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/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2008, 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 "algebra/nabeliangroup.h"
#include "manifold/ngraphloop.h"
#include "manifold/nsfs.h"
#include "maths/nmatrixint.h"
#include <cstdlib> // For labs().
namespace regina {
NGraphLoop::~NGraphLoop() {
delete sfs_;
}
bool NGraphLoop::operator < (const NGraphLoop& compare) const {
if (*sfs_ < *compare.sfs_)
return true;
if (*compare.sfs_ < *sfs_)
return false;
return simpler(matchingReln_, compare.matchingReln_);
}
NAbelianGroup* NGraphLoop::getHomologyH1() const {
// Just for safety (this should always be true anyway):
if (sfs_->punctures(false) != 2 || sfs_->punctures(true) != 0)
return 0;
// Construct a matrix.
// Generators: fibre, base curves, two base boundaries, exceptional
// fibre boundaries, obstruction boundary,
// reflector boundaries, reflector half-fibres, plus one
// for the loop created by the joining of boundaries.
// Relations: base curve relation, exception fibre relations,
// obstruction relation, reflector relations,
// fibre constraint, joining boundaries.
unsigned long genus = sfs_->baseGenus();
unsigned long fibres = sfs_->fibreCount();
unsigned long ref = sfs_->reflectors();
// If we have an orientable base space, we get two curves per genus.
// The easiest thing to do is just to double the genus now.
if (sfs_->baseOrientable())
genus *= 2;
NMatrixInt m(fibres + ref + 5, genus + fibres + 2 * ref + 5);
unsigned long i, f;
// The relation for the base orbifold:
for (i = 1 + genus; i < 1 + genus + 2 + fibres + 1 + ref; i++)
m.entry(0, i) = 1;
if (! sfs_->baseOrientable())
for (i = 1; i < 1 + genus; i++)
m.entry(0, i) = 2;
// A relation for each exceptional fibre:
NSFSFibre fibre;
for (f = 0; f < fibres; f++) {
fibre = sfs_->fibre(f);
m.entry(f + 1, 1 + genus + 2 + f) = fibre.alpha;
m.entry(f + 1, 0) = fibre.beta;
}
// A relation for the obstruction constant:
m.entry(1 + fibres, 1 + genus + 2 + fibres) = 1;
m.entry(1 + fibres, 0) = sfs_->obstruction();
// A relation for each reflector boundary:
for (i = 0; i < ref; i++) {
m.entry(2 + fibres + i, 0) = -1;
m.entry(2 + fibres + i, 1 + genus + 2 + fibres + 1 + ref + i) = 2;
}
// A relation constraining the fibre. This relationship only
// appears in some cases; otherwise we will just have a (harmless)
// zero row in the matrix.
if (sfs_->reflectors(true))
m.entry(2 + fibres + ref, 0) = 1;
else if (sfs_->fibreReversing())
m.entry(2 + fibres + ref, 0) = 2;
// Two relations for the joining of boundaries:
m.entry(3 + fibres + ref, 0) = -1;
m.entry(3 + fibres + ref, 0) += matchingReln_[0][0];
m.entry(3 + fibres + ref, 2 + genus) = matchingReln_[0][1];
m.entry(4 + fibres + ref, 1 + genus) = -1;
m.entry(4 + fibres + ref, 0) = matchingReln_[1][0];
m.entry(4 + fibres + ref, 2 + genus) = matchingReln_[1][1];
NAbelianGroup* ans = new NAbelianGroup();
ans->addGroup(m);
return ans;
}
std::ostream& NGraphLoop::writeName(std::ostream& out) const {
sfs_->writeName(out);
return out << " / [ " <<
matchingReln_[0][0] << ',' << matchingReln_[0][1] << " | " <<
matchingReln_[1][0] << ',' << matchingReln_[1][1] << " ]";
}
std::ostream& NGraphLoop::writeTeXName(std::ostream& out) const {
sfs_->writeTeXName(out);
return out << "_{\\homtwo{" <<
matchingReln_[0][0] << "}{" << matchingReln_[0][1] << "}{" <<
matchingReln_[1][0] << "}{" << matchingReln_[1][1] << "}}";
}
void NGraphLoop::reduce() {
/**
* Things to observe:
*
* 1. Inverting the matching matrix is harmless (it corresponds to
* rotating the space a half-turn to switch the two boundary tori).
*
* 2. If we add a (1,1) twist to the SFS we can compensate by either:
* - setting row 2 -> row 2 + row 1, or
* - setting col 1 -> col 1 - col 2.
*/
sfs_->reduce(false);
// Bring the SFS obstruction constant back to zero.
long b = sfs_->obstruction();
if (b != 0) {
sfs_->insertFibre(1, -b);
matchingReln_[0][0] += b * matchingReln_[0][1];
matchingReln_[1][0] += b * matchingReln_[1][1];
}
reduce(matchingReln_);
// See if we can do any better by reflecting the entire space
// and adding (1,1) twists to bring the obstruction constant back
// up to zero again.
// TODO: For non-orientable manifolds, reflect()/reduce() may yield
// better results.
NMatrix2 compMatch =
NMatrix2(1, 0, sfs_->fibreCount(), 1) *
NMatrix2(1, 0, 0, -1) *
matchingReln_ *
NMatrix2(1, 0, 0, -1);
reduce(compMatch);
if (simpler(compMatch, matchingReln_)) {
// Do it.
matchingReln_ = compMatch;
sfs_->complementAllFibres();
}
}
void NGraphLoop::reduce(NMatrix2& reln) {
// Reduce both the original and the inverse, and see who comes out
// on top.
reduceBasis(reln);
NMatrix2 inv = reln.inverse();
reduceBasis(inv);
if (simpler(inv, reln))
reln = inv;
}
void NGraphLoop::reduceBasis(NMatrix2& reln) {
// Use (1,1) / (1,-1) pairs to make the top-left element of the
// matrix as close to zero as possible.
if (reln[0][1] != 0 && reln[0][0] != 0) {
long nOps = (labs(reln[0][0]) + ((labs(reln[0][1]) - 1) / 2)) /
labs(reln[0][1]);
if ((reln[0][0] > 0 && reln[0][1] > 0) ||
(reln[0][0] < 0 && reln[0][1] < 0)) {
// Same signs.
for (long i = 0; i < nOps; i++) {
reln[0][0] -= reln[0][1];
reln[1][0] -= reln[1][1];
reln[1][0] -= reln[0][0];
reln[1][1] -= reln[0][1];
}
} else {
// Opposite signs.
for (long i = 0; i < nOps; i++) {
reln[0][0] += reln[0][1];
reln[1][0] += reln[1][1];
reln[1][0] += reln[0][0];
reln[1][1] += reln[0][1];
}
}
// If abs(0,0) is half abs(0,1) then we might do better with yet
// another operation. Check with simpler() in this case.
if (labs(reln[0][0]) * 2 == labs(reln[0][1])) {
NMatrix2 alt = reln;
if ((alt[0][0] > 0 && alt[0][1] > 0) ||
(alt[0][0] < 0 && alt[0][1] < 0)) {
// Same signs.
alt[0][0] -= alt[0][1];
alt[1][0] -= alt[1][1];
alt[1][0] -= alt[0][0];
alt[1][1] -= alt[0][1];
} else {
// Opposite signs.
alt[0][0] += alt[0][1];
alt[1][0] += alt[1][1];
alt[1][0] += alt[0][0];
alt[1][1] += alt[0][1];
}
if (simpler(alt, reln))
reln = alt;
}
} else {
// TODO: We can still do something here.
}
}
} // namespace regina
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