[[Figure-eight knot (mathematics)|Figure-eight knot]] <br> unknotting number 1 | |||
Image:Blue Cinquefoil Knot.png|[[Cinquefoil knot]] <br> unknotting number 2 | |||
Image:Blue Three-Twist Knot.png|[[Three-twist knot]] <br> unknotting number 1 | |||
Image:Blue Stevedore Knot.png|[[Stevedore knot (mathematics)|Stevedore knot]] <br> unknotting number 1 | |||
Image:Blue 6_2 Knot.png|[[6₂ knot]] <br> unknotting number 1 | |||
Image:Blue 6_3 Knot.png|[[6₃ knot]] <br> unknotting number 1 | |||
Image:Blue 7_1 Knot.png|[[7₁ knot]] <br> unknotting number 3 | |||
</gallery> | |||
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 <math>(p,q)</math>-[[torus knot]] is equal to <math>(p-1)(q-1)/2</math>. | |||
* The unknotting numbers of [[prime knot]]s with nine or fewer [[crossing number (knot theory)|crossings]] have all been determined.<ref>{{MathWorld|title=Unknotting Number|urlname=UnknottingNumber}}</ref> (The unknotting number of the 10<sub>11</sub> prime knot is unknown.) | |||
==Other numerical knot invariants== | |||
* [[Crossing number (knot theory)|Crossing number]] | |||
* [[Bridge number]] | |||
* [[Linking number]] | |||
* [[Stick number]] | |||
==See also== | |||
*[[Unknotting problem]] | |||
==References== | |||
{{reflist}} | |||
{{Knot theory|state=collapsed}} | |||
[[Category:Knot invariants]] | |||
{{knottheory-stub}} | |||
Revision as of 20:42, 15 April 2013
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:
-
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 .
- 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
See also
References
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