Census manifolds
Snappy comes with a large library of manifolds, which can be accessed individually through the Manifold and Triangulation constructors but can also be iterated through using the objects described on this page.
SnapPy’s iterators support several flexible methods for accessing manifolds. They can be sliced (i.e. restricted to subranges) either by index or by volume. Calling the iterator with keyword arguments such as num_tets=1, betti=2 or num_cusps=3 returns an iterator which is filtered by the specified conditions. In addition these iterators can determine whether they contain a given manifold. They support python’s “A in B” syntax, and also provide an identify method which will return a copy of the census manifold which is isometric to the manifold passed as an argument.
- snappy.OrientableCuspedCensus = OrientableCuspedCensus without filters
Iterator for all orientable cusped hyperbolic manifolds that can be triangulated with at most 9 ideal tetrahedra.
>>> for M in OrientableCuspedCensus[3:6]: print(M, M.volume()) ... m007(0,0) 2.56897060 m009(0,0) 2.66674478 m010(0,0) 2.66674478 >>> for M in OrientableCuspedCensus[-9:-6]: print(M, M.volume()) ... o9_44241(0,0) 8.96323909 o9_44242(0,0) 8.96736842 o9_44243(0,0) 8.96736842 >>> for M in OrientableCuspedCensus[4.10:4.11]: print(M, M.volume()) ... m217(0,0) 4.10795310 m218(0,0) 4.10942659 >>> for M in OrientableCuspedCensus(num_cusps=2)[:3]: ... print(M, M.volume(), M.num_cusps()) ... m125(0,0)(0,0) 3.66386238 2 m129(0,0)(0,0) 3.66386238 2 m202(0,0)(0,0) 4.05976643 2 >>> M = Manifold('m129') >>> M in LinkExteriors True >>> LinkExteriors.identify(M) 5^2_1(0,0)(0,0)
- snappy.OrientableClosedCensus = OrientableClosedCensus without filters
Iterator for 11,031 closed hyperbolic manifolds from the census by Hodgson and Weeks.
>>> len(OrientableClosedCensus) 11031 >>> len(OrientableClosedCensus(betti=2)) 1 >>> for M in OrientableClosedCensus(betti=2): ... print(M, M.homology()) ... v1539(5,1) Z + Z
- snappy.CensusKnots = CensusKnots without filters
Iterator for all of the knot exteriors in the SnapPea Census, as tabulated by Callahan, Dean, Weeks, Champanerkar, Kofman, Patterson, and Dunfield. These are the knot exteriors which can be triangulated by at most 9 ideal tetrahedra.
>>> for M in CensusKnots[3.4:3.5]: ... print(M, M.volume(), LinkExteriors.identify(M)) ... K4_3(0,0) 3.47424776 False K5_1(0,0) 3.41791484 False K5_2(0,0) 3.42720525 8_1(0,0) K5_3(0,0) 3.48666015 9_2(0,0)
>>> len(CensusKnots) 1267 >>> CensusKnots[-1].num_tetrahedra() 9
- snappy.LinkExteriors = LinkExteriors without filters
Iterator for all knots with at most 11 crossings and links with at most 10 crossings, using the Rolfsen notation. The triangulations were computed by Joe Christy.
>>> for K in LinkExteriors(num_cusps=3)[-3:]: ... print(K, K.volume()) ... 10^3_72(0,0)(0,0)(0,0) 14.35768903 10^3_73(0,0)(0,0)(0,0) 15.86374431 10^3_74(0,0)(0,0)(0,0) 15.55091438 >>> M = Manifold('8_4') >>> OrientableCuspedCensus.identify(M) s862(0,0)
By default, the ‘identify’ returns the first isometric manifold it finds; if the optional ‘extends_to_link’ flag is set, it insists that meridians are taken to meridians.
>>> M = Manifold('7^2_8') >>> LinkExteriors.identify(M) 5^2_1(0,0)(0,0) >>> LinkExteriors.identify(M, extends_to_link=True) 7^2_8(0,0)(0,0)
- snappy.HTLinkExteriors = HTLinkExteriors without filters
Iterator for all knots up to 14 or 15 crossings (see below for which) and links up to 14 crossings as tabulated by Jim Hoste and Morwen Thistlethwaite. In addition to the filter arguments supported by all ManifoldTables, this iterator provides alternating=<True/False>; knots_vs_links=<’knots’/’links’>; and crossings=N. These allow iterations only through alternating or non-alternating links with 1 or more than 1 component and a specified crossing number.
>>> HTLinkExteriors.identify(LinkExteriors['8_20']) K8n1(0,0) >>> Mylist = HTLinkExteriors(alternating=False,knots_vs_links='links')[8.5:8.7] >>> len(Mylist) 8 >>> for L in Mylist: ... print( L.name(), L.num_cusps(), L.volume() ) ... L11n138 2 8.66421454 L12n1097 2 8.51918360 L14n13364 2 8.69338342 L14n13513 2 8.58439465 L14n15042 2 8.66421454 L14n24425 2 8.60676092 L14n24777 2 8.53123093 L14n26042 2 8.64333782 >>> for L in Mylist: ... print( L.name(), L.DT_code() ) ... L11n138 [(8, -10, -12), (6, -16, -18, -22, -20, -2, -4, -14)] L12n1097 [(10, 12, -14, -18), (22, 2, -20, 24, -6, -8, 4, 16)] L14n13364 [(8, -10, 12), (6, -18, 20, -22, -26, -24, 2, -4, -28, -16, -14)] L14n13513 [(8, -10, 12), (6, -20, 18, -26, -24, -4, 2, -28, -16, -14, -22)] L14n15042 [(8, -10, 14), (12, -16, 18, -22, 24, 2, 26, 28, 6, -4, 20)] L14n24425 [(10, -12, 14, -16), (-18, 26, -24, 22, -20, -28, -6, 4, -2, 8)] L14n24777 [(10, 12, -14, -18), (2, 28, -22, 24, -6, 26, -8, 4, 16, 20)] L14n26042 [(10, 12, 14, -20), (8, 2, 28, -22, -24, -26, -6, -16, -18, 4)]
SnapPy comes with one of two versions of HTLinkExteriors. The smaller original one provides knots and links up to 14 crossings; the larger adds to that the knots (but not links) with 15 crossings. You can determine which you have by whether
>>> len(HTLinkExteriors(crossings=15))
gives 0 or 253293. To upgrade to the larger database, install the Python module ‘snappy_15_knots’ as discussed on the ‘installing SnapPy’ webpage.
- snappy.NonorientableCuspedCensus = NonorientableCuspedCensus without filters
Iterator for all orientable cusped hyperbolic manifolds that can be triangulated with at most 5 ideal tetrahedra.
>>> for M in NonorientableCuspedCensus(betti=2)[:3]: ... print(M, M.homology()) ... m124(0,0)(0,0)(0,0) Z/2 + Z + Z m128(0,0)(0,0) Z + Z m131(0,0) Z + Z
- snappy.NonorientableClosedCensus = NonorientableClosedCensus without filters
Iterator for 17 nonorientable closed hyperbolic manifolds from the census by Hodgson and Weeks.
>>> for M in NonorientableClosedCensus[:3]: print(M, M.volume()) ... m018(1,0) 2.02988321 m177(1,0) 2.56897060 m153(1,0) 2.66674478
There are also:
As instances of subclasses of ManifoldTable, the objects above support the following methods.
- class snappy.database.ManifoldTable(table='', db_path=None, mfld_hash=<function mfld_hash>, **filter_args)
Iterator for cusped manifolds in an sqlite3 table of manifolds.
Initialize with the table name. The table schema is required to include a text field called ‘name’ and a text field called ‘triangulation’. The text holds the result of M.triangulation_isosig(), M.triangulation_isosig(decorated = True), or M._to_string().
Both mapping from the manifold name, and lookup by index are supported. Slicing can be done either by numerical index or by volume.
The __contains__ method is supported, so M in T returns True if M is isometric to a manifold in the table T. The method T.identify(M) will return the matching manifold from the table.
- find(where=None, order_by='id', limit=None, offset=None)
Return a list of up to limit manifolds stored in this table, satisfying the where clause, and ordered by the order_by clause. If limit is None, all matching manifolds are returned. If the offset parameter is set, the first offset matches are skipped.
- identify(mfld, extends_to_link=False)
Look for a manifold in this table which is isometric to the argument.
Return the matching manifold, if there is one which SnapPea declares to be isometric.
Return False if no manifold in the table has the same hash.
Return None in all other cases (for now).
If the flag “extends_to_link” is True, requires that the isometry sends meridians to meridians. If the input manifold is closed this will result in no matches being returned.
- keys()
Return the list of column names for this manifold table.
- siblings(mfld)
Return all manifolds in the census which have the same hash value.
Because of the large size of their datasets, the classes below can only iterate through slices by index, and do not provide the identification methods.