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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 <set>
#include "triangulation/dim2.h"
#include "triangulation/dim3.h"
#include "triangulation/dim4.h"
namespace regina {
namespace {
// A helper routine that uses union-find to test whether a graph
// contains cycles. This is used by Triangulation<4>::collapseEdge().
//
// This routine returns true if the given edge connects two distinct
// components of the graph, or false if both endpoints of the edge
// are already in the same component (i.e., a cycle has been created).
//
// This routine is a direct clone of the corresponding routine for
// Triangulation<3>::collapseEdge(). It would be nice to reuse this
// routine instead of duplicate it, but that is for another day.
bool unionFindInsert(ssize_t* parent, size_t* depth,
size_t vtx1, size_t vtx2) {
// Find the root of the tree containing vtx1 and vtx2.
ssize_t top1, top2;
for (top1 = vtx1; parent[top1] >= 0; top1 = parent[top1])
;
for (top2 = vtx2; parent[top2] >= 0; top2 = parent[top2])
;
// Are both vertices in the same component?
if (top1 == top2)
return false;
// Join the two components.
// Insert the shallower tree beneath the deeper tree.
if (depth[top1] < depth[top2]) {
parent[top1] = top2;
} else {
parent[top2] = top1;
if (depth[top1] == depth[top2])
++depth[top1];
}
return true;
}
}
bool Triangulation<4>::internal44(Edge<4>* e, bool check, bool perform) {
const Triangulation<2>& edgeLink = e->buildLink();
Isomorphism<4> linkInc = e->buildLinkInclusion();
if (check) {
// e should meet four distinct pentachora.
if (e->degree() != 4) {
return false;
}
std::set< Pentachoron<4>* > pentSet;
for ( auto& emb : *e ) {
if ( not pentSet.insert( emb.pentachoron() ).second ) {
return false;
}
}
// The link of e should be a 2-2 move away from being combinatorially
// isomorphic to the boundary of a tetrahedron.
if ( edgeLink.isoSig() != "eLPbddaaa" ) {
return false;
}
}
for (auto& emb : *e) {
if (emb.simplex()->locks_) {
if (emb.simplex()->isLocked() ||
emb.simplex()->isFacetLocked(emb.vertices()[2]) ||
emb.simplex()->isFacetLocked(emb.vertices()[3]) ||
emb.simplex()->isFacetLocked(emb.vertices()[4])) {
if (check)
return false;
if (perform)
throw LockViolation("An attempt was made to perform a "
"4-4 move using a locked pentachoron and/or facet");
}
}
}
if (! perform)
return true;
// Perform the 4-4 move as a 2-4 move followed by a 4-2 move.
// Note that we use pachner(), which ensures that we preserve orientation
// (if the triangulation was originally oriented).
//
// The calls to pachner() will also manage any lock updates.
// Start by working out where the 2-4 and 4-2 moves should take place.
Vertex<2>* topVert = nullptr;
for ( int i = 0; i < 3; ++i ) {
if ( edgeLink.triangle(0)->vertex( i )->degree() == 2 ) {
topVert = edgeLink.triangle(0)->vertex( i );
break;
}
}
// Location of the (first) 2-4 move.
size_t linkFront = topVert->embedding(0).triangle()->index();
int vertFront = topVert->embedding(0).vertex();
Pentachoron<4>* frontPent = pentachoron( linkInc.pentImage(linkFront) );
Tetrahedron<4>* tet24 = frontPent->tetrahedron(
linkInc.facetPerm( linkFront )[ vertFront ] );
// Location of the (second) 4-2 move.
// We record this as a pentachoron-edge combination, since by the time we
// perform this second move the original skeleton will have been destroyed.
size_t linkBack = topVert->embedding(1).triangle()->index();
Pentachoron<4>* backPent = pentachoron( linkInc.pentImage(linkBack) );
int edge42;
for ( auto& emb : *e ) {
if ( emb.simplex() == backPent ) {
edge42 = emb.edge();
break;
}
}
PacketChangeGroup span(*this);
pachner(tet24, regina::unprotected);
pachner(backPent->edge(edge42), regina::unprotected);
// Done!
return true;
}
bool Triangulation<4>::internalOpenBook(Tetrahedron<4>* t, bool check,
bool perform) {
if (t->isLocked()) {
if (check)
return false;
if (perform)
throw LockViolation("An attempt was made to perform an "
"open book move using a locked tetrahedron");
}
const TetrahedronEmbedding<4>& emb = t->front();
Pentachoron<4>* pent = emb.pentachoron();
// Check that the triangle has exactly two boundary edges.
// Note that this will imply that the triangle joins two tetrahedra.
if (check) {
int i;
for (i = 0; i < 4; ++i)
if (! t->vertex(i)->isValid())
return false;
for (i = 0; i < 6; ++i)
if (! t->edge(i)->isValid())
return false;
for (i = 0; i < 4; ++i)
if (! t->triangle(i)->isValid())
return false;
int nBdry = 0;
int bdryTriangle[4];
for (i = 0; i < 4; ++i)
if (t->triangle(i)->isBoundary())
bdryTriangle[nBdry++] = i;
if (nBdry < 1 || nBdry > 3)
return false;
if (nBdry == 2) {
// Remaining edge is non-boundary.
int edge = Edge<3>::edgeNumber[bdryTriangle[0]][bdryTriangle[1]];
if (t->edge(edge)->isBoundary())
return false;
// Remaining two triangles are not identified.
if (t->triangle(Edge<3>::edgeVertex[5 - edge][0]) ==
t->triangle(Edge<3>::edgeVertex[5 - edge][1]))
return false;
} else if (nBdry == 1) {
// Remaining vertex is non-boundary.
if (t->vertex(bdryTriangle[0])->isBoundary())
return false;
// No two of the remaining three edges are identified.
Edge<4>* internal[3];
internal[0] = t->edge(
Edge<3>::edgeNumber[bdryTriangle[0]][(bdryTriangle[0] + 1) % 4]);
internal[1] = t->edge(
Edge<3>::edgeNumber[bdryTriangle[0]][(bdryTriangle[0] + 2) % 4]);
internal[2] = t->edge(
Edge<3>::edgeNumber[bdryTriangle[0]][(bdryTriangle[0] + 3) % 4]);
if (internal[0] == internal[1] || internal[1] == internal[2] ||
internal[2] == internal[0])
return false;
}
}
if (! perform)
return true;
// Actually perform the move.
// Don't bother with a change event group: this is very simple, and
// we will already get our change management bookkeeping via unjoin().
// We should however declare a topology lock here, since unjoin() does not
// know that the topology will be preserved.
TopologyLock lock(*this);
pent->unjoin(emb.tetrahedron());
return true;
}
bool Triangulation<4>::internalCollapseEdge(Edge<4>* e, bool check,
bool perform) {
// Find the pentachora to remove.
if (check) {
// We need a valid edge before we test anything else.
// From this we know that the edge link is a disc or sphere.
if (! e->isValid())
return false;
// CHECK 0: The pentachora around the edge must be distinct.
// We check this as follows:
//
// - None of the triangles containing edge e must contain e twice.
// We throw this into check 2 below (see points [0a] and [0b]).
//
// - The only remaining bad cases involve some tetrahedron with
// e as two opposite edges. In this case one can prove that
// we have a bad chain of bigons, which will be picked up in
// check 2 below. This works both for internal tetrahedra
// (which give a bad chain of internal bigons) and boundary
// tetrahedra (which give a bad chain of boundary bigons).
// CHECK 1: Can we collapse the edge to a point (which in turn
// collapses triangles to bigons and so on up the dimensions)?
// The vertices must be distinct.
if (e->vertex(0) == e->vertex(1))
return false;
// If both vertices are in the boundary then we must be collapsing a
// boundary edge, and both vertices must have plain old ball links.
// Recall that ideal vertices return isBoundary() == true.
if (e->vertex(0)->isBoundary() && e->vertex(1)->isBoundary()) {
if (! e->isBoundary())
return false;
// Since e is a boundary edge, both vertex links are bounded
// 3-manifolds. This means that the vertex links are balls
// if and only if the vertices are valid.
if (! e->vertex(0)->isValid())
return false;
if (! e->vertex(1)->isValid())
return false;
}
// CHECK 2: Triangles containing the edge have now become bigons.
// Can we flatten each bigon to an edge (leaving behind
// triangular pillows behind and so on up the dimensions)?
//
// This is trickier. Even if every individual bigon is okay, we
// don't want a _chain_ of bigons together to crush a sphere or
// projective plane.
//
// The way we do this is as follows. Consider each Edge<4>* to be
// a vertex of some graph G, and consider each bigon to be an edge
// in this graph G. The vertices at either end of the edge in G
// are the (Edge<4>*)s that bound the bigon.
//
// We can happily flatten each bigon if and only if the graph G
// contains no cycles. We shall test this using union-find,
// which should have log-linear complexity.
//
// We deal with boundary edges and invalid edges as follows.
// All boundary and/or invalid edges become the *same* vertex in
// the graph G. This means, for instance, that a bigon joining two
// distinct boundary edges is not allowed.
//
// If edge e is itself a boundary edge, things become more
// interesting again. In this case, the *boundary* bigons
// are not subject to the same restrictions -- crushing bigons
// along the boundary does no harm, *unless* the boundary bigon
// edges themselves form a cycle. This is essentially the same
// dilemma as before but one dimension down, and we must test this
// separately.
{
size_t nEdges = countEdges();
// The parent of each edge in the union-find tree, or -1 if
// an edge is at the root of a tree.
//
// This array is indexed by edge number in the triangulation.
// Although we might not use many of these edges, it's fast
// and simple. The "unified boundary" is assigned the edge
// number nEdges.
auto* parent = new ssize_t[nEdges + 1];
// The depth of each subtree in the union-find tree.
auto* depth = new size_t[nEdges + 1];
Edge<4> *upper, *lower;
if (e->isBoundary()) {
// Search for cycles in boundary bigons.
std::fill(parent, parent + nEdges + 1, -1);
std::fill(depth, depth + nEdges + 1, 0);
// Run through all boundary triangles containing e.
for (Triangle<4>* triangle : triangles()) {
if (! triangle->isBoundary())
continue;
int i;
for (i = 0; i < 3; ++i)
if (triangle->edge(i) == e)
break;
if (i == 3)
continue;
// This triangle contains edge e (specifically, as edge i
// of this triangle).
upper = triangle->edge((i + 1) % 3);
lower = triangle->edge((i + 2) % 3);
if (upper == e || lower == e) {
// [0a]: Check 0 fails; this triangle contains edge e
// more than once.
delete[] depth;
delete[] parent;
return false;
}
// This bigon joins nodes id1 and id2 in the graph G.
if (! unionFindInsert(parent, depth,
upper->index(), lower->index())) {
delete[] depth;
delete[] parent;
return false;
}
}
// All looks good on the boundary.
}
// Search for cycles in internal bigons.
std::fill(parent, parent + nEdges + 1, -1);
std::fill(depth, depth + nEdges + 1, 0);
// Run through all internal triangles containing e.
for (Triangle<4>* triangle : triangles()) {
if (triangle->isBoundary())
continue;
int i;
for (i = 0; i < 3; ++i)
if (triangle->edge(i) == e)
break;
if (i == 3)
continue;
// This triangle contains edge e (specifically, as edge i
// of this triangle).
upper = triangle->edge((i + 1) % 3);
lower = triangle->edge((i + 2) % 3);
if (upper == e || lower == e) {
// [0b]: Check 0 fails; this triangle contains edge e
// more than once.
delete[] depth;
delete[] parent;
return false;
}
size_t id1 = ((upper->isBoundary() || ! upper->isValid()) ?
nEdges : upper->index());
size_t id2 = ((lower->isBoundary() || ! lower->isValid()) ?
nEdges : lower->index());
// This bigon joins nodes id1 and id2 in the graph G.
if (! unionFindInsert(parent, depth, id1, id2)) {
delete[] depth;
delete[] parent;
return false;
}
}
// All looks good internally also.
// No bad chains of bigons!
delete[] depth;
delete[] parent;
}
// CHECK 3: Tetrahedra containing the edge have now become
// triangular pillows. Can we flatten each pillow to a triangle
// (leaving behind "tetrahedral 4-pillows" in higher dimensions)?
//
// We deal with this the same way we deal with flattening bigons
// to edges. Again, we must treat internal pillows and
// boundary pillows separately.
{
size_t nTriangles = countTriangles();
// The parent of each triangle in the union-find tree, or -1 if
// a triangle is at the root of a tree.
//
// This array is indexed by triangle number in the triangulation.
// Although we might not use many of these triangles, it's fast
// and simple. The "unified boundary" is assigned the triangle
// number nTriangles.
auto* parent = new ssize_t[nTriangles + 1];
// The depth of each subtree in the union-find tree.
auto* depth = new size_t[nTriangles + 1];
Triangle<4> *upper, *lower;
if (e->isBoundary()) {
// Search for cycles in boundary pillows.
std::fill(parent, parent + nTriangles + 1, -1);
std::fill(depth, depth + nTriangles + 1, 0);
// Run through all boundary tetrahedra containing e.
for (Tetrahedron<4>* tet : tetrahedra()) {
if (! tet->isBoundary())
continue;
int i;
for (i = 0; i < 6; ++i)
if (tet->edge(i) == e)
break;
if (i == 6)
continue;
// This tetrahedron contains edge e (specifically, as
// edge i of this tetrahedron).
upper = tet->triangle(Edge<3>::edgeVertex[i][0]);
lower = tet->triangle(Edge<3>::edgeVertex[i][1]);
if (! unionFindInsert(parent, depth,
upper->index(), lower->index())) {
delete[] depth;
delete[] parent;
return false;
}
}
// All looks good on the boundary.
}
// Search for cycles in internal pillows.
std::fill(parent, parent + nTriangles + 1, -1);
std::fill(depth, depth + nTriangles + 1, 0);
// Run through all internal tetrahedra containing e.
for (Tetrahedron<4>* tet : tetrahedra()) {
if (tet->isBoundary())
continue;
int i;
for (i = 0; i < 6; ++i)
if (tet->edge(i) == e)
break;
if (i == 6)
continue;
// This tetrahedron contains edge e (specifically, as edge i
// of this tetrahedron).
upper = tet->triangle(Edge<3>::edgeVertex[i][0]);
lower = tet->triangle(Edge<3>::edgeVertex[i][1]);
size_t id1 = ((upper->isBoundary() || ! upper->isValid()) ?
nTriangles : upper->index());
size_t id2 = ((lower->isBoundary() || ! lower->isValid()) ?
nTriangles : lower->index());
// This pillow joins nodes id1 and id2 in the graph G.
if (! unionFindInsert(parent, depth, id1, id2)) {
delete[] depth;
delete[] parent;
return false;
}
}
// All looks good internally also.
// No bad chains of bigons!
delete[] depth;
delete[] parent;
}
// CHECK 4: Can we flatten each tetrahedral 4-pillow to a tetrahedron?
//
// Again, even if each individual pillow is okay, we don't want
// a chain of pillows together to completely crush away a
// 4-manifold component.
//
// This means no cycles of pillows, and no chains of pillows
// that run from boundary to boundary.
//
// Test this in the same way that we tested edges. It's kind of
// overkill, since each vertex in the corresponding graph G will
// have degree <= 2, but it's fast so we'll do it.
{
size_t nTets = countTetrahedra();
// The parent of each tetrahedron in the union-find tree,
// or -1 if a tetrahedron is at the root of a tree.
//
// This array is indexed by tetrahedron number in the triangulation.
// The "unified boundary" is assigned the triangle number nTets.
auto* parent = new ssize_t[nTets + 1];
std::fill(parent, parent + nTets + 1, -1);
// The depth of each subtree in the union-find tree.
auto* depth = new size_t[nTets + 1];
std::fill(depth, depth + nTets + 1, 0);
Tetrahedron<4> *upper, *lower;
for (auto& emb : *e) {
upper = emb.simplex()->tetrahedron(emb.vertices()[0]);
lower = emb.simplex()->tetrahedron(emb.vertices()[1]);
size_t id1 = (upper->isBoundary() ? nTets : upper->index());
size_t id2 = (lower->isBoundary() ? nTets : lower->index());
// This 4-pillow joins nodes id1 and id2 in the graph G.
if (! unionFindInsert(parent, depth, id1, id2)) {
delete[] depth;
delete[] parent;
return false;
}
}
// No bad chains of 4-pillows!
delete[] depth;
delete[] parent;
}
}
// Finally, we search for potential lock violations:
size_t idx = 0;
for (auto& emb : *e) {
if (emb.simplex()->locks_) {
if (emb.simplex()->isLocked()) {
if (check)
return false;
if (perform)
throw LockViolation("An attempt was made to perform an "
"edge collapse that would remove a locked pentachoron");
}
for (int i = 2; i <= 4; ++i)
if (emb.simplex()->isFacetLocked(emb.vertices()[i])) {
if (check)
return false;
if (perform)
throw LockViolation("An attempt was made to perform an "
"edge collapse that would remove a locked "
"tetrahedron");
}
}
++idx;
}
if (! perform)
return true;
// Perform the move.
// The following ChangeAndClearSpan is essential, since we use
// "raw" routines (removeSimplexRaw, joinRaw, etc.) below.
ChangeAndClearSpan<ChangeType::PreserveTopology> span(*this);
// Clone the edge embeddings because we cannot rely on skeletal
// objects once we start changing the triangulation.
auto* embs = new EdgeEmbedding<4>[e->degree()];
std::copy(e->begin(), e->end(), embs);
for (size_t i = 0; i < e->degree(); ++i) {
const auto& emb = embs[i];
Simplex<4>* top = emb.simplex()->adjacentPentachoron(emb.vertices()[0]);
Perm<5> topPerm = emb.simplex()->adjacentGluing(emb.vertices()[0]);
Simplex<4>* bot = emb.simplex()->adjacentPentachoron(emb.vertices()[1]);
Perm<5> botPerm = emb.simplex()->adjacentGluing(emb.vertices()[1]);
if (emb.simplex()->locks_) {
if (bot && emb.simplex()->isFacetLocked(emb.vertices()[0]))
bot->lockFacetRaw(botPerm[emb.vertices()[1]]);
if (top && emb.simplex()->isFacetLocked(emb.vertices()[1]))
top->lockFacetRaw(topPerm[emb.vertices()[0]]);
}
removeSimplexRaw(emb.simplex());
if (top && bot)
top->joinRaw(topPerm[emb.vertices()[0]], bot,
botPerm * Perm<5>(emb.vertices()[0], emb.vertices()[1]) *
topPerm.inverse());
}
delete[] embs;
return true;
}
bool Triangulation<4>::internalSnapEdge(Edge<4>* e, bool check, bool perform) {
if (check &&
((e->vertex(0) == e->vertex(1)) ||
(e->vertex(0)->isBoundary() && e->vertex(1)->isBoundary())))
return false;
if (! perform)
return true;
// Our plan is to find a tetrahedron containing e, and then insert four
// pentachora in its place.
Pentachoron<4>* open = e->front().pentachoron();
Perm<5> vertices = e->front().vertices();
Pentachoron<4>* adj = open->adjacentPentachoron(vertices[2]);
Perm<5> glue = open->adjacentGluing(vertices[2]);
bool locked = open->isFacetLocked(vertices[2]);
// Actually perform the move.
// The following ChangeAndClearSpan is essential, since we use
// "raw" routines (newSimplicesRaw, joinRaw, etc.) below.
ChangeAndClearSpan<ChangeType::PreserveTopology> span(*this);
// The four pentachora that we insert together form a "pinched 4-ball".
// Combinatorially, the boundary of this pinched 4-ball is isomorphic to
// the 3-sphere that forms the boundary of a tetrahedral pillow; however,
// two adjacent boundary edges a and b in this pinched 4-ball are pinched
// together to form a single edge whose link becomes an annulus. We insert
// this pinched 4-ball into the opened-up tetrahedron in such a way that
// edges a, b and e together bound a triangle in the 3-sphere that used to
// form the boundary of the pinched 4-ball (this is possible because edges
// a and b are adjacent). For our purposes, the most important consequence
// of this is that the endpoints of e will become snapped together.
auto p = newSimplicesRaw<4>();
p[0]->joinRaw(0, p[1], {3, 4});
p[0]->joinRaw(2, p[1], {0, 2, 4, 1, 3});
p[0]->joinRaw(3, p[2], {3, 4});
p[0]->joinRaw(4, p[2], {3, 4});
p[1]->joinRaw(1, p[2], {1, 2});
p[1]->joinRaw(2, p[3], {3, 4});
p[1]->joinRaw(3, p[3], {3, 4});
p[2]->joinRaw(0, p[3], {3, 4});
p[2]->joinRaw(1, p[3], {3, 4});
// The boundary tetrahedra of this auxiliary structure are p[0]: 0234 and
// p[3]: 0214.
// The edges that glue to p[0]: 02, p[0]: 03, p[0]: 04 or p[0]: 23 will
// remain (topologically) unaffected.
// The edges that glue to p[0]: 24 and p[0]: 34 will be snapped together.
// A note for oriented triangulations: Simplex::faceMapping() guarantees
// that e->front().vertices() has a sign equal to the orientation of the
// relevant pentachoron, which for an oriented triangulation is always 1.
// Therefore all of the gluings that we make here use odd gluing
// permutations, and hence the orientation is preserved.
open->unjoinRaw(vertices[2]);
p[0]->joinRaw(1, open, vertices * Perm<5>(3, 2, 0, 1, 4));
p[3]->joinRaw(3, adj, glue * vertices * Perm<5>(3, 1, 0, 2, 4));
// If the tetrahedron that we popped open was locked, we will (arbitrarily)
// choose to move the lock to the tetrahedron that still belongs to open
// (as opposed to the ex-partner tetrahedron belonging to adj).
if (locked) {
// The lock is already present from open's side.
// Remove it from adj's side, and put it where it needs to be in p[0].
adj->unlockFacetRaw(glue[vertices[2]]);
p[0]->lockFacetRaw(1);
}
// Done!
return true;
}
} // namespace regina
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