## ::Cinquefoil knot

### ::concepts

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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 five, the other being the three-twist knot. It is listed as the **5 _{1} 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 but not amphichiral.<ref>Weisstein, Eric W., "Solomon's Seal Knot", *MathWorld*.</ref> Its Alexander polynomial is

- <math>\Delta(t) = t^2 - t + 1 - t^{-1} + t^{-2}</math>,

its Conway polynomial 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>

These are the same as the Alexander, Conway, and Jones polynomials of the knot 10_{132}. 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*.

**Cinquefoil knot sections**

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{{#invoke:Infobox|infobox}}
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 five, the other being the three-twist knot. It is listed as the **5 _{1} 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 but not amphichiral.<ref>Weisstein, Eric W., "Solomon's Seal Knot", *MathWorld*.</ref> Its Alexander polynomial is

- <math>\Delta(t) = t^2 - t + 1 - t^{-1} + t^{-2}</math>,

its Conway polynomial 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>

These are the same as the Alexander, Conway, and Jones polynomials of the knot 10_{132}. 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*.

**Cinquefoil knot sections**

Intro See also References Further reading

PREVIOUS: Intro | NEXT: See also |

<< | >> |