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Figure Eight Knot

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The figure eight knot, also known as the Flemish 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)=(1+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).

Helaman Ferguson's sculpture "Figure-Eight Complement II" illustrates the knot complement of the figure eight knot (Borwein and Bailey 2003, pp. 54-55, color plate IV, and front cover; Bailey et al. 2007, p. 37). Furthermore, Ferguson has carved the BBP-type formula for the hyperbolic volume of the knot complement (discussed below) on both figure eight knot complement sculptures commissioned by the Clay Mathematics Institute (Borwein and Bailey 2003, p. 56; Bailey et al. 2007, pp. 36-38).

The hyperbolic volume of the knot complement of the figure eight knot is approximately given by

V=2.0298832...
(8)

(OEIS A091518). Exact expressions are given by the infinite sums

V=2sqrt(2)sum_(k=1)^(infty)(psi_0(2k)-psi_0(k))/(k(2k; k))
(9)
=sum_(k=1)^(infty)(2sin(1/3kpi))/(k^3)
(10)
=2sum_(k=0)^(infty)((2k; k))/(16^k(2k+1)^2)
(11)
=(2pi)/3[1-ln(pi/3)+sum_(k=1)^(infty)(zeta(2k))/(k(2k+1)6^(2k))]
(12)
=1/2sqrt(3)sum_(k=0)^(infty)(H_(k+1/2)-H_k+2ln2)/((2k; k)(2k+1)),
(13)

where H_n is a harmonic number.

V has a variety of BBP-type formulas including

V=sqrt(3)sum_(k=0)^(infty)[1/((3k+1)^2)-2/((3k+2)^2)+4/((3k+3)^2)]
(14)
=(3sqrt(3))/2sum_(k=0)^(infty)[1/((3k+1)^2)-1/((3k+2)^2)]
(15)
=sqrt(3)sum_(k=0)^(infty)[1/((6k+1)^2)+1/((6k+2)^2)-1/((6k+4)^2)-1/((6k+5)^2)]
(16)
=sqrt(3)sum_(k=0)^(infty)[2/((6k+1)^2)-3/((6k+2)^2)-1/((6k+5)^2)]
(17)
=sqrt(3)sum_(k=0)^(infty)[1/((6k+1)^2)+3/((6k+4)^2)-2/((6k+5)^2)],
(18)

with additional identities for coefficients of k of the form 3l (E. W. Weisstein, Sep. 30, 2007). Higher-order identities are

V=(2sqrt(3))/(243)sum_(k=0)^infty1/(729^k)[(243)/((12k+1)^2)-(243)/((12k+2)^2)-(324)/((12k+3)^2)-(81)/((12k+4)^2)+(27)/((12k+5)^2)-9/((12k+7)^2)+9/((12k+8)^2) +(12)/((12k+9)^2)+3/((12k+10)^2)-1/((12k+11)^2)] =(2sqrt(3))/(177147)sum_(k=0)^infty1/(531441^k)[(177147)/((24k+1)^2)-(177147)/((24k+2)^2)-(236196)/((24k+3)^2)-(59049)/((24k+4)^2)+(19683)/((24k+5)^2)-(6561)/((24k+7)^2)+(6561)/((24k+8)^2)+(8748)/((24k+9)^2)+(2187)/((24k+10)^2)-(729)/((24k+11)^2)+(243)/((24k+13)^2)-(243)/((24k+14)^2)-(324)/((24k+15)^2)-(81)/((24k+16)^2)+(27)/((24k+17)^2)-9/((24k+19)^2)+9/((24k+20)^2)+(12)/((24k+21)^2)+3/((24k+22)^2) -1/((24k+23)^2)]
(19)

(E. W. Weisstein, Aug. 11, 2008).

Additional classes of identities are given by

V=sqrt(3)sum_(k=0)^(infty)(-1)^k[1/((3k+1)^2)+1/((3k+2)^2)]
(20)
=sqrt(3)sum_(k=0)^(infty)(-1)^k[1/((9k+1)^2)+1/((9k+2)^2)-1/((9k+4)^2)-1/((9k+5)^2)+1/((9k+7)^2)+1/((9k+8)^2)],
(21)

with additional identities for coefficients of k of the form 6l+3 (E. W. Weisstein, Sep. 30, 2007). Another BBP-type formula is given by

V=(2sqrt(3))/9sum_(k=0)^(infty)((-1)^k)/(27^k)[9/((6k+1)^2)-9/((6k+2)^2)-(12)/((6k+3)^2)-3/((6k+4)^2)+1/((6k+5)^2)].
(22)

V is also given by the integrals

V=-2int_0^1(lny)/(sqrt(1-(1/2y)^2))dy
(23)
=-sqrt(3)int_0^1(lny)/(1-y+y^2)dy
(24)
=2sqrt(3)int_0^(1/2)((1+s)ln(1+s)-(1-s)ln(1-s))/((1-s^2)sqrt(1-4s^2))ds,
(25)

and the analytic expressions

V=2_3F_2(1/2,1/2,1/2;3/2,3/2;1/4)
(26)
=1/6sqrt(3)[psi_1(1/3)-psi_1(2/3)]
(27)
=1/9sqrt(3)[3psi_1(1/3)-2pi^2]
(28)
=1/(36)sqrt(3)[psi_1(1/6)+psi_1(1/3)-psi_1(2/3)-psi_1(5/6)]
(29)
=i[Li_2(e^(-ipi/3))-Li_2(e^(ipi/3))]
(30)
=2I[Li_2(e^(ipi/3))]
(31)
=2Cl_2(1/3pi)
(32)
=3Cl_2(2/3pi),
(33)

(Broadhurst 1998; Borwein and Bailey 2003, pp. 54 and 88-92; Bailey et al. 2007, pp. 36-38 and 265-266), where _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function, psi_1(z) is the trigamma function, Li_2(z) is the dilogarithm and Cl_2(x) is Clausen's integral.

See also

BBP-Type Formula, Knot, Prime Knot

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References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Bar-Natan, D. "The Knot 4_1." http://www.math.toronto.edu/~drorbn/KAtlas/Knots/4.1.html.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Broadhurst, D. J. "Massive 3-Loop Feynman Diagrams Reducible to SC^* Primitives of Algebras of the Sixth Root of Unity." March 11, 1998. http://arxiv.org/abs/hep-th/9803091.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.KnotPlot. "4_1." http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?ncomp=1&ncross=4&id=1.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.

Cite this as:

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

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