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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/link.h"
#include "triangulation/detail/retriangulate-impl.h"
// NOLINTNEXTLINE(modernize-concat-nested-namespaces)
namespace regina {
namespace detail {
/**
* Provides domain-specific details for the link rewriting process.
*
* For link propagation, we do make use of the options type
* `Retriangulator::PropagationOptions`. This type should be one of:
*
* - `std::true_type`, to indicate that only classical Reidemeister moves
* should be allowed;
*
* - `std::false_type`, to indicate that both classical and virtual
* Reidemeister moves should be allowed.
*/
template <>
struct RetriangulateParams<Link> {
static std::string sig(const Link& link) {
return link.sig();
}
static std::string rigidSig(const Link& link) {
// Do not allow reflection, reversal and/or rotation.
return link.sig(false, false, false);
}
static constexpr const char* progressStage = "Exploring diagrams";
template <class Retriangulator>
static void propagateFrom(const std::string& sig, size_t maxSize,
Retriangulator* retri) {
constexpr bool classicalOnly =
Retriangulator::PropagationOptions::value;
Link t = Link::fromSig(sig);
if (t.size() == 0) {
// We have a zero-crossing unlink (possibly empty).
if (t.isEmpty() || maxSize == 0) {
// No moves are available at all.
return;
}
// The link is non-empty, and we are allowed to add crossings.
{
// Add a twist to a single unknot component.
// The side does not matter, since both options are
// equivalent under reversal of individual link components.
// The sign does not matter either, since there no
// pre-existing crossings, and so the two options are
// equivalent under reflection of the entire diagram.
Link alt(t, false);
alt.r1(StrandRef(), 0 /* side */, 1 /* sign */);
if (retri->candidate(std::move(alt), sig))
return;
}
if constexpr (! classicalOnly) {
if (maxSize > 1) {
// There are only two essentially different diagrams
// that we can obtain from a zero-crossing unknot
// using a virtual R2. These are obtained via
// (firstSide == firstStrand) and
// (firstSide != firstStrand).
for (int firstSide = 0; firstSide < 2; ++firstSide) {
Link alt(t, false);
alt.r2Virtual({}, firstSide, 1 /* firstStrand */);
if (retri->candidate(std::move(alt), sig))
return;
}
}
}
// We promise not to merge diagram components, so we do not
// consider moves that pass one unknot component over another.
return;
}
// From here we assume >= 1 crossing.
// Moves that reduce the number of crossings:
for (size_t i = 0; i < t.size(); ++i)
if (auto alt = t.withR1(t.crossing(i)))
if (retri->candidate(std::move(*alt), sig))
return;
for (size_t i = 0; i < t.size(); ++i)
if (auto alt = t.withR2(t.crossing(i)))
if (retri->candidate(std::move(*alt), sig))
return;
// Moves that preserve the number of crossings:
for (size_t i = 0; i < t.size(); ++i)
for (int side = 0; side < 2; ++side)
if (auto alt = t.withR3(t.crossing(i), side))
if (retri->candidate(std::move(*alt), sig))
return;
// All that remains is moves that increase the number of crossings.
if (t.size() >= maxSize)
return;
// We need to know whether there are any 0-crossing link components.
bool hasTrivial = false;
for (auto c : t.components())
if (! c) {
hasTrivial = true;
break;
}
// R1 twist moves on arcs are always valid.
for (size_t i = 0; i < t.size(); ++i)
for (int strand = 0; strand < 2; ++strand)
for (int side = 0; side < 2; ++side)
for (int sign = -1; sign <= 1; sign += 2) {
Link alt(t, false);
alt.r1(alt.crossing(i)->strand(strand), side, sign);
if (retri->candidate(std::move(alt), sig))
return;
}
if (hasTrivial) {
for (int sign = -1; sign <= 1; sign += 2) {
// The side does not matter, since both options are
// equivalent under reversal of individual link components.
Link alt(t, false);
alt.r1(StrandRef(), 0, sign);
if (retri->candidate(std::move(alt), sig))
return;
}
}
if (t.size() + 1 < maxSize) {
if constexpr (! classicalOnly) {
// Testing for virtual R2 moves is very fast, and these
// moves (as enumerated below) are always valid.
// However, we do have to be sure not to mix different
// diagram components.
// Moves that work on different strands:
auto [ comp, nComp ] = t.diagramComponentIndices();
for (size_t cr1 = 0; cr1 < t.size(); ++cr1) {
for (size_t cr2 = 0; cr2 < t.size(); ++cr2) {
if (comp[cr1] != comp[cr2])
continue;
for (int str1 = 0; str1 < 2; ++str1) {
for (int str2 = 0; str2 < 2; ++str2) {
// Do not operate on the same strand.
if (cr1 == cr2 && str1 == str2)
continue;
for (int side1 = 0; side1 < 2; ++side1) {
for (int side2 = 0; side2 < 2; ++side2) {
Link alt(t, false);
alt.r2Virtual(
alt.crossing(cr1)->strand(str1),
side1,
alt.crossing(cr2)->strand(str2),
side2);
if (retri->candidate(
std::move(alt), sig))
return;
}
}
}
}
}
}
// Moves that work on the same strand:
for (size_t cr = 0; cr < t.size(); ++cr)
for (int strand = 0; strand < 2; ++strand)
for (int fSide = 0; fSide < 2; ++fSide)
for (int fStrand = 0; fStrand < 2; ++fStrand) {
Link alt(t, false);
alt.r2Virtual(
alt.crossing(cr)->strand(strand),
fSide, fStrand);
if (retri->candidate(std::move(alt), sig))
return;
}
if (hasTrivial) {
// There are only two possible diagrams that can come
// from a virtual R2 on a zero-crossing unknot:
// one with firstSide == firstStrand, and
// one with firstSide != firstStrand.
for (int firstSide = 0; firstSide < 2; ++firstSide) {
Link alt(t, false);
alt.r2Virtual({}, firstSide, 1 /* firstStrand */);
if (retri->candidate(std::move(alt), sig))
return;
}
}
} else {
// We are restricting ourselves to classical moves.
for (size_t i = 0; i < t.size(); ++i)
for (int strand = 0; strand < 2; ++strand) {
StrandRef uArc = t.crossing(i)->strand(strand);
for (int uSide = 0; uSide < 2; ++uSide) {
// Walk around the 2-cell containing uArc.
// This code follows the (better documented)
// code in reidemeister.cpp for testing r2
// validity.
//
// We walk around the 2-cell from upper,
// ensuring that we always turn left.
//
// At each stage we consider an edge of this
// 2-cell:
//
// - ref points to the strand of the crossing at
// the beginning of the edge, with respect to
// the direction in which we are walking
// around the cell;
// - lArc points to the strand of the crossing
// at the beginning of the edge, with respect
// to the orientation of the link;
// - fwd indicates whether these two
// directions are the same.
//
// Note that we don't actually set lArc
// until we get near the end of the while loop.
//
StrandRef ref = uArc;
bool fwd;
if (uSide == 0) {
fwd = true;
} else {
// We are traversing the arc backwards, so
// we need to jump to the other endpoint.
ref = ref.next();
fwd = false;
}
while (true) {
// Move to the next edge around this 2-cell.
if (fwd) {
ref = ref.next();
ref.jump();
// fwd remains true iff
// (sign, strand) == (+, 0) or (-, 1).
if (ref.crossing()->sign() > 0)
fwd = (0 == ref.strand());
else
fwd = (0 != ref.strand());
} else {
ref = ref.prev();
ref.jump();
// fwd becomes true iff
// (sign, strand) == (-, 0) or (+, 1).
if (ref.crossing()->sign() > 0)
fwd = (0 != ref.strand());
else
fwd = (0 == ref.strand());
}
StrandRef lArc = (fwd ? ref : ref.prev());
int lSide = (fwd ? 0 : 1);
if (lArc == uArc && lSide == uSide) {
// We completed the cycle.
break;
}
// The r2() check is expensive when adding
// two crossings. We already know this move
// is legal (in the classical sense), so use
// r2Virtual() instead which avoids the
// expensive planarity test.
Link alt(t, false);
alt.r2Virtual(
alt.translate(uArc), uSide,
alt.translate(lArc), lSide);
if (retri->candidate(std::move(alt), sig))
return;
}
}
}
// We promise not to merge diagram components, so we do not
// consider moves that pass an unknot component over some
// other component.
}
}
}
};
} // namespace detail
// Instantiate all necessary rewriting/retriangulation template functions
// so the full implementation can stay out of the headers.
template bool detail::retriangulateInternal<Link, true,
detail::RetriangulateDefault, std::true_type>(
const Link&, bool, int, int, ProgressTrackerOpen*,
regina::detail::RetriangulateActionFunc<Link, true>&&);
template bool detail::retriangulateInternal<Link, false,
detail::RetriangulateDefault, std::true_type>(
const Link&, bool, int, int, ProgressTrackerOpen*,
regina::detail::RetriangulateActionFunc<Link, false>&&);
template bool detail::retriangulateInternal<Link, true,
detail::RetriangulateDefault, std::false_type>(
const Link&, bool, int, int, ProgressTrackerOpen*,
regina::detail::RetriangulateActionFunc<Link, true>&&);
template bool detail::retriangulateInternal<Link, false,
detail::RetriangulateDefault, std::false_type>(
const Link&, bool, int, int, ProgressTrackerOpen*,
regina::detail::RetriangulateActionFunc<Link, false>&&);
template bool detail::retriangulateInternal<Link, false,
detail::RetriangulateNoLocks | detail::RetriangulateNotFinished,
std::true_type>(const Link&, bool, int, int, ProgressTrackerOpen*,
regina::detail::RetriangulateActionFunc<Link, false>&&);
template bool detail::retriangulateInternal<Link, false,
detail::RetriangulateNoLocks | detail::RetriangulateNotFinished,
std::false_type>(const Link&, bool, int, int, ProgressTrackerOpen*,
regina::detail::RetriangulateActionFunc<Link, false>&&);
} // namespace regina
|
