Gieseking manifold explained

In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately

V ≈ 1 . 0149416 . It was discovered by .

The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0, 1, 2 to the face with vertices 3, 1, 0 in that order. Glue the face 0, 2, 3 to the face 3, 2, 1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner. Moreover, the angle made by the faces is

\pi/ 3 . The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.

The Gieseking manifold has a double cover homeomorphic to the figure-eight knot complement. The underlying compact manifold has a Klein bottle boundary, and the first homology group of the Gieseking manifold is the integers.

The Gieseking manifold is a fiber bundle over the circle with fiber the once-punctured torus and monodromy given by

(x,y) \to (x + y,x). The square of this map is Arnold's cat map and this gives another way to see that the Gieseking manifold is double covered by the complement of the figure-eight knot.

Gieseking constant

The volume of the Gieseking manifold is called the Gieseking constant^[1] and has a numeral value of approximately:

V = 1 . 01494 16064 09653 62502 12025 ... ^[2] It can be given as in a closed form^[3] with the Clausen function \operatorname{Cl} _{2\left(\varphi\right)} as:V = \operatorname{Cl} _{2 \left( (\pi)/(3)} \right) G , which also manifests as a volume and can be expressed in terms of the Clausen function:G = \operatorname{Cl} _{2 \left( (\pi)/(2)} \right) = 0 . 91596559...

Another closed form expression may be given in terms of the trigamma function:

V = (\sqrt 3)/(\left( 3) (\psi _{1(1/3)})/(- 2) (\pi ^{2})/(3) \right)

Integrals for the Gieseking constant are given by

V = \int ^{2 \pi/ 3} _{0} ln \left( 2 \cos\left(\tfrac 12 x\right)\right) dx V =2 \int ^{1} _{0} (ln ( 1+ x))/(\sqrt{( 1- x)( 3+ x)) }\mathrm dx

which follow from its definition through the Clausen function and^[4]

V = (\sqrt{ 3)/() }2 \int_0^\infty\int_0^\infty\int_0^\infty \frac

A further expression is:

V = (3 \sqrt 3)/(4) \left(\sum ^{inf ty} _{k =0} (1)/(( 3 k +1 ) ^{2}) - \sum ^{inf ty} _{k =0} (1)/(( 3 k +2 ) ^{2}) \right)

This gives:

\sum ^{inf ty} _{k =0} (1)/(= ( 3 k +1 ) ^{2}) (2 \pi ^{2})/(+ 27) (2 \sqrt 3)/(9) V \sum ^{inf ty} _{k =0} (1)/(= ( 3 k +2 ) ^{2}) (2 \pi ^{2})/(- 27) (2 \sqrt 3)/(9) V

which is similar to:

\sum ^{inf ty} _{k =0} (1)/(= ( 4 k +1 ) ^{2}) (\pi ^{2})/(+ 16) (1)/(2) G \sum ^{inf ty} _{k =0} (1)/(= ( 4 k +3 ) ^{2}) (\pi ^{2})/(- 16) (1)/(2) G

for Catalan's constant

G .

See also

• List of mathematical constants
• Catalan's constant

References

• • • David B.A. . Epstein . David B. A. Epstein. Robert C. . Penner. Euclidean decompositions of noncompact hyperbolic manifolds. . 27 . 1988. 1. 67–80. 0918457 . 10.4310/jdg/1214441650. free.

Notes and References

1. Book: Finch, Steven R. . Mathematical Constants . 2003-08-18 . Cambridge University Press . 978-0-521-81805-6 . en.
2. Web site: Gieseking constant - A143298 - OEIS . 2024-09-24 . oeis.org.
3. Web site: Weisstein . Eric W. . Gieseking's Constant . 2024-09-24 . mathworld.wolfram.com . en.
4. Bailey . D H . Borwein . J M . Crandall . R E . 2006-09-19 . Integrals of the Ising class . Journal of Physics A: Mathematical and General . 39 . 40 . 12271–12302 . 10.1088/0305-4470/39/40/001 . 1959.13/803609 . 0305-4470. free .