mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Gieseking manifold is a cusped hyperbolic

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

of finite volume. It is

non-orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately

V \approx 1.0149416

. It was discovered by . The volume is called Gieseking constant and has a closed-form, :

V = \operatorname_2\left(\tfrac13\pi\right) =\frac \left(\sum_^\infty\frac-\sum_^\infty \frac \right) = 1.0149416\dots

with

Clausen function In mathematics, the Clausen function, introduced by , is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimate ...

\operatorname_2\left(\varphi\right)

. Compare to the related

Catalan's constant In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is irra ...

which also manifests as a volume, :

K=\operatorname_2\left(\tfrac12\pi\right) = \sum_^\infty\frac-\sum_^\infty \frac = \sum_^ \frac = 0.91596559\dots

The Gieseking manifold can be constructed by removing the vertices from a

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

, 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 David Bernard Alper Epstein Fellow of the Royal Society, FRS (born 1937) is a mathematician known for his work in hyperbolic geometry, 3-manifolds, and group theory, amongst other fields. He co-founded the University of Warwick mathematics depa ...

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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

to the

figure-eight knot The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

. The underlying compact manifold has a

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...

boundary, and the first

homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

of the Gieseking manifold is the integers. The Gieseking manifold is a fiber bundle over the circle with fiber the

once-punctured In topology, puncturing a manifold is removing a finite set of point (topology), points from that manifold. The set of points can be small as a single point. In this case, the manifold is known as once-punctured. With the removal of a second point ...

torus and monodromy given by

(x,y) \to (x+y,x).

The square of this map is

Arnold's cat map In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name. Thinking of the torus \mathbb^2 as the quotient space ...

and this gives another way to see that the Gieseking manifold is double covered by the complement of the figure-eight knot.

References

* * * 3-manifolds Geometric topology Hyperbolic geometry {{hyperbolic-geometry-stub

See also

References