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This database consists of the fundamental groups of the 10,986
small-volume closed hyperbolic manifolds in the Hodgson--Weeks census.
The presentations included were generated by Jeffrey Weeks' program
SnapPea http://www.geometrygames.org/SnapPea/. Information about
finite-index subgroups with homology
was generated by Dunfield and Thurston in [DT03].
The basic access functions for the database are described in this section.
The result returned by the Manifold function is a record with a number
of fields containing information about the manifold and its fundamental group.
The fields of the records are as follows:
Field Name: A string giving a name to the manifold M.
Field Volume: The volume of M as a floating point number.
Field Homology: A sequence of integers describing the first homology
group of M.
Field Group; The fundamental group of M as a finitely presented group.
Field GoodCoverImage: A possibly empty sequence of permutations or
integers 1 representing the identity permutation. These permutations define
a homomorphism from the fundamental group to Sn, such that the kernel
of the homomorphism has infinite abelianization.
Field GoodCover: A list describing the construction of the good cover.
Field Degree: A positive integer, the degree of the GoodCoverImage permutation
representation.
Field KnownPosBettiCover: A boolean value, always true in the current
database.
Field KnownWeakPosBettiCover: A boolean value, always true in the current
database.
Field Reason: A string, one of "AbelianInvariants", "RationalReconstruction"
or "MAGMA".
Field Rank: A positive integer.
Field GoodCoverImageU: A possibly empty sequence of permutations or
integers 1 representing the identity permutation.
Open the database and return a reference to it.
Manifold(D, name) : DB, MonStgElt -> Rec
Extract the associated record from the database of fundamental groups
of 3-dimensional manifolds.
The manifold may be specified by the index i (1 ≤i ≤11126)
or a name from the Hodgson--Weeks census.
The intrinsic Manifold is one way to access the data in the database.
It may be more convenient to iterate over the database object returned
by ManifoldDatabase. The following examples show how this may be done.
We extract a record from the database.
> D := ManifoldDatabase();
> r := Manifold(D, 100);
> r`Name;
m019(1,4)
> r`Homology;
[ 2, 31 ]
> r`Group;
Finitely presented group on 2 generators
Relations
$.1 * $.2^3 * $.1 * $.2 * $.1^4 * $.2 * $.1 * $.2 * $.1^4
* $.2 = Id($)
$.1 * $.2 * $.1 * $.2^2 * $.1^-3 * $.2^2 = Id($)
> r`GoodCoverImage;
[
(1, 2, 4, 6, 5, 8, 7, 9, 3),
(1, 3, 5, 4, 7, 6, 9, 8, 2)
]
In [DT03], Dunfield and Thurston note that they found 132
manifolds with positive Betti number. We find them in the database as
those records where the Degree is 1. We then search the database
for one of these, but by name. Both searches use the facility to iterate
over the database that was mentioned above.
> D := ManifoldDatabase();
> pos_betti := {r`Name:r in D|r`Degree eq 1};
> #pos_betti;
132
> Random(pos_betti);
s527(-5,1)
> exists(r){r:r in D|r`Name eq "s527(-5,1)"};
true
> F := r`Group; F;
Finitely presented group F on 2 generators
Relations
F.1^2 * F.2^2 * F.1^2 * F.2^-1 * F.1^2 * F.2^2 * F.1^2 *
F.2^2 * F.1^-1 * F.2^2 = Id(F)
F.1^2 * F.2^2 * F.1^2 * F.2 * F.1^2 * F.2^2 * F.1^2 * F.2
* F.1^2 * F.2^2 * F.1^2 * F.2 * F.1^2 * F.2^2 * F.1^2 *
F.2 * F.1^2 * F.2^2 * F.1^2 * F.2 * F.1^2 * F.2^2 * F.1^2
* F.2^2 * F.1^-3 * F.2^2 = Id(F)
> AbelianQuotientInvariants(F);
[ 7, 0 ]
> r`Homology;
[ 0, 7 ]
As expected, we see that the fundamental group has infinite abelianization.
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