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Figure Eight Knot Complement

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The figure eight knot complement is the knot complement S^3\K of the figure eight knot K in the 3-sphere S^3. It admits a complete finite-volume Riemannian metric of constant sectional curvature -1, making the figure eight knot a hyperbolic knot.

FigureEightComplementII

Helaman Ferguson's sculpture "Figure-Eight Complement II" depicts a thickening of a punctured torus bounded by the figure eight knot, rather than the entire three-dimensional knot complement (Borwein and Bailey 2003, pp. 54-55, color plate IV, and front cover; Bailey et al. 2007, p. 37; photograph courtesy of Helaman Ferguson). This Seifert surface is topologically a torus with an open disk removed. Its thickening has a closed genus-two boundary on which the figure eight knot separates the two copies of the Seifert surface. Ferguson's "Figure-Eight Complement III" instead thickens a punctured Klein bottle (Edmonds).

FigureEightThickenedSurface

The plotted model uses a disk with two full-twist bands of opposite handedness, attached along alternating pairs of boundary arcs (Baker 2004, p. 143). The red curve traces the figure eight knot along the boundary of the thickening.

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...
(1)

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

V=2sqrt(3)sum_(k=1)^(infty)(psi_0(2k)-psi_0(k))/(k(2k; k))
(2)
=sum_(k=1)^(infty)(2sin(1/3kpi))/(k^2)
(3)
=2sum_(k=0)^(infty)((2k; k))/(16^k(2k+1)^2)
(4)
=(2pi)/3[1-ln(pi/3)+sum_(k=1)^(infty)(zeta(2k))/(k(2k+1)6^(2k))]
(5)
=1/2sqrt(3)sum_(k=0)^(infty)(H_(k+1/2)-H_k+2ln2)/((2k; k)(2k+1)),
(6)

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)]
(7)
=(3sqrt(3))/2sum_(k=0)^(infty)[1/((3k+1)^2)-1/((3k+2)^2)]
(8)
=sqrt(3)sum_(k=0)^(infty)[1/((6k+1)^2)+1/((6k+2)^2)-1/((6k+4)^2)-1/((6k+5)^2)]
(9)
=sqrt(3)sum_(k=0)^(infty)[2/((6k+1)^2)-3/((6k+2)^2)-1/((6k+5)^2)]
(10)
=sqrt(3)sum_(k=0)^(infty)[1/((6k+1)^2)+3/((6k+4)^2)-2/((6k+5)^2)],
(11)

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)]
(12)

(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)]
(13)
=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)],
(14)

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)].
(15)

V is also given by the integrals

V=-2int_0^1(lny)/(sqrt(1-(1/2y)^2))dy
(16)
=-sqrt(3)int_0^1(lny)/(1-y+y^2)dy
(17)
=2sqrt(3)int_0^(1/2)((1+s)ln(1+s)-(1-s)ln(1-s))/((1-s^2)sqrt(1-4s^2))ds,
(18)

and the analytic expressions

V=2_3F_2(1/2,1/2,1/2;3/2,3/2;1/4)
(19)
=1/6sqrt(3)[psi_1(1/3)-psi_1(2/3)]
(20)
=1/9sqrt(3)[3psi_1(1/3)-2pi^2]
(21)
=1/(36)sqrt(3)[psi_1(1/6)+psi_1(1/3)-psi_1(2/3)-psi_1(5/6)]
(22)
=i[Li_2(e^(-ipi/3))-Li_2(e^(ipi/3))]
(23)
=2Im[Li_2(e^(ipi/3))]
(24)
=2Cl_2(1/3pi)
(25)
=3Cl_2(2/3pi),
(26)

(Broadhurst 1999; 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, Figure Eight Knot, Hyperbolic Knot, Hyperbolic Volume, Knot Complement, Seifert Surface

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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.Baker, K. L. "Knots on Once-Punctured Torus Fibers." PhD dissertation. Austin, TX: University of Texas at Austin, 2004. https://www.math.miami.edu/~kenken/Baker-Diss-07-02-04.pdf.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." Eur. Phys. J. C 8, 311-333, 1999. https://doi.org/10.1007/s100529900935.Edmonds, A. "Knots." Department of Mathematics, Indiana University Bloomington. https://math.indiana.edu/research/gallery/knots.html.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 58 and 224, 1976.Sloane, N. J. A. Sequence A091518 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

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

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