Census Building

Using Twister in conjunction with SnapPy by Marc Culler and Nathan Dunfield, I have produced a census of the hyperbolic surface bundles over the circle for the surface S_{1,2}, the twice punctured torus. The following generating set of the mapping class group of S_{1,2} was used:

The census building code operates in 4 stages, known as ‘grow’, ‘load’, ‘thin’ and ‘validate’. Firstly, in the ‘grow’ phase, a list of possible words to check is generated. This list is produced in such a way as to ensure that it contains at least one representative of each isometry class but also contains as few words as possible. Secondly, in the ‘load’ phase, for each word on the possible word list Twister and SnapPy are used to build the corresponding surface bundle over the surface. Thirdly, in the ‘thin’ phase, the manifolds are compared to each other and duplicates are discarded until only one representative of each isometry class (the one with lexicographically first monodromy) remains. Finally, in the ‘validate’ phase, the manifolds and census are checked once again to make sure that the data produced is plausible.

The results obtained from this were:

Census Generate Time (s) Number of Distinct Mapping Tori
3 0.8 5
4 2.9 23
5 8.8 89
6 33.5 300
7 129.3 939
8 461.9 2903
9 1723.0 9030
10 7006.8 28809
11 26789.2 94429

On a 1.9GHz AMD TL-58 dual core CPU the script takes approximately 7.3 hours and a little over 1.7 GB of RAM to construct and validate the census of hyperbolic surface bundles over the surface with monodromy with length at most 11.

Download census or scripts (Last Updated: 23/5/2012)

Download Precomputed Manifold Data (Caution: This is a 160MB compressed rar archive of 160,000 files, please download sparingly.)

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