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{{Infobox knot theory | |||
| name= Stevedore knot | |||
| practical name= Stevedore knot | |||
| image= Blue Stevedore Knot.png | |||
| caption= | |||
| arf invariant= 0 | |||
| braid length= 7 | |||
| braid number= 4 | |||
| bridge number= 2 | |||
| crosscap number= 2 | |||
| crossing number= 6 | |||
| hyperbolic volume= 3.16396 | |||
| linking number= | |||
| stick number= 8 | |||
| unknotting number= 1 | |||
| conway_notation= [42] | |||
| ab_notation= 6<sub>1</sub> | |||
| dowker notation= 4, 8, 12, 10, 2, 6 | |||
| thistlethwaite= | |||
| last crossing= 5 | |||
| last order= 2 | |||
| next crossing= 6 | |||
| next order= 2 | |||
| alternating= alternating | |||
| class= hyperbolic | |||
| fibered= | |||
| prime= prime | |||
| slice= slice | |||
| symmetry= reversible | |||
| pretzel= pretzel | |||
| tricolorable= | |||
| twist= twist | |||
}} | |||
[[Image:Double eight -1.JPG|thumb|The common [[stevedore knot]]. If the ends were joined together, the result would be equivalent to the mathematical knot.]] | |||
In [[knot theory]], the '''stevedore knot''' is one of three [[prime knot]]s with [[crossing number (knot theory)|crossing number]] six, the others being the [[6₂ knot|6<sub>2</sub> knot]] and the [[6₃ knot|6<sub>3</sub> knot]]. The stevedore knot is listed as the '''6<sub>1</sub> knot''' in the [[Alexander–Briggs notation]], and it can also be described as a [[twist knot]] with four twists, or as the (5,−1,−1) [[pretzel link|pretzel knot]]. | |||
The mathematical stevedore knot is named after the common [[stevedore knot]], which is often used as a [[stopper knot|stopper]] at the end of a [[rope]]. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted [[loop (topology)|loop]]. | |||
The stevedore knot is [[invertible knot|invertible]] but not [[amphichiral knot|amphichiral]]. Its [[Alexander polynomial]] is | |||
:<math>\Delta(t) = -2t+5-2t^{-1}, \,</math> | |||
its [[Conway polynomial]]{{dn|date=January 2014}} is | |||
:<math>\nabla(z) = 1-2z^2, \, </math> | |||
and its [[Jones polynomial]] is | |||
:<math>V(q) = q^2-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}. \, </math><ref>{{Knot Atlas|6_1}}</ref> | |||
The Alexander polynomial and Conway polynomial are the same as those for the knot 9<sub>46</sub>, but the Jones polynomials for these two knots are different.<ref>{{MathWorld|title=Stevedore's Knot|urlname=StevedoresKnot}}</ref> Because the Alexander polynomial is not [[monic polynomial|monic]], the stevedore knot is not [[fibered knot|fibered]]. | |||
The stevedore knot is a [[ribbon knot]], and is therefore also a [[slice knot]]. | |||
The stevedore knot is a [[hyperbolic knot]], with its complement having a [[Hyperbolic volume (knot)|volume]] of approximately 3.16396. | |||
==See also== | |||
* [[Figure-eight knot (mathematics)]] | |||
==References== | |||
{{reflist}} | |||
{{Knot theory|state=collapsed}} | |||
{{knottheory-stub}} | |||
Revision as of 00:54, 11 December 2013
In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, and it can also be described as a twist knot with four twists, or as the (5,−1,−1) pretzel knot.
The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper at the end of a rope. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted loop.
The stevedore knot is invertible but not amphichiral. Its Alexander polynomial is
its Conway polynomialTemplate:Dn is
and its Jones polynomial is
The Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different.[2] Because the Alexander polynomial is not monic, the stevedore knot is not fibered.
The stevedore knot is a ribbon knot, and is therefore also a slice knot.
The stevedore knot is a hyperbolic knot, with its complement having a volume of approximately 3.16396.
See also
References
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- ↑ Template:Knot Atlas
- ↑
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