1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
|
/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2009, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public *
* License along with this program; if not, write to the Free *
* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, *
* MA 02110-1301, USA. *
* *
**************************************************************************/
/* end stub */
#include <algorithm>
#include <cmath>
#include <complex>
#include "regina-config.h"
#include "maths/approx.h"
#include "maths/numbertheory.h"
#include "triangulation/ntriangulation.h"
namespace regina {
namespace {
/**
* Allows calculation of [n]! for arbitrary n.
* Values are cached as they are calculated.
*/
class BracketFactorial {
private:
double* fact;
/**< The cached values [0]!, [1]!, ..., [r-1]! . */
double* inv;
/**< The cached inverses of the values stored in fact[]. */
double angle;
/**< The angle arg(q0). */
unsigned long r;
/**< The integer r, for which 2r * angle is some integer
multiple of 2 * Pi. */
public:
/**
* Precalculate all values [0]!, ..., [r-1]!.
*
* Requires r >= 3.
*/
BracketFactorial(double newAngle, unsigned long newR) :
fact(new double[newR]), inv(new double[newR]),
angle(newAngle), r(newR) {
fact[0] = fact[1] = inv[0] = inv[1] = 1.0;
for (unsigned long i = 2; i < r; i++) {
fact[i] = fact[i - 1] * sin(angle * i) / sin(angle);
inv[i] = inv[i - 1] * sin(angle) / sin(angle * i);
}
}
/**
* Clean up memory.
*/
~BracketFactorial() {
delete[] fact;
delete[] inv;
}
/**
* Calculates the single value [n] (note that there is no
* factorial symbol included).
* These values are individually easy to calculate and so
* are not cached.
*/
double bracket(unsigned long index) const {
if (index == 0 || index == 1)
return 1;
return sin(angle * index) / sin(angle);
}
/**
* Returns the value [index]!.
*/
double operator [] (unsigned long index) const {
return (index < r ? fact[index] : 0.0);
}
/**
* Returns the value [index]! ^ -1.
*
* Requires index < r.
*/
double inverse(unsigned long index) const {
return inv[index];
}
};
/**
* Represents the initial data as described in Section 7 of Turaev
* and Viro's paper.
*/
struct InitialData {
unsigned long r;
/**< The integer r. */
double angle;
/**< The angle arg(q0). */
BracketFactorial fact;
/**< The cached values [n]!. */
double vertexContrib;
/**< The vertex-based contribution to the Turaev-Viro invariant;
this is the inverse square of the distinguished value w. */
InitialData(unsigned long newR, double newAngle) :
r(newR), angle(newAngle), fact(angle, r) {
vertexContrib = 2.0 * sin(angle) * sin(angle) / r;
}
/**
* Determines whether (i/2, j/2, k/2) is an admissible triple.
*/
bool isAdmissible(unsigned long i, unsigned long j,
unsigned long k) const {
return ((i + j + k) % 2 == 0) &&
(i <= j + k) && (j <= i + k) && (k <= i + j) &&
(i + j + k <= 2 * (r - 2));
}
/**
* Determines the face-based contribution to the Turaev-Viro
* invariant. This corresponds to +/- Delta(i/2, j/2, k/2)^2.
*/
std::complex<double> faceContrib(unsigned long i, unsigned long j,
unsigned long k) const {
// By admissibility, (i + j + k) is guaranteed to be even.
std::complex<double> ans =
fact[(i + j - k) / 2] *
fact[(j + k - i) / 2] *
fact[(k + i - j) / 2] *
fact.inverse((i + j + k + 2) / 2);
return ((i + j + k) % 4 == 0 ? ans : -ans);
}
/**
* Determines the edge-based contribution to the Turaev-Viro
* invariant. This corresponds to w(i/2)^2.
*/
std::complex<double> edgeContrib(unsigned long i) const {
return (i % 2 == 0 ? fact.bracket(i + 1) : - fact.bracket(i + 1));
}
/**
* Determines the tetrahedron-based contribution to the Turaev-Viro
* invariant. This combines with the square roots of the face-based
* contributions for the four tetrahedron faces to give the symbol
*
* | i/2 j/2 k/2 |
* | l/2 m/2 n/2 | .
*/
std::complex<double> tetContrib(unsigned long i, unsigned long j,
unsigned long k, unsigned long l, unsigned long m,
unsigned long n) {
std::complex<double> ans = 0.0;
unsigned long minZ = i + j + k;
if (minZ < i + m + n)
minZ = i + m + n;
if (minZ < j + l + n)
minZ = j + l + n;
if (minZ < k + l + m)
minZ = k + l + m;
unsigned long maxZ = i + j + l + m;
if (maxZ > i + k + l + n)
maxZ = i + k + l + n;
if (maxZ > j + k + m + n)
maxZ = j + k + m + n;
double term;
for (unsigned long z = minZ; z <= maxZ; z++) {
if (z % 2 != 0)
continue;
// We are guaranteed that z / 2 is an integer.
term = fact[(z + 2) / 2] *
fact.inverse((z - i - j - k) / 2) *
fact.inverse((z - i - m - n) / 2) *
fact.inverse((z - j - l - n) / 2) *
fact.inverse((z - k - l - m) / 2) *
fact.inverse((i + j + l + m - z) / 2) *
fact.inverse((i + k + l + n - z) / 2) *
fact.inverse((j + k + m + n - z) / 2);
if (z % 4 == 0)
ans += term;
else
ans -= term;
}
return ans;
}
};
}
double NTriangulation::turaevViro(unsigned long r, unsigned long whichRoot)
const {
// Have we already calculated this invariant?
std::pair<unsigned long, unsigned long> tvParams(r, whichRoot);
TuraevViroSet::const_iterator it = turaevViroCache.find(tvParams);
if (it != turaevViroCache.end())
return (*it).second;
// Do some basic parameter checks.
if (r < 3)
return 0;
if (whichRoot >= 2 * r)
return 0;
if (gcd(r, whichRoot) > 1)
return 0;
// Set up our initial data.
double angle = (M_PI * whichRoot) / r;
InitialData init(r, angle);
// Run through all admissible colourings.
std::complex<double> ans = 0.0;
unsigned long nEdges = getNumberOfEdges();
unsigned long nFaces = getNumberOfFaces();
unsigned long* colour = new unsigned long[nEdges];
std::fill(colour, colour + nEdges, 0);
long curr = 0;
std::complex<double> valColour;
bool admissible;
std::deque<NEdgeEmbedding>::const_iterator embit;
long index1, index2;
unsigned long i;
while (curr >= 0) {
// Have we found an admissible colouring?
if (curr >= static_cast<long>(nEdges)) {
// Increment ans appropriately.
valColour = 1.0;
for (i = 0; i < vertices.size(); i++)
valColour *= init.vertexContrib;
for (i = 0; i < nEdges; i++)
valColour *= init.edgeContrib(colour[i]);
for (i = 0; i < nFaces; i++)
valColour *= init.faceContrib(
colour[edgeIndex(faces[i]->getEdge(0))],
colour[edgeIndex(faces[i]->getEdge(1))],
colour[edgeIndex(faces[i]->getEdge(2))]);
for (i = 0; i < tetrahedra.size(); i++)
valColour *= init.tetContrib(
colour[edgeIndex(tetrahedra[i]->getEdge(0))],
colour[edgeIndex(tetrahedra[i]->getEdge(1))],
colour[edgeIndex(tetrahedra[i]->getEdge(3))],
colour[edgeIndex(tetrahedra[i]->getEdge(5))],
colour[edgeIndex(tetrahedra[i]->getEdge(4))],
colour[edgeIndex(tetrahedra[i]->getEdge(2))]
);
ans += valColour;
// Step back down one level.
curr--;
if (curr >= 0)
colour[curr]++;
continue;
}
// Have we run out of values to try at this level?
if (colour[curr] > r - 2) {
colour[curr] = 0;
curr--;
if (curr >= 0)
colour[curr]++;
continue;
}
// Does the current value for colour[curr] preserve admissibility?
admissible = true;
const std::deque<NEdgeEmbedding>& embs(edges[curr]->getEmbeddings());
for (embit = embs.begin(); embit != embs.end(); embit++) {
index1 = edgeIndex((*embit).getTetrahedron()->getEdge(
NEdge::edgeNumber[(*embit).getVertices()[0]]
[(*embit).getVertices()[2]]));
index2 = edgeIndex((*embit).getTetrahedron()->getEdge(
NEdge::edgeNumber[(*embit).getVertices()[1]]
[(*embit).getVertices()[2]]));
if (index1 <= curr && index2 <= curr) {
// We've decided upon colours for all three edges of
// this face containing the current edge.
if (! init.isAdmissible(colour[index1], colour[index2],
colour[curr])) {
admissible = false;
break;
}
}
}
// Use the current value for colour[curr] if appropriate;
// otherwise step forwards to the next value.
if (admissible)
curr++;
else
colour[curr]++;
}
delete[] colour;
if (isNonZero(ans.imag())) {
// This should never happen, since the Turaev-Viro invariant is the
// square of the modulus of the Witten invariant for sl_2.
std::cerr <<
"WARNING: The Turaev-Viro invariant has an imaginary component.\n"
" This should never happen.\n"
" Please report this (along with the 3-manifold that"
" was used) to " << PACKAGE_BUGREPORT << "." << std::endl;
}
turaevViroCache[tvParams] = ans.real();
return ans.real();
}
} // namespace regina
|
