The main data file from which all others are derived is called comm_data.
This contains data on the commensurability classes of all manifolds in the
Callahan-Hildebrand-Weeks census of cusped hyperbolic manifolds with
up to $7$ tetrahedra, and for complements of hyperbolic knots and links
up to 12 crossings, supplied by Morwen Thistlethwaite.
There are 8557 hyperbolic links and 4929 manifolds in the cusped censi
giving a total of 13486 entries in comm_data, comm_grouped, and comm_grouped_cd.
Column headings for comm_grouped
--------------------------------
1-2: manifold (c for census, a/n for alternating/non-alternating)
3: num cusps
4-7: field (degree, index, root#, chirality) -> info in file link_fields
8: G means triangulation was geometric.
9: integer/noninteger traces
10: arithmetic/nonarithmetic (a7 means discriminant -7 of inv. trace field).
11-12: symmetry group (order, chirality)
13: = means no hidden symmetries, ! means hidden symmetries
14-15: quotient (covering degree, orientability (a=nonor,c=or))
16: tilt polytope dimension (see below)
17: C means quotient is commensurator, Q means arbitrary minimal quotient
18: volume of quotient
19: commensurability class (within volume for non-arithmetic) (see below)
20-22: cusp (commensurability, equivalence, covering degrees) (see below)
To find all hidden symmetries of non-arith manifold we have to find
maximal symmetry group of all ideal cell decomps lifted to H^3.
There are finitely many cell decomps parametrized by relative areas of
a set of cusp cross sections. For n cusps there is an n-1 dimensional
polytope of different cell decomps called the tilt polytope (name may
change). As soon as a quotient is found we need only vary cusp
cross sections in the quotient. If cusps have incommensurable shapes
then they cannot be identified in any quotient, so we do not need to
vary their relative areas. Therefore the dimension of tilt polytope
required to find the commensurator is
#(cusps in best quotient found) - #(commensurability classes of shape).
This will be nonzero only when the best (i.e. smallest) quotient has
distinct commensurable shaped cusps. Then we need to check all cell
decomps in a tilt polytope of this dimension in order to verify that this
quotient is the commensurator.
Commensurability classes within volume are numbered from zero. If the
number is followed by an = sign it means the manifold is actually isometric
with the one on the previous line. (Since nonorientable manifolds in the
5-census are replaced by their orientable double covers before testing this
= sign will occasionally be incorrect.)
The final 3 columns give: commensurability class of cusp shapes
numbered from zero; equivalence classes of cusp -- with regard to
their images in the quotient; relative degree of covering restricted
to this cusp (i.e. if all cusps cover with the same degree, this will
be 1:...:1). Thus the tilt polytope dimension equals the maximum
number in column 18 minus the maximum number in column 17. Column 19
tells us that in order to find the cell decomp with maximal symmetry
we should take cusp cross sections in this area ratio. It will also be
the case that the sum of the relative covering degrees over each
equivalence class of cusps will be the same and will divide the
overall covering degree (column 14).
comm_data
---------
Column headings as above except that there is no commensurability class
column (19 in comm_grouped). comm_data tells us about the
commensurator of each listed manifold, and how to recompute it quickly
(via the last column). After sorting by quotient volume (and filtering
out the arithmetic manifolds) they can be arranged into commensurability
class by (essentially) comparing commensurators.
comm_grouped_cd
---------------
Column headings as for comm_grouped except that cusp density is inserted
after column 19 of the former. Cusp density can be defined for
multi-cusped non-arithmetic manifolds as the cusp density of maximal,
equal area cusp cross sections in the commensurator quotient.
link_fields
-----------
This gives additional information on the fields occurring in the above
tables. Entries are: degree, index, minimal polynomial, signature,
discriminant.