Cinquefoil knot

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

${\displaystyle \Delta (t)=t^{2}-t+1-t^{-1}+t^{-2}}$,

its Conway polynomial is

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

and its Jones polynomial is

${\displaystyle V(q)=q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}.}$

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.

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

Edible cinquefoil knot.

• Pentagram
• Trefoil knot
• 7₁ knot
• Skein relation

• A Pentafoil Knot at the Wayback Machine (archived June 4, 2004)