Three-manifolds up to complexity
10
Classification of closed orientable
irreducible 3-manifolds
having a triangulation with at most
10 tetrahedra.
You can see here the tables of the
manifolds we have found in complexity 1 to 9.
For each complexity we list the manifolds
according to their geometry and JSJ
decomposition, and we provide a summarizing table. See below for important
conventions used in the tables, and
here for the computer programs
used to find the data.
- Complexity 1: tables
- Complexity 2:
tables
- Complexity 3:
tables
- Complexity 4:
tables
- Complexity 5:
tables
- Complexity 6:
tables
- Complexity 7:
tables
- Complexity 8:
tables
- Complexity 9:
tables
- Complexity
10: tables
(this census refers to the following list
of small hyperbolic manifolds
not included in SnapPea's list, because
they have very short geodesics)
Conventions used in the tables:
- the manifolds of complexity 0, i.e. the sphere,
projective 3-space, and L(3,1), are not listed
- `elliptic' means `elliptic and not a lens space'
- when Seifert manifolds are involved
- the base surface of the fibration
is specified either before the table or in the first line of the table
- D, S, A stand respectively for the disc, the Moebius strip, the annulus
- the parameters (p,q) of a fibre are the filling parameters,
not the orbital parameters
- the additional twisting parameter b is equivalent to a fiber of type (1,b)
- for the twisted circle bundle over the Moebius strip we have always
used
the alternative fibration (D,(2,1),(2,1))
- when non-trivial graph manifolds are involved, the
gluing (or self-gluing) matrices
are expressed with respect to the homology bases
described in [34]
-
the `census' referred to for closed hyperbolic manifolds is a
list of
144
Dehn surgeries on the chain link with 3 components, that we propose as
the
candidate list of all smallest closed manifolds with volume < 1.96.
(It
contains the 39 manifolds of the census of Callahan, Hildebrand, and
Weeks
having volume < 1.96 and geodesics longer than 0.3).
- we have organized the tables in order to show which
bricks are used to realize the exact
value of complexity of each manifold
(see [34]), namely
- for lens spaces, the value is always realized using B2 and B3
- for all other Seifert spaces and graph manifolds, unless otherwise indicated,
the value is realized using B2, B3 and B4
(sometimes without B3)
- the only exceptions to the point just stated are the closed bricks
C(i,j), E(k)
and some manifolds for which the brick B5 is employed
- the hyperbolic bricks are only used for the hyperbolic
manifolds
Page last updated on November 18, 2004