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Figure Eight Knot

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FigureEightKnot
FigureEightKnot3D

The figure eight knot, also known as the Flemish knot, Listing knot, and savoy knot, is the unique prime knot of four crossings 04-001. It has braid word sigma_1sigma_2^(-1)sigma_1sigma_2^(-1).

The figure eight knot is implemented in the Wolfram Language as KnotData["FigureEight"].

It is a 2-embeddable knot, and is amphichiral as well as invertible. It has Arf invariant 1. It is not a slice knot (Rolfsen 1976, p. 224).

The Alexander polynomial Delta(x), BLM/Ho polynomial Q(x), Conway polynomial del (x), HOMFLY polynomial P(l,m), Jones polynomial V(t), and Kauffman polynomial F F(a,z) of the figure eight knot are

Delta(x)=-x^(-1)+3-x
(1)
Q(x)=2x^3+4x^2-2x-3
(2)
del (x)=1-x^2
(3)
P(l,m)=m^2-(l^2+1/(l^2)+1)
(4)
V(t)=t^2-t+1-t^(-1)+t^(-2)
(5)
F(a,z)=(a+a^(-1))z^3+(a^2+2+a^(-2))z^2-(a+a^(-1))z-(a^2+1+a^(-2)).
(6)

There are no other knots on 10 or fewer crossings sharing the same Alexander polynomial, BLM/Ho polynomial, bracket polynomial, HOMFLY polynomial, Jones polynomial, or Kauffman polynomial F.

The figure eight knot has knot group

<x,y|x^(-1)yxy^(-1)xy=yx^(-1)yx>
(7)

(Rolfsen 1976, p. 58).

The figure eight knot complement admits a complete finite-volume hyperbolic metric. Helaman Ferguson's sculpture "Figure-Eight Complement II" is based on a thickened Seifert surface bounded by the knot.

See also

Figure Eight Knot Complement, Knot, Prime Knot

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References

Bar-Natan, D. "The Knot 4_1." https://katlas.org/wiki/4_1.Encyclopedia of Mathematics. "Knot Theory." https://encyclopediaofmath.org/wiki/Knot_theory.Francis, G. K. A Topological Picture Book. New York: Springer-Verlag, 1987.Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, pp. 8, 12, and 35, 1991.Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., pp. 21 and 153, 1993.Owen, P. Knots. Philadelphia, PA: Courage, p. 16, 1993.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 58 and 224, 1976.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 78-79, 1991.

Referenced on Wolfram|Alpha

Figure Eight Knot

Cite this as:

Weisstein, Eric W. "Figure Eight Knot." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FigureEightKnot.html

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