Cinquefoil knot

Cinquefoil

Common name	Double overhand knot
Arf invariant	1
Braid length	5
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	5
Genus	2
Hyperbolic volume	0
Stick no.	8
Unknotting no.	2
Conway notation	[5]
A-B notation	5₁
Dowker notation	6, 8, 10, 2, 4
Last /Next	4₁ / 5₂
Other
alternating, torus, fibered, prime, 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.

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

\Delta (t)=t^{2}-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}}.

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.

References

↑ Weisstein, Eric W. "Solomon's Seal Knot". MathWorld.

Knot theory (knots and links)

Hyperbolic	Figure-eight (4₁) Three-twist (5₂) Stevedore (6₁) 6₂ 6₃ Endless (7₄) Carrick mat (8₁₈) Perko pair (10₁₆₁) (−2,3,7) pretzel (12n242) Whitehead (52 1) Borromean rings (63 2) L10a140

Satellite	Composite knots Granny Square Knot sum

Torus	Unknot (0₁) Trefoil (3₁) Cinquefoil (5₁) Septafoil (7₁) Unlink (02 1) Hopf (22 1) Solomon's (42 1)

Invariants	Alternating Arf invariant Bridge no. 2-bridge Brunnian Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability Unknotting no. & problem

Notation & operations	Alexander–Briggs notation Conway notation Dowker notation Flype Mutation Reidemeister move Skein relation Tabulation

Other	Alexander's theorem Berge Braid theory Conway sphere Complement Double torus Fibered Knot List of knots and links Ribbon Slice Sum Tait conjectures Twist Wild Writhe

Category Commons

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Cinquefoil knot

See also

References

Further reading