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/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2016, 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, write to the Free *
* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, *
* MA 02110-1301, USA. *
* *
**************************************************************************/
#include "algebra/abeliangroup.h"
#include "maths/matrix.h"
#include "manifold/torusbundle.h"
namespace regina {
AbelianGroup* TorusBundle::homology() const {
MatrixInt relns(2, 2);
relns.entry(0, 0) = monodromy_[0][0] - 1;
relns.entry(0, 1) = monodromy_[0][1];
relns.entry(1, 0) = monodromy_[1][0];
relns.entry(1, 1) = monodromy_[1][1] - 1;
AbelianGroup* ans = new AbelianGroup();
ans->addGroup(relns);
ans->addRank();
return ans;
}
std::ostream& TorusBundle::writeName(std::ostream& out) const {
if (monodromy_.isIdentity())
return out << "T x I";
else
return out << "T x I / [ " << monodromy_[0][0] << ','
<< monodromy_[0][1] << " | " << monodromy_[1][0] << ','
<< monodromy_[1][1] << " ]";
}
std::ostream& TorusBundle::writeTeXName(std::ostream& out) const {
if (monodromy_.isIdentity())
return out << "T^2 \\times I";
else
return out << "T^2 \\times I / \\homtwo{" << monodromy_[0][0] << "}{"
<< monodromy_[0][1] << "}{" << monodromy_[1][0] << "}{"
<< monodromy_[1][1] << "}";
}
void TorusBundle::reduce() {
// Make the monodromy prettier.
// In general we are allowed to:
//
// - Replace M with A M A^-1
// - Replace M with M^-1
//
// Some specific tricks we can pull include:
//
// - Rotate the matrix 180 degrees (A = [ 0 1 | 1 0 ])
// - Negate the off-diagonal (A = [ 1 0 | 0 -1 ])
//
// If det == +1, we can also:
//
// - Swap either diagonal individually (invert, then negate the
// off-diagonal, then optionally rotate by 180 degrees)
//
// If det == -1, we can also:
//
// - Simultaneously swap and negate the main diagonal (invert)
// The determinant should be +/-1 according to our preconditions,
// but we'd better check that anyway.
long det = monodromy_.determinant();
if (det != 1 && det != -1) {
// Something is very wrong. Don't touch it.
std::cerr << "ERROR: TorusBundle monodromy does not have "
"determinant +/-1.\n";
return;
}
// Deal with the case where the main diagonal has strictly opposite
// signs.
long x;
if (monodromy_[0][0] < 0 && monodromy_[1][1] > 0) {
// Rotate 180 degrees to put the positive element up top.
rotate();
}
while (monodromy_[0][0] > 0 && monodromy_[1][1] < 0) {
// Set x to the greatest absolute value of any main diagonal element.
if (monodromy_[0][0] >= - monodromy_[1][1])
x = monodromy_[0][0];
else
x = - monodromy_[1][1];
// If we catch any of the following four cases, the main diagonal
// will either no longer have opposite signs, or it will have a
// strictly smaller maximum absolute value.
if (0 < monodromy_[0][1] && monodromy_[0][1] <= x) {
addRCDown();
continue;
} else if (0 < - monodromy_[0][1] && - monodromy_[0][1] <= x) {
subtractRCDown();
continue;
} else if (0 < monodromy_[1][0] && monodromy_[1][0] <= x) {
subtractRCUp();
continue;
} else if (0 < - monodromy_[1][0] && - monodromy_[1][0] <= x) {
addRCUp();
continue;
}
// Since the determinant is +/-1 and neither element of the
// main diagonal is zero, we cannot have both elements of the
// off-diagonal with absolute value strictly greater than x.
// The only remaining possibility is that some element of the
// off-diagonal is zero (and therefore the main diagonal
// contains +1 and -1).
// The non-zero off-diagonal element (if any) can be reduced
// modulo 2. This leaves us with the following possibilities:
// [ 1 0 | 0 -1 ] , [ 1 1 | 0 -1 ], [ 1 0 | 1 -1 ].
// The final two possibilities are both equivalent to [ 0 1 | 1 0 ].
if ((monodromy_[0][1] % 2) || (monodromy_[1][0] % 2)) {
monodromy_[0][0] = monodromy_[1][1] = 0;
monodromy_[0][1] = monodromy_[1][0] = 1;
} else {
monodromy_[0][1] = monodromy_[1][0] = 0;
// The main diagonal elements stay as they are (1, -1).
}
// In these cases we are completely finished.
return;
}
// We are now guaranteed that the main diagonal does not have
// strictly opposite signs.
// Time to arrange the same for the off-diagonal.
// If the off-diagonal has strictly opposite signs, the elements
// must be +1 and -1, and the main diagonal must contain a zero.
// Otherwise there is no way we can get determinant +/-1.
if (monodromy_[0][1] < 0 && monodromy_[1][0] > 0) {
// We have [ a -1 | 1 d ].
// Move the -1 to the bottom left corner by negating the off-diagonal.
monodromy_[0][1] = 1;
monodromy_[1][0] = -1;
}
if (monodromy_[0][1] > 0 && monodromy_[1][0] < 0) {
// We have [ a 1 | -1 d ], where one of a or d is zero.
// Rotate by 180 degrees to move the 0 to the bottom right
// corner, negating the off-diagonal if necessary to preserve
// the 1/-1 positions.
if (monodromy_[1][1]) {
monodromy_[0][0] = monodromy_[1][1];
monodromy_[1][1] = 0;
}
// Now we have [ a 1 | -1 0 ].
if (monodromy_[0][0] > 1) {
addRCDown();
// Everything becomes non-negative.
} else if (monodromy_[0][0] < -1) {
subtractRCUp();
// Everything becomes non-positive.
} else {
// We have [ 1 1 | -1 0 ], [ 0 1 | -1 0 ] or [ -1 1 | -1 0 ].
// All of these are canonical.
return;
}
}
// Neither diagonal has strictly opposite signs.
// Time to give all elements of the matrix the same sign (or zero).
bool allNegative = false;
if (det == 1) {
// Either all non-negative or all non-positive, as determined by
// the main diagonal.
// If it's going to end up negative, just switch the signs for
// now and remember this fact for later on.
if (monodromy_[0][0] < 0 || monodromy_[1][1] < 0) {
allNegative = true;
monodromy_[0][0] = - monodromy_[0][0];
monodromy_[1][1] = - monodromy_[1][1];
}
if (monodromy_[0][1] < 0 || monodromy_[1][0] < 0) {
// We're always allowed to do this.
monodromy_[0][1] = - monodromy_[0][1];
monodromy_[1][0] = - monodromy_[1][0];
}
} else {
// The determinant is -1.
// The entire matrix can be made non-negative.
if (monodromy_[0][0] < 0 || monodromy_[1][1] < 0) {
// Invert (swap and negate the main diagonal).
x = monodromy_[0][0];
monodromy_[0][0] = - monodromy_[1][1];
monodromy_[1][1] = -x;
}
if (monodromy_[0][1] < 0 || monodromy_[1][0] < 0) {
// Negate the off-diagonal as usual.
monodromy_[0][1] = - monodromy_[0][1];
monodromy_[1][0] = - monodromy_[1][0];
}
}
// We now have a matrix whose entries are all non-negative.
// Run through a cycle of equivalent matrices, and choose the nicest.
// I'm pretty sure I can prove that this is a cycle, but the proof
// really should be written down.
Matrix2 start = monodromy_;
Matrix2 best = monodromy_;
while (1) {
// INV: monodromy has all non-negative entries.
// INV: best contains the best seen matrix, including the current one.
// It can be proven (via det = +/-1) that one row must dominate
// another, unless we have [ 1 0 | 0 1 ] or [ 0 1 | 1 0 ].
if (monodromy_.isIdentity()) {
if (allNegative)
monodromy_.negate();
return;
}
if (monodromy_[0][0] == 0 && monodromy_[0][1] == 1 &&
monodromy_[1][0] == 1 && monodromy_[1][1] == 0) {
if (allNegative)
monodromy_.negate();
return;
}
// We know at this point that one row dominates the other.
if (monodromy_[0][0] >= monodromy_[1][0] &&
monodromy_[0][1] >= monodromy_[1][1])
subtractRCUp();
else
subtractRCDown();
// Looking at a new matrix.
if (monodromy_ == start)
break;
if (TorusBundle::simplerNonNeg(monodromy_, best))
best = monodromy_;
}
// In the orientable case, run this all again for the rotated matrix.
// This is not necessary in the non-orientable case since the
// rotated matrix belongs to the same cycle as the original.
if (det > 0) {
rotate();
if (TorusBundle::simplerNonNeg(monodromy_, best))
best = monodromy_;
start = monodromy_;
while (1) {
if (monodromy_.isIdentity()) {
if (allNegative)
monodromy_.negate();
return;
}
if (monodromy_[0][0] == 0 && monodromy_[0][1] == 1 &&
monodromy_[1][0] == 1 && monodromy_[1][1] == 0) {
if (allNegative)
monodromy_.negate();
return;
}
// We know at this point that one row dominates the other.
if (monodromy_[0][0] >= monodromy_[1][0] &&
monodromy_[0][1] >= monodromy_[1][1])
subtractRCUp();
else
subtractRCDown();
// Looking at a new matrix.
if (monodromy_ == start)
break;
if (TorusBundle::simplerNonNeg(monodromy_, best))
best = monodromy_;
}
}
monodromy_ = best;
// Don't forget that negative case.
if (allNegative)
monodromy_.negate();
}
bool TorusBundle::simplerNonNeg(const Matrix2& m1, const Matrix2& m2) {
// Value symmetric matrices above all else.
if (m1[0][1] == m1[1][0] && m2[0][1] != m2[1][0])
return true;
if (m1[0][1] != m1[1][0] && m2[0][1] == m2[1][0])
return false;
// Go for the smallest possible bottom-right element, then so on
// working our way up.
if (m1[1][1] < m2[1][1])
return true;
if (m1[1][1] > m2[1][1])
return false;
if (m1[1][0] < m2[1][0])
return true;
if (m1[1][0] > m2[1][0])
return false;
if (m1[0][1] < m2[0][1])
return true;
if (m1[0][1] > m2[0][1])
return false;
if (m1[0][0] < m2[0][0])
return true;
return false;
}
} // namespace regina
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