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.

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

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

since is a possible Seifert surface, or because of its Conway polynomial, which is

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

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.

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

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See also

• Pentagram
• Trefoil knot
• 7₁ knot
• Skein relation

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Further reading

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