In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Gieseking manifold is a cusped
hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It ...
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 "anticlockwise". A space is ori ...
and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately
. It was discovered by .
The Gieseking manifold can be constructed by removing the vertices from a
tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, 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 (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 department with Christopher Zeeman a ...
and Robert C. Penner. Moreover, the angle made by the faces is
. 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 mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
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 sailing, rock climbing and caving as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under ...
complement
Complement may refer to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class collections into complementary sets
* Complementary color, in the visu ...
. The underlying compact manifold has a
Klein bottle
In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
boundary, and the first
homology group
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
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
The square of this map is
Arnold's cat map
In mathematics, Arnold's cat map is a chaos theory, 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. It is a simple and pedagogical example for ...
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 In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is orientability, non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately V \approx 1.0149416. It ...
and has a numeral value of approximately:
:
It can be given as in a closed form with the
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 intimatel ...
as:
This is similar to
Catalan's constant
In mathematics, Catalan's constant , is the alternating sum of the reciprocals of the odd square numbers, being defined by:
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function ...
, which also manifests as a volume and can be expressed in terms of the Clausen function:
There is a related expression in terms of a special value of a
Dirichlet L-function
In mathematics, a Dirichlet L-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By anal ...
given by the identity
whereas
Catalan's constant
In mathematics, Catalan's constant , is the alternating sum of the reciprocals of the odd square numbers, being defined by:
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function ...
is equal to
Another closed form expression may be given in terms of the
trigamma function
In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by
: \psi_1(z) = \frac \ln\Gamma(z).
It follows from this definition that
: \psi_1(z) = \frac \psi(z)
where is the digamma functi ...
:
Integrals for the Gieseking constant are given by
which follow from its definition through the Clausen function and
A further expression is:
This gives:
which is similar to:
for Catalan's constant
.
In 2024,
Frank Calegari
Francesco Damien "Frank" Calegari is a professor of mathematics at the University of Chicago working in number theory and the Langlands program.
Early life and education
Frank Calegari was born on December 15, 1975. He has both Australian and A ...
, Vesselin Dimitrov, and
Yunqing Tang
Yunqing Tang is a mathematician specialising in number theory and arithmetic geometry and an assistant professor at University of California, Berkeley. She was awarded the SASTRA Ramanujan Prize in 2022 for "having established, by herself and in co ...
proved that
are linearly independent over the rationals. This proves that
is irrational as well as the special values
of the trigamma function. The irrationality of
itself is still open.
See also
*
List of mathematical constants
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. For e ...
*
Catalan's constant
In mathematics, Catalan's constant , is the alternating sum of the reciprocals of the odd square numbers, being defined by:
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function ...
References
*
*
*
{{Manifolds
3-manifolds
Geometric topology
Hyperbolic geometry
Mathematical constants