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
|
/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2025, 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. *
* *
* As an exception, when this program is distributed through (i) the *
* App Store by Apple Inc.; (ii) the Mac App Store by Apple Inc.; or *
* (iii) Google Play by Google Inc., then that store may impose any *
* digital rights management, device limits and/or redistribution *
* restrictions that are required by its terms of service. *
* *
* 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, see <https://www.gnu.org/licenses/>. *
* *
**************************************************************************/
#include "link.h"
#include <algorithm>
namespace regina {
bool Link::hasReducingPass() const {
// Get rid of 0-crossing links.
if (crossings_.empty())
return false;
// The planarity test is expensive (linear time), but finding a reducing
// pass is more expensive (cubic time), and it's best to tell users if
// they're doing things that will give mysteriously wrong answers.
if (! isClassical())
throw FailedPrecondition("Regina can only work with pass moves "
"in classical link diagrams, not virtual link diagrams");
// We consider sides of arcs: for crossing i, we denote:
// 4i = left side of upper outgoing arc
// 4i+1 = right side of upper outgoing arc
// 4i+2 = left side of lower outgoing arc
// 4i+3 = right side of lower outgoing arc
//
// Thus the index is:
// 4 * crossing index + (2 if lower) + (1 if right side)
// We will build an all-pairs shortest paths matrix between sides of
// arcs, where the distance represents the number of strands we must
// cross to get from one (side of arc) to the other. Here we only
// allow crossing a single strand (i.e., we cannot move between
// regions by passing through a crossing).
//
// For links diagrams with multiple disjoint components, our shortest paths
// algorithm will fail to realise that you can jump between components
// (since it essentially works by crawling around the boundary curves of
// cells in the link diagram). As a result, the "shortest paths" between
// disjoint components will be infinity. This does not matter, since a
// pass move always begins in a single component of the diagram, and (as
// we want a _reducing_ pass) it is never beneficial for such a move to
// interact with other components.
size_t nSides = 4 * crossings_.size();
// A distance that is larger than anything we should legitimately see:
size_t infinity = nSides + 1;
// dist[nSides*i + j] denotes the distance from (side of arc) i to j.
FixedArray<size_t> dist(nSides * nSides, infinity);
size_t upperOutgoing, lowerOutgoing, upperIncoming, lowerIncoming;
StrandRef prev;
for (Crossing* c : crossings_) {
upperOutgoing = 4 * c->index();
lowerOutgoing = upperOutgoing + 2;
prev = c->upper().prev();
upperIncoming = 4 * prev.crossing()->index() +
(prev.strand() == 0 ? 2 : 0);
prev = c->lower().prev();
lowerIncoming = 4 * prev.crossing()->index() +
(prev.strand() == 0 ? 2 : 0);
// Identify the initial distance 0 pairs:
if (c->sign() > 0) {
// Clockwise order:
// upper outgoing; lower incoming; upper incoming; lower outgoing
dist[nSides * (upperOutgoing + 1) + (lowerIncoming + 1)] =
dist[nSides * (lowerIncoming + 1) + (upperOutgoing + 1)] =
dist[nSides * (lowerIncoming) + (upperIncoming + 1)] =
dist[nSides * (upperIncoming + 1) + (lowerIncoming)] =
dist[nSides * (upperIncoming) + (lowerOutgoing)] =
dist[nSides * (lowerOutgoing) + (upperIncoming)] =
dist[nSides * (lowerOutgoing + 1) + (upperOutgoing)] =
dist[nSides * (upperOutgoing) + (lowerOutgoing + 1)] = 0;
} else {
// Clockwise order:
// upper outgoing; lower outgoing; upper incoming; lower incoming
dist[nSides * (upperOutgoing + 1) + (lowerOutgoing)] =
dist[nSides * (lowerOutgoing) + (upperOutgoing + 1)] =
dist[nSides * (lowerOutgoing + 1) + (upperIncoming + 1)] =
dist[nSides * (upperIncoming + 1) + (lowerOutgoing + 1)] =
dist[nSides * (upperIncoming) + (lowerIncoming + 1)] =
dist[nSides * (lowerIncoming + 1) + (upperIncoming)] =
dist[nSides * (lowerIncoming) + (upperOutgoing)] =
dist[nSides * (upperOutgoing) + (lowerIncoming)] = 0;
}
}
// Identify the initial distance 1 pairs, which are just the
// matching left/right pairs:
size_t i, j, k;
for (i = 0; i < nSides; i += 2)
dist[nSides * i + (i + 1)] =
dist[nSides * (i + 1) + i] = 1;
// Run Floyd-Warshall to compute all-pairs shortest paths:
for (k = 0; k < nSides; ++k)
for (i = 0; i < nSides; ++i)
for (j = 0; j < nSides; ++j) {
// To get from i to j, go via k if this helps.
if (dist[nSides * i + j] >
dist[nSides * i + k] + dist[nSides * k + j])
dist[nSides * i + j] =
dist[nSides * i + k] + dist[nSides * k + j];
}
#if 0
for (i = 0; i < nSides; ++i)
for (j = i + 1; j < nSides; ++j) {
std::cout << ((i & 2) ? '_' : '^') << (i / 4)
<< ((i & 1) ? '>' : '<') << " <--> ";
std::cout << ((j & 2) ? '_' : '^') << (j / 4)
<< ((j & 1) ? '>' : '<') << " : ";
std::cout << dist[nSides * i + j] << " == "
<< dist[nSides * j + i] << std::endl;
}
#endif
// Now we need to identify maximal sequences of strands that either
// (i) only involve over-crossings, or (ii) only involve under-crossings;
// and then see compare the number of crossings in these sequences
// against the corresponding shortest-paths entries in dist[].
// We make two passes through each link component: once looking for
// sequences of over-crossings, and then once looking for sequences of
// under-crossings. This is wasteful, but I've been flying for 28 hours
// now and this increases my chances of getting it right the first time. :/
for (auto comp : components_) {
// Zero-crossing components do not have reducing pass moves.
if (! comp)
continue;
// Start with over-crossings. This requires us to begin our
// traversal from an under-crossing.
StrandRef start = underForComponent(comp);
if (! start) {
// This component is a zero-crossing knot placed above the rest of
// the diagram, which means the entire knot can be slid away.
// This can be viewed as a reducing pass.
return true;
}
StrandRef s = start;
StrandRef beginSeq; // Arc from under-crossing to over-crossing
StrandRef endSeq; // Arc from over-crossing to under-crossing
size_t seqLen;
do {
if (s.strand() == 0) {
// This is always run in the first iteration of the loop,
// thereby initialising beginSeq and seqLen.
beginSeq = s;
seqLen = 0;
} else
++seqLen;
++s;
if (s.strand() == 0) {
endSeq = s.prev();
if (seqLen > 0) {
// Compare this to the shortest path in dist[].
// Note: we know that beginSeq is a (lower outgoing) arc,
// and that endSeq is an (upper outgoing) arc.
i = 4 * beginSeq.crossing()->index() + 2;
j = 4 * endSeq.crossing()->index();
if (dist[nSides * i + j] < seqLen ||
dist[nSides * i + j + 1] < seqLen ||
dist[nSides * (i + 1) + j] < seqLen ||
dist[nSides * (i + 1) + j + 1] < seqLen) {
return true;
}
}
}
} while (s != start);
// Now look for sequences of under-crossings.
// This time we must begin our traversal from an over-crossing.
start = overForComponent(comp);
if (! start) {
// This component is a zero-crossing knot placed below the rest of
// the diagram. Again this can be viewed as a reducing pass.
return true;
}
s = start;
do {
if (s.strand() == 1) {
// This is always run in the first iteration of the loop,
// thereby initialising beginSeq and seqLen.
beginSeq = s;
seqLen = 0;
} else
++seqLen;
++s;
if (s.strand() == 1) {
endSeq = s.prev();
if (seqLen > 0) {
// Compare this to the shortest path in dist[].
// Note: we know that beginSeq is an (upper outgoing) arc,
// and that endSeq is a (lower outgoing) arc.
i = 4 * beginSeq.crossing()->index();
j = 4 * endSeq.crossing()->index() + 2;
if (dist[nSides * i + j] < seqLen ||
dist[nSides * i + j + 1] < seqLen ||
dist[nSides * (i + 1) + j] < seqLen ||
dist[nSides * (i + 1) + j + 1] < seqLen) {
return true;
}
}
}
} while (s != start);
}
// Nothing found!
return false;
}
} // namespace regina
|
