Cinquefoil knot - Alchetron, The Free Social Encyclopedia

Common name Double overhand knot Braid length 5 Bridge no. 2		Arf invariant 1 Braid no. 2 Crosscap no. 1

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. Its Alexander polynomial is

Δ ( t ) = t 2 − t + 1 − t − 1 + t − 2 ,

its Conway polynomial is

∇ ( z ) = z 4 + 3 z 2 + 1 ,

and its Jones polynomial is

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.

References

Cinquefoil knot Wikipedia

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