-
Notifications
You must be signed in to change notification settings - Fork 0
/
esch_space.cpp
321 lines (299 loc) · 12.7 KB
/
esch_space.cpp
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
#include "esch_space.h"
using std::array;
const boost::rational<INT_KS> Space::KS_UNKNOWN = boost::rational<INT_KS>(-1,1);
const boost::rational<INT_KS> Space::KS_UNCOMPUTABLE = boost::rational<INT_KS>(1,1);
#include "aux_math.h"
#include <boost/math/constants/constants.hpp>
#include <boost/math/special_functions/round.hpp>
#include <boost/math/special_functions/sin_pi.hpp>
#include <boost/math/special_functions/cos_pi.hpp>
//////////////////////////////////////////////////
void Space::print(FILE* file) const
{
//M2//fprintf(file, " [%4ld,%4ld,%4ld, %4ld,%4ld,%4ld] -> r = %5ld, s = %7ld, Sigma = %2d, m2 = %d, p1 = %5ld",
//M2// (long)k_[0],(long)k_[1],(long)k_[2],(long)l_[0],(long)l_[1],(long)l_[2],(long)(abs(r_)), (long)s_, (int)(Sigma_), (int)(m2_), (long)p1_);
fprintf(file, " [%4ld,%4ld,%4ld, %4ld,%4ld,%4ld] -> r = %5ld, s = %7ld, Sigma = %2d, p1 = %5ld", //M2//
(long)k_[0],(long)k_[1],(long)k_[2],(long)l_[0],(long)l_[1],(long)l_[2],(long)(abs(r_)), (long)s_, (int)(Sigma_), (long)p1_); //M2//
if (s2_ == KS_UNKNOWN)
fprintf(file, "\n");
else if (s2_ == KS_UNCOMPUTABLE)
fprintf(file, " |!| WARNING: condition C not satisfied |!|\n");
else
{
boost::rational<INT_KS> s22 = this->s22();
fprintf(file, ", s22 = %3lld/%lld, s2 = %7lld/%lld\n", (long long)s22.numerator(), (long long)s22.denominator(), (long long)s2_.numerator(), (long long)s2_.denominator());
}
}
void Space::print(void) const
{
if (is_space())
{
printf("\nInvariants of the Eschenburg space with parameters [%ld,%ld,%ld, %ld,%ld,%ld]:\n",
(long)k_[0],(long)k_[1],(long)k_[2],(long)l_[0],(long)l_[1],(long)l_[2]);
//M2//printf(" r = %ld, s = %ld, Sigma = %d, m2 = %d\n p1 = %ld\n",(long)(abs(r_)), (long)s_, (int)(Sigma_), (int)(m2_), (long)p1_);
printf(" r = %ld, s = %ld, Sigma = %d \n p1 = %ld\n",(long)(abs(r_)), (long)s_, (int)(Sigma_), (long)p1_); //M2//
if (s2_ == KS_UNCOMPUTABLE)
printf("! Condition C is not satisfied !\n");
else
{
boost::rational<INT_KS> s22 = this->s22();
printf(" s2 = %lld/%lld, s22 = %lld/%lld\n", (long long)s2_.numerator(), (long long)s2_.denominator(), (long long)s22.numerator(), (long long)s22.denominator());
}
if (is_positively_curved())
printf(" The space is positively curved.\n\n");
else
printf(" The space is NOT positively curved.\n\n");
}
else
printf("\nThe supplied parameters do not describe an Eschenburg space.\n\n");
}
//////////////////////////////////////////////////
Space::Space(array<INT_P,3> kkk, array<INT_P,3> lll){
k_ = kkk;
l_ = lll;
INT_R sigma1_k = (INT_R)k_[0] + (INT_R)k_[1] + (INT_R)k_[2];
INT_R sigma2_k = (INT_R)k_[0]*(INT_R)k_[1] + (INT_R)k_[0]*(INT_R)k_[2] + (INT_R)k_[1]*(INT_R)k_[2];
INT_R sigma3_k = (INT_R)k_[0]*(INT_R)k_[1]*(INT_R)k_[2];
INT_R sigma1_l = (INT_R)l_[0] + (INT_R)l_[1] + (INT_R)l_[2];
INT_R sigma2_l = (INT_R)l_[0]*(INT_R)l_[1] + (INT_R)l_[0]*(INT_R)l_[2] + (INT_R)l_[1]*(INT_R)l_[2];
INT_R sigma3_l = (INT_R)l_[0]*(INT_R)l_[1]*(INT_R)l_[2];
r_ = sigma2_k - sigma2_l;
s_ = signed_mod(sigma3_k-sigma3_l, abs(r_))*sign(r_);
Sigma_ = signed_mod(sigma1_l,3);
//M2//m2_ = absolute_mod(sigma1_l + sigma2_l,2);
p1_ = absolute_mod(2*sigma1_k*sigma1_k - 6*sigma2_k, abs(r_));
good_col_or_row = GOOD_CoR_UNKNOWN;
s2_ = KS_UNKNOWN;
}
bool Space::is_space(void) const {
// Is it an Eschenburg space?
// -- Test conditions of [CEZ07] (1.1):
using boost::math::gcd;
if(k_[0]+k_[1]+k_[2] != l_[0]+l_[1]+l_[2]) return false;
if(gcd(k_[0] - l_[0], k_[1] - l_[1]) > 1) return false;
if(gcd(k_[0] - l_[0], k_[1] - l_[2]) > 1) return false;
if(gcd(k_[0] - l_[2], k_[1] - l_[0]) > 1) return false;
if(gcd(k_[0] - l_[1], k_[1] - l_[0]) > 1) return false;
if(gcd(k_[0] - l_[1], k_[1] - l_[2]) > 1) return false;
if(gcd(k_[0] - l_[2], k_[1] - l_[1]) > 1) return false;
return true;
}
bool Space::is_positively_curved(void) const {
// Is the space positively curved?
// -- Test conditions of [CEZ07] (1.2):
INT_P min_l = std::min(l_[0],std::min(l_[1],l_[2]));
INT_P max_l = std::max(l_[0],std::max(l_[1],l_[2]));
if (min_l <= k_[0] && k_[0] <= max_l) return false;
if (min_l <= k_[1] && k_[1] <= max_l) return false;
if (min_l <= k_[2] && k_[2] <= max_l) return false;
return true;
}
//////////////////////////////////////////////////
// KS-invariants
boost::rational<INT_KS> Space::s22() const
{
if (s2_ == KS_UNKNOWN || s2_ == KS_UNCOMPUTABLE)
return s2_;
else
return reduce_mod_ZZ( 2*abs(r_)*s2_ );
}
bool Space::compute_KS_invariants() // compute s2 & s22
{
if (test_condition_C())
{
if (good_col_or_row < 3)
compute_s2_from_col(good_col_or_row);
else
compute_s2_from_row(good_col_or_row-3);
return true;
}
else
{
s2_ = KS_UNCOMPUTABLE;
return false;
}
}
bool Space::test_condition_C(void) // check condition C & find good column or row
{
// see "condition C" in [CEZ07]
if (good_col_or_row == GOOD_CoR_UNKNOWN) // condition has not yet been checked
{
good_col_or_row = GOOD_CoR_NONEXISTENT; // if value stays, condition C is not satisfied
for(int i = 0; i < 3; ++i)
{
if(gcdA(i,0,i,1) == 1 && gcdA(i,0,i,2) == 1 && gcdA(i,1,i,2) == 1)
{
good_col_or_row = i + 3;// 3/4/5 = row 1/2/3 satisfies condition C
break;
}
else if(gcdA(0,i,1,i) == 1 && gcdA(0,i,2,i) == 1 && gcdA(1,i,2,i) == 1)
{
good_col_or_row = i;// 0/1/2 = col 1/2/3 satisfies condition C
break;
}
}
}
return (good_col_or_row != GOOD_CoR_NONEXISTENT);
}
int Space::gcdA(int i, int j, int ii, int jj) // helper function for testing condition C
{
// auxiliary function for find_good_row_or_col
/*std::cout << k_[i] << " " << l_[j] << " "
<< k_[ii] << " " << l_[jj] << "--->"
<< boost::math::gcd(k_[i]-l_[j],k_[ii]-l_[jj]) << std::endl;*/
return boost::math::gcd(k_[i]-l_[j],k_[ii]-l_[jj]);
}
void Space::compute_s2_from_col(int j)
{
// see [CEZ07] (2.1):
//
// s2 = (q-2)/d + SUM,
// s22 = 2|r|s2
//
int jp1 = absolute_mod(j+1,2);
//printf("I'm using column %d: ",j+1);
//printf("(%ld,%ld,%ld)\n",k_[0]-l_[j],k_[1]-l_[j],k_[2]-l_[j]);
//printf("jp1 = %d",jp1+1);
INT_KS q =
square(k_[0]-l_[j]) + square(k_[1]-l_[j]) + square(k_[2]-l_[j]) +
square(k_[0]-l_[jp1]) + square(k_[1]-l_[jp1]) + square(k_[2]-l_[jp1])
- square(l_[j]-l_[jp1]);
INT_KS d = 16*3*(INT_KS)(r_)*(k_[0]-l_[j])*(k_[1]-l_[j])*(k_[2]-l_[j]);
s2_.assign(q-2,d);
//printf("q = %lld, r = %lld, d = %lld\n", q, (long long)(r_), d);
//printf("=> (q-2)/d = %lld/%lld\n",s2_.numerator(),s2_.denominator());
for(int i = 0; i < 3; ++i)
{
int ip1 = absolute_mod(i+1,3);
int ip2 = absolute_mod(i+2,3);
array<INT_P,4> params = {k_[ip1]-l_[j], k_[ip2]-l_[j], k_[ip1]-l_[jp1], k_[ip2]-l_[jp1]};
s2_ += -lens_s2(k_[i]-l_[j], params);
}
s2_ = reduce_mod_ZZ(s2_);
//printf("=> s2(E) = %lld/%lld\n",s2_.numerator(),s2_.denominator());
}
void Space::compute_s2_from_row(int j)
{
int jp1 = absolute_mod(j+1,2);
//printf("I'm using row %d",j+1);
//printf(" (%ld,%ld,%ld)\n",k_[j]-l_[0],k_[j]-l_[1],k_[j]-l_[2]);
//printf("jp1 = %d",jp1+1);
INT_KS q =
square(k_[j]-l_[0]) + square(k_[j]-l_[1]) + square(k_[j]-l_[2]) +
square(k_[jp1]-l_[0]) + square(k_[jp1]-l_[1]) + square(k_[jp1]-l_[2])
- square(k_[j]-k_[jp1]);
INT_KS d = 16*3*(INT_KS)(r_)*(k_[j]-l_[0])*(k_[j]-l_[1])*(k_[j]-l_[2]);
s2_.assign(q-2,d);
//printf("q = %lld, d = %lld\n", q, d);
// printf("=> (q-2)/d = %lld/%lld\n",s2_.numerator(),s2_.denominator());
for(int i = 0; i < 3; ++i)
{
// printf("Lens space invariant s_2 for i=%d: \n",i+1);
int ip1 = absolute_mod(i+1,3);
int ip2 = absolute_mod(i+2,3);
array<INT_P,4> params = {k_[j]-l_[ip1], k_[j]-l_[ip2], k_[jp1]-l_[ip1], k_[jp1]-l_[ip2]};
// printf(" parameters: %ld; %ld, %ld, %ld, %ld\n", k_[j]-l_[i], params[0], params[1], params[2], params[3]);
s2_ += lens_s2(k_[j]-l_[i], params);
}
s2_ = reduce_mod_ZZ(s2_);
// printf("=> s2(E) = %lld/%lld\n",s2_.numerator(),s2_.denominator());
}
//////////////////////////////////////////////////
// Lens space invariants:
rational<INT_KS> Space::lens_s2(INT_P p, array<INT_P,4> param)
{
using boost::math::cos_pi;
using boost::math::sin_pi;
FLOAT_KS a = 0;
for(INT_P k = 1; k < abs(p); ++k)
{
FLOAT_KS s =
(cos_pi((FLOAT_KS)2*k/abs(p)) - 1)
/sin_pi((FLOAT_KS)k*param[0]/p)
/sin_pi((FLOAT_KS)k*param[1]/p)
/sin_pi((FLOAT_KS)k*param[2]/p)
/sin_pi((FLOAT_KS)k*param[3]/p);
a += s;
}
/*std::cout << " ";
std::cout << std::setprecision(std::numeric_limits<FLOAT_KS>::max_digits10)
<< a*45
<< std::endl; // print a*/
INT_KS rounded_45_a = (INT_KS)boost::math::round(a*45);
//printf(" round(...) = %lld\n",rounded_45_a);
rational<INT_KS> s2(rounded_45_a,45*16*p);
//printf(" s2 = %lld/%lld\n",s2.numerator(),s2.denominator());
return s2;
}
//////////////////////////////////////////////////
//M2//Space::comp Space::compareHomotopyType(const Space& E1, const Space& E2)
//M2//{
//M2// if (abs(E1.r_ ) > abs(E2.r_ )) return comp::GREATER;
//M2// if (abs(E1.r_ ) < abs(E2.r_ )) return comp::SMALLER;
//M2// if (abs(E1.s_ ) > abs(E2.s_ )) return comp::GREATER;
//M2// if (abs(E1.s_ ) < abs(E2.s_ )) return comp::SMALLER;
//M2// if (abs(E1.Sigma_) > abs(E2.Sigma_)) return comp::GREATER;
//M2// if (abs(E1.Sigma_) < abs(E2.Sigma_)) return comp::SMALLER;
//M2// if ( E1.m2_ > E2.m2_ ) return comp::GREATER;
//M2// if ( E1.m2_ < E2.m2_ ) return comp::SMALLER;
//M2// if (sign(E1.Sigma_)*sign(E1.s_) > sign(E2.Sigma_)*sign(E2.s_)) return comp::GREATER;
//M2// if (sign(E1.Sigma_)*sign(E1.s_) < sign(E2.Sigma_)*sign(E2.s_)) return comp::SMALLER;
//M2// return comp::EQUAL;
//M2//}
//M2//Space::comp Space::compareHomotopyType_using_KS(const Space& E1, const Space& E2)
Space::comp Space::compareHomotopyType(const Space& E1, const Space& E2)
{
if (abs(E1.r_) > abs(E2.r_)) return comp::GREATER;
if (abs(E1.r_) < abs(E2.r_)) return comp::SMALLER;
if (abs(E1.s_) > abs(E2.s_)) return comp::GREATER;
if (abs(E1.s_) < abs(E2.s_)) return comp::SMALLER;
if (abs(E1.Sigma_) > abs(E2.Sigma_)) return comp::GREATER; //M2//
if (abs(E1.Sigma_) < abs(E2.Sigma_)) return comp::SMALLER; //M2//
if (sign(E1.Sigma_)*sign(E1.s_) > sign(E2.Sigma_)*sign(E2.s_)) return comp::GREATER; //M2//
if (sign(E1.Sigma_)*sign(E1.s_) < sign(E2.Sigma_)*sign(E2.s_)) return comp::SMALLER; //M2//
// If both KS-invariants are unkown, return MAYBE_EQUAL;
// otherwise, the unknown KS-invariant is MAYBE_GREATER
// than the known one.
if (E1.s22() == KS_UNKNOWN || E1.s22() == KS_UNCOMPUTABLE)
if (E2.s22() == KS_UNKNOWN || E2.s22() == KS_UNCOMPUTABLE)
return comp::MAYBE_EQUAL;
else return comp::MAYBE_GREATER;
if (E2.s22() == KS_UNKNOWN || E2.s22() == KS_UNCOMPUTABLE)
return comp::MAYBE_SMALLER;
if (abs(E1.s22()) > abs(E2.s22())) return comp::GREATER;
if (abs(E1.s22()) < abs(E2.s22())) return comp::SMALLER;
if (sign(E1.s22())*sign(E1.s_) > sign(E2.s22())*sign(E2.s_)) return comp::GREATER;
if (sign(E1.s22())*sign(E1.s_) < sign(E2.s22())*sign(E2.s_)) return comp::SMALLER;
return comp::EQUAL;
}
Space::comp Space::compareTangentialHomotopyType(const Space& E1, const Space& E2)
{
comp homotopy = compareHomotopyType(E1,E2);
//M2//if (homotopy != comp::EQUAL) return homotopy;
//M2//if (abs(E1.p1_) > abs(E2.p1_)) return comp::GREATER;
//M2//if (abs(E1.p1_) < abs(E2.p1_)) return comp::SMALLER;
//M2//return comp::EQUAL;
if (homotopy == comp::GREATER || homotopy == comp::SMALLER) return homotopy; //M2//
if (abs(E1.p1_) > abs(E2.p1_)) return comp::GREATER; //M2//
if (abs(E1.p1_) < abs(E2.p1_)) return comp::SMALLER; //M2//
return homotopy; //M2//
}
Space::comp Space::compareHomeomorphismType(const Space& E1, const Space& E2)
{
comp tangential_homotopy = compareTangentialHomotopyType(E1,E2);
if (tangential_homotopy != comp::EQUAL) return tangential_homotopy;
// If KS-invariants of both spaces are unkown, return MAYBE_EQUAL;
// otherwise, the unknown KS-invariant is MAYBE_GREATER
// than the known one.
if (E1.s2_ == KS_UNKNOWN || E1.s2_ == KS_UNCOMPUTABLE)
if (E2.s2_ == KS_UNKNOWN || E2.s2_ == KS_UNCOMPUTABLE)
return comp::MAYBE_EQUAL;
else return comp::MAYBE_GREATER;
if (E2.s2_ == KS_UNKNOWN || E2.s2_ == KS_UNCOMPUTABLE)
return comp::MAYBE_SMALLER;
if (abs(E1.s2_) > abs(E2.s2_)) return comp::GREATER;
if (abs(E1.s2_) < abs(E2.s2_)) return comp::SMALLER;
if (sign(E1.s2_)*sign(E1.s_) > sign(E2.s2_)*sign(E2.s_)) return comp::GREATER;
if (sign(E1.s2_)*sign(E1.s_) < sign(E2.s2_)*sign(E2.s_)) return comp::SMALLER;
return comp::EQUAL;
}