2016-06-28-132830.ipynb

Kernel: Python 2 (SageMath)

Random group (to test hyperbolicity)

Note: the groups produced by L.knot_group() are fundamental group of non-compact manifolds. In order to fill the cusps, one needs to add more relations: e.g. x^k for each generator x and k>1.

import snappy
snappy.version()

u'2.4.1'

L = snappy.random_link(50)

G = L.knot_group()

gens = G.gens()

k = 2
rels = list(G.relations())
rels.extend(x**k for x in gens)

F = G.free_group() / rels

F

Finitely presented group < x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29 | x5*x14*x5^-1*x0^-1, x14*x5*x14^-1*x1^-1, x9*x10*x9^-1*x2^-1, x14^-1*x7*x14*x3^-1, x9^-1*x19*x9*x4^-1, x7^-1*x1*x7*x5^-1, x21*x16*x21^-1*x6^-1, x5^-1*x28*x5*x7^-1, x21^-1*x17*x21*x8^-1, x4^-1*x12*x4*x9^-1, x17^-1*x27*x17*x10^-1, x28*x24*x28^-1*x11^-1, x28*x3*x28^-1*x12^-1, x2*x9*x2^-1*x13^-1, x28^-1*x6*x28*x14^-1, x28*x23*x28^-1*x15^-1, x6*x18*x6^-1*x16^-1, x24^-1*x20*x24*x17^-1, x9^-1*x4*x9*x18^-1, x4^-1*x15*x4*x19^-1, x2*x26*x2^-1*x20^-1, x6*x29*x6^-1*x21^-1, x10*x25*x10^-1*x22^-1, x0^-1*x8*x0*x23^-1, x26*x2*x26^-1*x24^-1, x17^-1*x21*x17*x25^-1, x20^-1*x13*x20*x26^-1, x21*x0*x21^-1*x27^-1, x9*x22*x9^-1*x28^-1, x9^-1*x11*x9*x29^-1, x0^2, x1^2, x2^2, x3^2, x4^2, x5^2, x6^2, x7^2, x8^2, x9^2, x10^2, x11^2, x12^2, x13^2, x14^2, x15^2, x16^2, x17^2, x18^2, x19^2, x20^2, x21^2, x22^2, x23^2, x24^2, x25^2, x26^2, x27^2, x28^2, x29^2 >