Cinquefoil knot

A cinquefoil knot.

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.

The cinquefoil is a prime knot. Its writhe is 5, and it is invertible but not amphichiral.^[1] Its Alexander polynomial is

Δ(t) = t² − t + 1 − t ^{− 1} + t ^{− 2},

its Conway polynomial is

$\nabla(z) = z^4 + 3z^2 + 1$,

and its Jones polynomial is

V(q) = q ^{− 2} + q ^{− 4} − q ^{− 5} + q ^{− 6} − q ^{− 7}.^[2]

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.

Edible cinquefoil knot.

Students can be introduced to the essentials of knot theory by being challenged to create edible cinquefoil pretzels.

See also

• Pentagram
• Trefoil knot
• 7₁ knot
• Skein relation

References

1. ^ Weisstein, Eric W., "Solomon's Seal Knot" from MathWorld.
2. ^ 5 1 - Knot Atlas