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/**************************************************************************
* *
* 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"
namespace regina {
GroupPresentation Link::internalGroup(bool flip, bool simplify) const {
if (crossings_.empty()) {
// This is a zero-crossing unlink.
return { components_.size() };
}
// We have a non-zero number of crossings.
// Build the Wirtinger presentation.
//
// We start with just the generators corresponding to sections of
// the diagram that include crossings; we will pick up any additional
// generators for zero-crossing unknot components when we traverse the
// link shortly.
GroupPresentation g(crossings_.size());
// We will need to number the "segments" - contiguous sections of the link
// that consist entirely of over-crossings (if flip is false), or entirely
// of under-crossings (if flip is true).
// Construct a map from arc IDs to segment IDs, by traversing each
// component one at a time.
FixedArray<size_t> strandToSegment(2 * crossings_.size());
size_t currSegment = 0;
// The kind of strand at which we start a new segment as we traverse.
int breakAt = (flip ? 1 : 0);
for (StrandRef comp : components_) {
if (! comp) {
// This is a zero-crossing unknot component.
g.addGenerator();
continue;
}
// Start our traversal of each component at the beginning of a segment.
StrandRef start = comp;
if (start.strand() != breakAt) {
if (components_.size() == 1) {
// Just jump immediately to the other strand at this crossing.
start.jump();
} else {
// There is no guarantee that the other strand is part of the
// same component. Instead, walk along the component until we
// find a viable starting point.
StrandRef s = start;
do {
++s;
} while (s.strand() != breakAt && s != start);
start = s;
// It is possible that we never found a good starting point.
// This happens when the entire component is an knot with
// no self-crossings that is overlaid above the diagram
// (if flip is false) or under the diagram (if flip is true).
//
// How this affects us now is that the total number of
// segments (i.e., the number of generators in our
// group presentation) goes up by one.
//
// We will adjust this later.
}
}
StrandRef s = start;
do {
strandToSegment[s.id()] = currSegment;
++s;
if (s.strand() == breakAt) {
// We just passed through a crossing that ended a segment.
++currSegment;
}
} while (s != start);
if (start.strand() != breakAt) {
// This is the scenario noted above where some component
// consists entirely of a closed segment that never gets broken.
// We need to make two adjustments:
//
// - increment currSegment, since we are about to move to a new
// component but we did not increment it at the end of the
// loop just now; and
//
// - increment the total number of group generators, since we
// based our original count on the number of crossings, which
// only counts those segments with start and end points.
++currSegment;
g.addGenerator();
}
}
// Now build the presentation.
for (Crossing* c : crossings_) {
GroupExpression exp;
if (flip) {
if (c->sign() < 0) {
exp.addTermLast(strandToSegment[c->lower().id()], 1);
exp.addTermLast(strandToSegment[c->upper().id()], 1);
exp.addTermLast(strandToSegment[c->lower().id()], -1);
exp.addTermLast(strandToSegment[c->upper().prev().id()], -1);
} else {
exp.addTermLast(strandToSegment[c->lower().id()], 1);
exp.addTermLast(strandToSegment[c->upper().prev().id()], 1);
exp.addTermLast(strandToSegment[c->lower().id()], -1);
exp.addTermLast(strandToSegment[c->upper().id()], -1);
}
} else {
if (c->sign() > 0) {
exp.addTermLast(strandToSegment[c->upper().id()], 1);
exp.addTermLast(strandToSegment[c->lower().id()], 1);
exp.addTermLast(strandToSegment[c->upper().id()], -1);
exp.addTermLast(strandToSegment[c->lower().prev().id()], -1);
} else {
exp.addTermLast(strandToSegment[c->upper().id()], 1);
exp.addTermLast(strandToSegment[c->lower().prev().id()], 1);
exp.addTermLast(strandToSegment[c->upper().id()], -1);
exp.addTermLast(strandToSegment[c->lower().id()], -1);
}
}
g.addRelation(std::move(exp));
}
if (simplify)
g.simplify();
return g;
}
GroupPresentation Link::internalExtendedGroup(bool flip, bool simplify) const {
if (crossings_.empty()) {
// This is a zero-crossing unlink.
return { components_.size() + 1 };
}
// We have a non-zero number of crossings.
// Build the Wirtinger-like presentation as given by Silver and Williams.
//
// For strand s, we use generator number s.id() + 1.
// For the special generator x, we use generator number 0.
// For zero-crossing unknot components, we use additional generators
// beyond these indices (which will never appear in any relations).
GroupPresentation g(2 * crossings_.size() + 1 + countTrivialComponents());
for (Crossing* c : crossings_) {
// The first relation is the same regardless of whether we flip.
GroupExpression r1;
if (c->sign() > 0) {
r1.addTermLast(c->upper().id() + 1, 1);
r1.addTermLast(c->lower().id() + 1, 1);
r1.addTermLast(c->upper().prev().id() + 1, -1);
r1.addTermLast(c->lower().prev().id() + 1, -1);
} else {
r1.addTermLast(c->upper().prev().id() + 1, 1);
r1.addTermLast(c->lower().prev().id() + 1, 1);
r1.addTermLast(c->upper().id() + 1, -1);
r1.addTermLast(c->lower().id() + 1, -1);
}
g.addRelation(std::move(r1));
// The second relation changes.
GroupExpression r2;
if (flip) {
if (c->sign() > 0) {
r2.addTermLast(c->lower().id() + 1, 1);
r2.addTermLast(0, 1);
r2.addTermLast(c->lower().prev().id() + 1, -1);
r2.addTermLast(0, -1);
} else {
r2.addTermLast(c->lower().prev().id() + 1, 1);
r2.addTermLast(0, 1);
r2.addTermLast(c->lower().id() + 1, -1);
r2.addTermLast(0, -1);
}
} else {
if (c->sign() > 0) {
r2.addTermLast(c->upper().prev().id() + 1, 1);
r2.addTermLast(0, 1);
r2.addTermLast(c->upper().id() + 1, -1);
r2.addTermLast(0, -1);
} else {
r2.addTermLast(c->upper().id() + 1, 1);
r2.addTermLast(0, 1);
r2.addTermLast(c->upper().prev().id() + 1, -1);
r2.addTermLast(0, -1);
}
}
g.addRelation(std::move(r2));
}
if (simplify)
g.simplify();
return g;
}
} // namespace regina
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