Geometric and material buckling

2013-11-07T09:45:41Z

196.35.252.210:

{{Infobox knot theory
| name= Cinquefoil
| practical name= Double overhand knot
| image= Blue Cinquefoil Knot.png
| caption=
| arf invariant= 1
| braid length= 5
| braid number= 2
| bridge number= 2
| crosscap number= 1
| crossing number= 5
| hyperbolic volume= 0
| linking number=
| stick number= 8
| unknotting number= 2
| conway_notation= [5]
| ab_notation= 51
| dowker notation= 6, 8, 10, 2, 4
| thistlethwaite=
| last crossing= 4
| last order= 1
| next crossing= 5
| next order= 2
| alternating= alternating
| class= torus
| fibered= fibered
| prime= prime
| slice=
| symmetry= reversible
| tricolorable=
}}

In [[knot theory]], the '''cinquefoil knot''', also known as '''Solomon's seal knot''' or the '''pentafoil knot''', is one of two knots with [[crossing number (knot theory)|crossing number]] five, the other being the [[three-twist knot]]. It is listed as the '''51 knot''' in the [[Alexander-Briggs notation]], and can also be described as the (5,2)-[[torus knot]]. The cinquefoil is the closed version of the [[double overhand knot]].

The cinquefoil is a [[prime knot]]. Its [[writhe]] is 5, and it is [[invertible knot|invertible]] but not [[amphichiral knot|amphichiral]].<ref>{{MathWorld|title=Solomon's Seal Knot|urlname=SolomonsSealKnot}}</ref> Its [[Alexander polynomial]] is
:<math>\Delta(t) = t^2 - t + 1 - t^{-1} + t^{-2}</math>,
its [[Conway polynomial]]{{dn|date=January 2014}} is
:<math>\nabla(z) = z^4 + 3z^2 + 1</math>,
and its [[Jones polynomial]] is
:<math>V(q) = q^{-2} + q^{-4} - q^{-5} + q^{-6} - q^{-7}.</math><ref>{{Knot Atlas|5_1}}</ref>
Surprisingly, these are the same as the Alexander, Conway, and Jones polynomials of the knot 10132. However, the [[Kauffman polynomial]] can be used to distinguish between these two knots.

The name “cinquefoil” comes from the five-petaled flowers of plants in the genus ''[[Potentilla]]''.

[[File:Cinquefoil Knot.jpg|right|thumb|Edible cinquefoil knot.]]

==See also==
*[[Pentagram]]
*[[Trefoil knot]]
*[[7₁ knot]]
*[[Skein relation]]

==References==
{{reflist}}

==Further reading==
*{{Wayback|url=http://wwwhome.cs.utwente.nl/~jagersaa/Knopen/IndexP.html|title=A Pentafoil Knot|date=20040604232208}}

{{Knot theory|state=collapsed}}

{{knottheory-stub}}

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Geometric and material buckling