Cinquefoil knot explained

Cinquefoil
Practical Name:	Double overhand knot
Arf Invariant:	1
Braid Length:	5
Braid Number:	2
Bridge Number:	2
Crosscap Number:	1
Crossing Number:	5
Genus:	2
Hyperbolic Volume:	0
Stick Number:	8
Unknotting Number:	2
Writhe:	5
Conway Notation:	[5]
Ab Notation:	51
Dowker Notation:	6, 8, 10, 2, 4
Last Crossing:	4
Last Order:	1
Next Crossing:	5
Next Order:	2
Alternating:	alternating
Class:	torus
Fibered:	fibered
Prime:	prime
Symmetry:	reversible

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 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.

Properties

The cinquefoil is a prime knot. Its writhe is 5, and it is invertible but not amphichiral. Its Alexander polynomial is

\Delta(t) = t ^{2} - t + 1 - t ^{-1} + t ^{-2} ,its Conway polynomial is\nabla(z) = z ^{4} + 3 z ^{2} + 1 ,and its Jones polynomial isV(q) = q ^{-2} + q ^{-4} - q ^{-5} + q ^{-6} - q ^{-7} . 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.

History

The name "cinquefoil" comes from the five-petaled flowers of plants in the genus Potentilla.

See also

• Pentagram
• Trefoil knot
• 7₁ knot
• Skein relation