.. Documentation of the Snap part of SnapPy .. automodule:: snappy.snap Number theory of hyperbolic 3-manifolds ============================================= SnapPy has support for arbitrary-precision computation and for identifying number fields associated to hyperbolic 3-manifolds. While this functionality is less than that of `Snap `_, it is already useful. Except for the first example, one currently needs to use SnapPy inside of `Sage `_ to have access to these features. Here's how to find the tetrahedra shapes to high-precision:: sage: import snappy sage: M = snappy.Manifold('m004') sage: M.tetrahedra_shapes('rect', bits_prec=100) [0.50000000000000000000000000000 + 0.86602540378443864676372317075*I, 0.50000000000000000000000000000 + 0.86602540378443864676372317075*I] sage: M.tetrahedra_shapes('rect', dec_prec=100) 0.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 0.866025403784438646763723170752936183471402626905190314027903489725966508454400018540573093378624288*I One can also compute the holonomy representation to any precision:: sage: G = M.polished_holonomy(bits_prec=100) sage: G.SL2C('a') [0.50000000000000000000000000000 + 0.86602540378443864676372317075*I -1.0000000000000000000000000000*I] [ 1.0000000000000000000000000000*I 1.0000000000000000000000000000 - 1.7320508075688772935274463415*I] You can also try to guess the shapes exactly using an LLL-based algorithm of the type pioneered by Snap:: sage: T = M.tetrahedra_field_gens() sage: T.find_field(prec=100, degree=10, optimize=True) (Number Field in z with defining polynomial x^2 - x + 1, , [x, x]) You can do the same for various other fields via the methods ``trace_field_gens``, ``invariant_trace_field_gens``, and ``holonomy_matrix_entries``. In more complicated examples, one needs to use higher precision and/or degree to actually find the exact values:: sage: N = snappy.Manifold('m004(1,3)') sage: K = N.trace_field_gens() sage: K.find_field(prec=100, degree=10, optimize=True) # Fails, so no output sage: K.find_field(prec=200, degree=20, optimize=True) Number Field in z with defining polynomial x^11 - 2*x^10 - 8*x^9 + 16*x^8 + 22*x^7 - 44*x^6 - 25*x^5 + 50*x^4 + 11*x^3 - 22*x^2 - x + 1 We can also compute various hyperbolicly-twisted Alexander polynomials, as described `here `_:: sage: M = snappy.Manifold('5_2') sage: M.alexander_polynomial() 2*a^2 - 3*a + 2 sage: M.hyperbolic_torsion(bits_prec=100) (2.3376410213776269870195455729 - 0.56227951206230124389918214504*I)*a^2 - 4.0000000000000000000000000003*a + 2.3376410213776269870195455731 - 0.56227951206230124389918214477*I sage: M.hyperbolic_SLN_torsion(3, 100) # Dubois-Yamagachi adjoint torsion (0.40431358073618481197132660504 + 0.75939451500971650241038772223*I)*a^3 + (2.9032849613891083021420278850 - 4.1185388389935516999882632998*I)*a^2 + (-2.9032849613891083021420278809 + 4.1185388389935516999882633007*I)*a - 0.40431358073618481197132661847 - 0.75939451500971650241038771418*I sage: M.hyperbolic_SLN_torsion(4, 100) # Why not? (2.5890988184099251088892745185 + 3.5126610817613336586374292713*I)*a^4 + (10.357403823939297224437742077 - 13.378446302375451727042633120*I)*a^3 + (-26.821802363180149782221451472 + 7.0253221635226673172748587283*I)*a^2 + (10.357403823939297224437738856 - 13.378446302375451727042631346*I)*a + 2.5890988184099251088892549440 + 3.5126610817613336586374448040*I You can find out more about each of these methods using introspection:: sage: M.hyperbolic_torsion? Definition: M.hyperbolic_torsion(M, bits_prec=100, all_lifts=False, wada_conventions=False, phi=None) Docstring: Computes the hyperbolic torision polynomial as defined in [DFJ]. >>> M = Manifold('K11n42') >>> M.alexander_polynomial() 1 >>> tau = M.hyperbolic_torsion(bits_prec=200) >>> tau.degree() 6