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nloop_num.py
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from snappy import *
from random import randrange
from collections import OrderedDict
import multiprocessing
class NeumannZagierDatum():
"""
description of class, comment entire class
"""
def __init__(self, manifold, engine=None, verbose=False, file_name=None):
self.manifold = manifold
self.engine = engine
self.verbose = verbose
self.file_name = file_name
self._raw_gluing_equations = manifold.gluing_equations()
self.num_shapes = self._raw_gluing_equations.ncols() // 3
self.num_eqns = self._raw_gluing_equations.nrows()
self._eliminated_shapes = self.num_shapes * [1, ]
self.computed_ptolemy = False
self.nz = None
pari.set_real_precision(100)
def every_third_elem(L, shift):
return [L[i] for i in range(shift, len(L), 3)]
def epsilon(CC, d):
return 2**(-CC.precision()//d)
def all_shape_parameters(self, z):
return [z, 1 / (1 - z), 1 - 1 / z]
def in_threes(self, L):
return [L[3 * i : 3 * (i + 1)] for i in range(len(L) // 3)]
def shift_in_threes(self, L, shifts):
return sum([X[s:] + X[:s] for X, s in zip(self.in_threes(L),\
shifts)], [])
def is_geom(self, M, c):
vol = M.volume()
for v in c.volume_numerical():
if abs(v - vol) < 1e-10:
return True
return False
def gluing_equations(self):
eqns = self._raw_gluing_equations
new_cols = self.shift_in_threes(eqns.columns(), [(i - 1) % 3 for\
i in self._eliminated_shapes])
return matrix(new_cols).transpose()
def ABCbar(self):
eqns = self.gluing_equations()
n = self.num_shapes
return [eqns.matrix_from_columns(range(i, 3 * n, 3))\
for i in range(3)]
def target_vector(self):
"""
Answer times pi*i is right-hand side of
gluing equations.
"""
m = self.num_eqns
c = self.manifold.num_cusps()
return vector(ZZ, [2 for i in range(m - 2 * c)] + (2 * c) * [0])
def ABv(self):
A, B, C = self.ABCbar()
one = vector(B.base_ring(), B.ncols() * [1])
return A - B, C - B, self.target_vector() - B * one
def shapes(self, precision=None, shapes=None):
if shapes:
base_shapes = shapes
else:
if precision == None:
base_shapes = [CC(z) for z in\
self.manifold.tetrahedra_shapes(part='rect')]
else:
base_shapes = hypertorsion.snap.\
polished_tetrahedra_shapes(self.manifold,\
precision)
return [self.all_shape_parameters(z)[(i - 1) % 3]
for z, i in zip(base_shapes, self._eliminated_shapes)]
def ABv_square(self, rows_to_eliminate=None):
A, B, v = self.ABv()
M = block_matrix([[B, A, v.column()]])
c = self.manifold.num_cusps()
rows = range(M.nrows())[:-2 * c]
rows += [2 * i + rows[-1] + 1 for i in range(c)]
M = M.matrix_from_rows(rows)
M = M.hermite_form(include_zero_rows=False)
n = A.ncols()
return M.matrix_from_columns(range(n, 2 * n)),\
M.matrix_from_columns(range(n)), M.columns()[-1]
def f_and_fddot(self):
A, B, v = self.ABv_square()
n = A.ncols()
M = block_matrix([[A, B]])
S, U, V = M.smith_form()
d = S.diagonal()
f = V * vector(ZZ, [x / y for x, y in zip(U * v, d)] + n * [0])
assert M * f == v
return vector(f[:n]), vector(f[n:])
def make_B_nondegenerate(self):
while det(self.ABv_square()[1]) == 0:
self._eliminated_shapes = [randrange(3) for\
i in range(self.num_shapes)]
def compute_ptolemy_field_and_embedding(self):
vol = self.manifold.volume()
p = self.manifold.ptolemy_variety(2, 'all')
if self.computed_ptolemy == False:
if self.engine == "retrieve":
try:
s = p.retrieve_solutions(verbose=self.verbose)\
.flatten(depth=2)
except:
s = p.compute_solutions(engine=self.engine,\
verbose=self.verbose).flatten(depth=2)
else:
s = p.compute_solutions(engine=self.engine,\
verbose=self.verbose).flatten(depth=2)
self.computed_ptolemy = s
else:
s = self.computed_ptolemy
for sol in s:
rsol = zip(pari('polroots(%s)' %sol.number_field()),\
sol.numerical())
for root, numerical_sol in rsol:
if abs(vol - numerical_sol.volume_numerical()) < 1e-10:
return sol.number_field(), root
def check(self):
shapes = vector(self.all_log_shapes())
CC = shapes[0].parent()
eqns = self.gluing_equations()
pi_I = CC.pi() * CC.gen()
error = eqns * shapes - pi_I * self.target_vector()
assert error.norm(Infinity) < epsilon(CC, 2)
A, B, v = self.ABv()
z = vector(every_third_elem(shapes, 0))
z_ddot = vector(every_third_elem(shapes, 2))
error = A * z + B * z_ddot - pi_I * v
assert error.norm(Infinity) < epsilon(CC, 2)
def generate_nz_data(self):
self.make_B_nondegenerate()
temp_ABv = self.ABv_square()
A = temp_ABv[0]
B = temp_ABv[1]
nu = temp_ABv[2]
temp_fs = self.f_and_fddot()
f = temp_fs[0]
f_ddot = temp_fs[1]
new_shapes = self.shapes()
pol, embedding = self.compute_ptolemy_field_and_embedding()
new_nz = (A, B, nu, f, f_ddot, pol, new_shapes, embedding)
self.nz = new_nz
if self.file_name != None:
save(new_nz, self.file_name)
class nloop():
"""
Compute the n-loop invariant S_n.
Reference: ``The Quantum Content of the Gluing Equations'' by
Dimofte and Garoufalidis.
"""
def __init__(self, nzdata, n, diagrams):
"""Initializes class variables."""
(A, B, nu, f, f_ddot, _, zees, _) = nzdata
self.A = A
self.B = B
self.nu = nu
self.f = f
self.f_ddot = f_ddot
self.zees = zees
self.CC = self.zees[0].parent()
self.prec = self.zees[0].prec()
self.n = n
self.diagrams = [g for g in diagrams if \
self.feynman_loop_number(g) <= self.n]
self.ver_factor = None
self.prev = OrderedDict()
def exponentiate_list(self, L, E):
return prod([l ** e for l, e in zip(L, E)])
def one_loop(self, precision=None, shapes=None):
CC = self.zees[0].parent()
shapes_dd = [1 - 1 / z for z in self.zees]
D1 = diagonal_matrix(shapes_dd)
D2 = diagonal_matrix([1 / z for z in self.zees])
return (1 / CC(2)) * det(self.A * D1 + self.B * D2) *\
self.exponentiate_list(self.zees, self.f_ddot) *\
self.exponentiate_list(shapes_dd, -self.f)
def pre_comp_polylog(self, index, z):
"""
This function contains the polylogs commonly used
in the calculation of the n-loop invariant.
These polylogs are used a large number of times
in a computation so they are saved to minimize waste.
"""
if index == 1:
return -ln(1 - z)
if index == 0:
return z / (1 - z)
if index == -1:
return z / (z ** 2 - 2 * z + 1)
if index == -2:
return (-z ** 2 - z) / (z ** 3 - 3 * z ** 2 + 3 * z - 1)
if index == -3:
return (z ** 3 + 4 * z ** 2 + z) / (z ** 4 - 4 * z ** 3 +\
6 * z ** 2 - 4 * z + 1)
if index == -4:
return (-z ** 4 - 11 * z ** 3 - 11 * z ** 2 - z) / (z **\
5 - 5 * z ** 4 + 10 * z ** 3 - 10 * z ** 2 + 5 * z - 1)
if index == -5:
return (z ** 5 + 26 * z ** 4 + 66 * z ** 3 + 26 * z ** 2 +\
z) / (z ** 6 - 6 * z ** 5 + 15 * z ** 4 - 20 * z ** 3 +\
15 * z ** 2 - 6 * z + 1)
if index == -6:
return (-z ** 6 - 57 * z ** 5 - 302 * z ** 4 - 302 * z **\
3 - 57 * z ** 2 - z) / (z ** 7 - 7 * z ** 6 + 21 *\
z ** 5 - 35 * z ** 4 + 35 * z ** 3 -\
21 * z ** 2 + 7 * z - 1)
if index == -7:
return (z ** 7 + 120 * z ** 6 + 1191 * z ** 5 + 2416 *\
z ** 4 + 1191 * z ** 3 + 120 * z ** 2 + z) / (z **\
8 - 8 * z ** 7 + 28 * z ** 6 - 56 * z ** 5 + 70 *\
z ** 4 - 56 * z ** 3 + 28 * z ** 2 - 8 * z + 1)
if index == -8:
return (-z ** 8 - 247 * z ** 7 - 4293 * z ** 6 -\
15619 * z ** 5 - 15619 * z ** 4 - 4293 * z **\
3 - 247 * z ** 2 - z) / (z ** 9 - 9 * z ** 8 +\
36 * z ** 7 - 84 * z ** 6 + 126 * z ** 5 - 126 *\
z ** 4 + 84 * z ** 3 - 36 * z ** 2 + 9 * z - 1)
if index == -9:
return (z ** 9 + 502 * z ** 8 + 14608 * z ** 7 +\
88234 * z ** 6 + 156190 * z ** 5 + 88234 * z ** 4 +\
14608 * z ** 3 + 502 * z ** 2 + z) / (z ** 10 -\
10 * z ** 9 + 45 * z ** 8 - 120 * z ** 7 + 210 * z **\
6 - 252 * z ** 5 + 210 * z ** 4 - 120 * z ** 3 + 45 *\
z ** 2 - 10 * z + 1)
if index == -10:
return (-z ** 10 - 1013 * z ** 9 - 47840 * z ** 8 -\
455192 * z ** 7 - 1310354 * z ** 6 - 1310354 * z ** 5 -\
455192 * z ** 4 - 47840 * z ** 3 - 1013 * z ** 2 - z) /\
(z ** 11 - 11 * z ** 10 + 55 * z ** 9 - 165 * z ** 8 +\
330 * z ** 7 - 462 * z ** 6 + 462 * z ** 5 - 330 * z **\
4 + 165 * z ** 3 - 55 * z ** 2 + 11 * z - 1)
if index == -11:
return (z ** 11 + 2036 * z ** 10 + 152637 * z ** 9 +\
2203488 * z ** 8 + 9738114 * z**7 + 15724248 *\
z ** 6 + 9738114 * z ** 5 + 2203488 * z ** 4 +\
152637 * z ** 3 + 2036 * z ** 2 + z) / (z ** 12 - 12 *\
z ** 11 + 66 * z ** 10 - 220 * z ** 9 + 495 * z ** 8 -\
792 * z ** 7 + 924 * z ** 6 - 792 * z ** 5 + 495 * z **\
4 - 220 * z ** 3 + 66 * z ** 2 - 12 * z + 1)
if index == -12:
return (-z ** 12 - 4083 * z ** 11 - 478271 * z ** 10 -\
10187685 * z ** 9 - 66318474 * z ** 8 - 162512286 *\
z ** 7 - 162512286 * z ** 6 - 66318474 * z ** 5 -\
10187685 * z ** 4 - 478271 * z ** 3 - 4083 * z **\
2 - z) / (z ** 13 - 13 * z ** 12 + 78 * z ** 11 -\
286 * z ** 10 + 715 * z ** 9 - 1287 * z ** 8 + 1716 *\
z ** 7 - 1716 * z ** 6 + 1287 * z ** 5 - 715 * z **\
4 + 286 * z ** 3 - 78 * z ** 2 + 13 * z - 1)
if index == -13:
return (z ** 13 + 8178 * z ** 12 + 1479726 * z ** 11 +\
45533450 * z ** 10 + 423281535 * z ** 9 + 1505621508 *\
z ** 8 + 2275172004 * z ** 7 + 1505621508 * z ** 6 +\
423281535 * z ** 5 + 45533450 * z ** 4 + 1479726 *\
z ** 3 + 8178 * z ** 2 + z) / (z ** 14 - 14 * z ** 13 +\
91 * z ** 12 - 364 * z ** 11 + 1001 * z ** 10 - 2002 *\
z ** 9 + 3003 * z ** 8 - 3432 * z ** 7 + 3003 * z **\
6 - 2002 * z ** 5 + 1001 * z ** 4 - 364 * z ** 3 +\
91 * z ** 2 - 14 * z + 1)
if index == -14:
return (-z ** 14 - 16369 * z ** 13 - 4537314 * z ** 12 -\
198410786 * z ** 11 - 2571742175 * z ** 10 -\
12843262863 * z ** 9 - 27971176092 * z ** 8 -\
27971176092 * z ** 7 - 12843262863 * z ** 6 -\
2571742175 * z ** 5 - 198410786 * z ** 4 - 4537314 *\
z ** 3 - 16369 * z ** 2 - z) / (z ** 15 - 15 * z **\
14 + 105 * z ** 13 - 455 * z ** 12 + 1365 * z ** 11 -\
3003 * z ** 10 + 5005 * z ** 9 - 6435 * z ** 8 +\
6435 * z ** 7 - 5005 * z ** 6 + 3003 * z ** 5 -\
1365 * z ** 4 + 455 * z ** 3 - 105 * z ** 2 + 15 *\
z - 1)
if index == -15:
return (z ** 15 + 32752 * z ** 14 + 13824739 * z ** 13 +\
848090912 * z ** 12 + 15041229521 * z ** 11 +\
102776998928 * z ** 10 + 311387598411 * z ** 9 +\
447538817472 * z ** 8 + 311387598411 * z ** 7 +\
102776998928 * z ** 6 + 15041229521 * z ** 5 +\
848090912 * z ** 4 + 13824739 * z ** 3 + 32752 * z **\
2 + z) / (z ** 16 - 16 * z ** 15 + 120 * z ** 14 -\
560 * z ** 13 + 1820 * z ** 12 - 4368 * z ** 11 +\
8008 * z ** 10 - 11440 * z ** 9 + 12870 * z ** 8 -\
11440 * z ** 7 + 8008 * z ** 6 - 4368 * z ** 5 +\
1820 * z ** 4 - 560 * z ** 3 + 120 * z ** 2 -\
16 * z + 1)
if index == -16:
return (-z ** 16 - 65519 * z ** 15 - 41932745 * z ** 14 -\
3572085255 * z ** 13 - 85383238549 * z ** 12 -\
782115518299 * z ** 11 - 3207483178157 * z ** 10 -\
6382798925475 * z ** 9 - 6382798925475 * z ** 8 -\
3207483178157 * z ** 7 - 782115518299 * z ** 6 -\
85383238549 * z ** 5 - 3572085255 * z ** 4 -\
41932745 * z ** 3 - 65519 * z ** 2 - z) / (z **\
17 - 17 * z ** 16 + 136 * z ** 15 - 680 * z **\
14 + 2380 * z ** 13 - 6188 * z ** 12 + 12376 *\
z ** 11 - 19448 * z ** 10 + 24310 * z ** 9 - 24310 *\
z ** 8 + 19448 * z ** 7 - 12376 * z ** 6 + 6188 * z **\
5 - 2380 * z ** 4 + 680 * z ** 3 - 136 * z ** 2 +\
17 * z - 1)
if index == -17:
return (z ** 17 + 131054 * z ** 16 + 126781020 * z ** 15 +\
14875399450 * z ** 14 + 473353301060 * z ** 13 +\
5717291972382 * z ** 12 + 31055652948388 * z ** 11 +\
83137223185370 * z ** 10 + 114890380658550 * z ** 9 +\
83137223185370 * z ** 8 + 31055652948388 * z ** 7 +\
5717291972382 * z ** 6 + 473353301060 * z ** 5 +\
14875399450 * z ** 4 + 126781020 * z ** 3 + 131054 *\
z ** 2 + z) / (z ** 18 - 18 * z ** 17 + 153 * z **\
16 - 816 * z ** 15 + 3060 * z ** 14 - 8568 * z **\
13 + 18564 * z ** 12 - 31824 * z ** 11 + 43758 *\
z ** 10 - 48620 * z ** 9 + 43758 * z ** 8 - 31824 *\
z ** 7 + 18564 * z ** 6 - 8568 * z ** 5 + 3060 *\
z ** 4 - 816 * z ** 3 + 153 * z ** 2 - 18 * z + 1)
if index == -18:
return (-z ** 18 - 262125 * z ** 17 - 382439924 * z ** 16 -\
61403313100 * z ** 15 - 2575022097600 * z ** 14 -\
40457344748072 * z ** 13 - 285997074307300 * z ** 12 -\
1006709967915228 * z ** 11 - 1865385657780650 * z **\
10 - 1865385657780650 * z ** 9 - 1006709967915228 *\
z ** 8 - 285997074307300 * z ** 7 - 40457344748072 *\
z ** 6 - 2575022097600 * z ** 5 - 61403313100 * z **\
4 - 382439924 * z ** 3 - 262125 * z ** 2 - z) / (z **\
19 - 19 * z ** 18 + 171 * z ** 17 - 969 * z ** 16 +\
3876 * z ** 15 - 11628 * z ** 14 + 27132 * z ** 13 -\
50388 * z ** 12 + 75582 * z ** 11 - 92378 * z ** 10 +\
92378 * z ** 9 - 75582 * z ** 8 + 50388 * z ** 7 -\
27132 * z ** 6 + 11628 * z ** 5 - 3876 * z ** 4 +\
969 * z ** 3 - 171 * z ** 2 + 19 * z - 1)
if index == -19:
return (z ** 19 + 524268 * z ** 18 + 1151775897 * z ** 17 +\
251732291184 * z ** 16 + 13796160184500 * z ** 15 +\
278794377854832 * z ** 14 + 2527925001876036 * z **\
13 + 11485644635009424 * z ** 12 + 27862280567093358 *\
z ** 11 + 37307713155613000 * z ** 10 +\
27862280567093358 * z ** 9 + 11485644635009424 * z **\
8 + 2527925001876036 * z ** 7 + 278794377854832 * z **\
6 + 13796160184500 * z ** 5 + 251732291184 * z ** 4 +\
1151775897 * z ** 3 + 524268 * z ** 2 + z) / (z **\
20 - 20 * z ** 19 + 190 * z ** 18 - 1140 * z ** 17 +\
4845 * z ** 16 - 15504 * z ** 15 + 38760 * z ** 14 -\
77520 * z ** 13 + 125970 * z ** 12 - 167960 * z **\
11 + 184756 * z ** 10 - 167960 * z ** 9 + 125970 *\
z ** 8 - 77520 * z ** 7 + 38760 * z ** 6 - 15504 *\
z ** 5 + 4845 * z ** 4 - 1140 * z ** 3 + 190 * z **\
2 - 20 * z + 1)
def feynman_loop_number(self, diagram):
"""
Calculate the Feynman Loop Number of a Diagram.
The Feynman Loop Number of a connected looped multigraph
is the number of 1-vertices+2-vertices + the number of loops
"""
if diagram.num_edges() == 0:
return 0
return diagram.degree().count(1) + diagram.degree().count(2) +\
diagram.num_edges() - diagram.num_verts() + 1
def symmetry_factor(self, diag):
"""
Calculate the symmetry factor of a diagram.
This is equal to the order of the group of vertex
permutations preserving edges times k! for each
k-multiedge times 2^number of loops
"""
symfactor = diag.automorphism_group().cardinality()
for foo in diag.vertices():
for bar in range(foo, diag.num_verts()):
conecs = diag.adjacency_matrix()[foo][bar]
symfactor = kronecker_delta(foo, bar) * symfactor * 2 **\
conecs * factorial(conecs) + (1 - kronecker_delta(\
foo, bar)) * symfactor * factorial(conecs)
return QQ(1) / symfactor
def bernoulli_plus_half(self, m):
"""Return bernoulli number with convention B1=+1/2."""
return bernoulli(m) * (-1) ** m
def polylogarithm(self, index, z):
"""Give the nth polylogarithm evaluated at z."""
_ = gp.set_precision(self.prec)
if (index, z) in self.prev:
return self.prev[(index, z)]
if index <= 1 and index >= -19:
tmp = self.CC(self.pre_comp_polylog(index, z))
self.prev[(index, z)] = tmp
if len(self.prev) > 1000:
self.prev.popitem(last=False)
return tmp
return self.CC(gp.subst(gp(polylog(index, x)), x, z))
def gamma(self, eye, kay, ell):
"""Return the gamma equation for vertex_factor_tensor."""
if kay == 0:
return sum([self.polylogarithm(2 - self.n, 1 / z) for z \
in self.zees]) * self.bernoulli_plus_half(self.n) /\
factorial(self.n) + kronecker_delta(self.n, 2) * \
(self.f * self.B.inverse() * self.A *\
self.f / 8)[0][0]
return (-1) ** kay * sum([h ** (bar - 1) / factorial(bar) *
self.bernoulli_plus_half(bar) * self.polylogarithm(\
2 - bar - kay, 1 / self.zees[eye]) for bar in \
range((kronecker_delta(kay, 1) + kronecker_delta(kay, 2)),\
1 + (kronecker_delta(kay, 1) + kronecker_delta(kay, 2)) + \
self.n - ell)]) - kronecker_delta(kay, 1) * self.CC(0.5) * \
(self.B.inverse() * self.nu)[eye]
def vertex_factor_tensor(self):
"""
Generate vertex gamma as a tensor access values.
Output is in the form
vertexgamma[feynman_loop_number][vertex_degree][ith_shape_parameter]
"""
var('h')
return [[[self.gamma(eye, kay, ell) for eye in
range(len(self.zees))] for kay in range(2 * self.n + 1)]\
for ell in range(self.n + 1)]
def diagram_contribution_to_nloop(self, diagram):
"""The diagram contribution to the n-loop invariant."""
N = len(self.zees)
hamil = -self.B.inverse() * self.A + diagonal_matrix([1 / (1 - z)\
for z in self.zees])
prop = h * hamil.inverse()
temp_sum = self.CC(0)
for foo in range(N ** diagram.num_verts()):
indices = [floor(foo / (N ** bar)) % N for bar in
range(diagram.num_verts())]
temp_sum += prod([prop[indices[eee[0]]]
[indices[eee[1]]] for eee in diagram.edges
(labels=False)] +
[self.ver_factor[self.feynman_loop_number(diagram)]
[diagram.degree()[vee]][indices[vee]] for vee in
diagram.vertices()]).expand()
return self.symmetry_factor(diagram) *\
temp_sum.coeff(h, self.n - 1)
def nloop_invariant(self):
"""The Dimofte-Garoufalidis n-loop invariant."""
self.ver_factor = self.vertex_factor_tensor()
PROCESSES = multiprocessing.cpu_count()
#print 'cpu_count() = %d\n' % multiprocessing.cpu_count()
pool = multiprocessing.Pool(PROCESSES)
collect_results = pool.map(self.diagram_contribution_to_nloop,\
self.diagrams)
loop_invar = sum(collect_results) + self.ver_factor[self.n][0][0]
return loop_invar
def nloop_from_manifold(manifold, n, diagrams, engine=None, verbose=False,\
file_name=None):
D = NeumannZagierDatum(manifold, engine, verbose, file_name)
D.generate_nz_data()
E = nloop(D.nz, n, all_diagrams)
if n == 1:
return [E.one_loop(), D.nz]
return [E.nloop_invariant(), D.nz]
def nloop_from_nzdatum(nz, n, diagrams):
E = nloop(nz, n, all_diagrams)
if n == 1:
return [E.one_loop(), nz]
return [E.nloop_invariant(), nz]