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knot_diagram.cpp
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knot_diagram.cpp
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#include <knotkit.h>
knot_diagram::knot_diagram (const planar_diagram &pd)
: name(pd.name),
n_crossings(pd.crossings.size ()),
marked_edge(0),
crossings(n_crossings),
ept_crossing(num_epts ()),
ept_index(num_epts ()),
nminus(0), nplus(0)
{
basedvector<basedvector<pair<unsigned, unsigned>, 1>, 1> edge_ci (num_edges ());
for (unsigned c = 1; c <= n_crossings; c ++)
{
for (unsigned i = 1; i <= 4; i ++)
{
unsigned e = pd.crossings[c][i];
edge_ci[e].append (pair<unsigned, unsigned> (c, i));
}
crossings[c] = basedvector<unsigned, 1> (4);
}
set<unsigned> done;
for (unsigned c0 = 1; c0 <= n_crossings; c0 ++)
{
for (unsigned i0 = 1; i0 <= 2; i0 ++)
{
unsigned e = pd.crossings[c0][i0];
if (done % e)
continue;
for (unsigned c = c0, i = i0;;)
{
crossings[c][i] = edge_to_ept (e);
done.push (e);
ept_crossing[edge_to_ept (e)] = c;
ept_index[edge_to_ept (e)] = i;
i = add_base1_mod4 (i, 2);
e = pd.crossings[c][i];
crossings[c][i] = edge_from_ept (e);
ept_crossing[edge_from_ept (e)] = c;
ept_index[edge_from_ept (e)] = i;
if (edge_ci[e][1].first == c
&& edge_ci[e][1].second == i)
{
c = edge_ci[e][2].first;
i = edge_ci[e][2].second;
}
else
{
assert (edge_ci[e][2].first == c && edge_ci[e][2].second == i);
c = edge_ci[e][1].first;
i = edge_ci[e][1].second;
}
if (c == c0 && i == i0)
break;
}
}
}
assert (done.card () == num_edges ());
calculate_smoothing_orientation ();
calculate_nminus_nplus ();
}
class dt_layout
{
public:
knot_diagram &kd;
unsigned n_components;
basedvector<unsigned, 1> comp_first_edge,
comp_last_edge;
basedvector<unsigned, 1> edge_to_crossing;
smallbitset edge_to_under;
smallbitset placed;
public:
dt_layout (const dt_code &dt, knot_diagram &kd_);
dt_layout (const dt_layout &) = delete;
~dt_layout () { }
dt_layout &operator = (const dt_layout &) = delete;
unsigned find_crossing (unsigned prevc, unsigned previ, unsigned target,
bool under, unsigned dir);
bool layout_2 (unsigned e,
unsigned prevc, unsigned previ,
unsigned c, unsigned i);
bool layout_1 (unsigned e,
unsigned prevc, unsigned previ);
unsigned comp_next_edge (unsigned e) const;
void layout ();
};
dt_layout::dt_layout (const dt_code &dt, knot_diagram &kd_)
: kd(kd_),
n_components(dt.num_components ()),
comp_first_edge(n_components),
comp_last_edge(n_components),
edge_to_crossing(kd.num_edges ()),
edge_to_under(kd.num_edges ()),
placed(kd.n_crossings)
{
unsigned k = 1;
for (unsigned i = 1; i <= dt.even_labels.size (); i ++)
{
comp_first_edge[i] = 2 * k - 1;
comp_last_edge[i] = 2 * (k + dt.even_labels[i].size () - 1);
for (unsigned j = 1; j <= dt.even_labels[i].size (); j ++, k ++)
{
int as = dt.even_labels[i][j];
unsigned a = abs (as),
b = 2 * k - 1;
if (as > 0)
edge_to_under.push (a);
else
edge_to_under.push (b);
edge_to_crossing[a] = edge_to_crossing[b] = k;
}
}
}
unsigned
dt_layout::find_crossing (unsigned prevc, unsigned previ, unsigned target,
bool under, unsigned dir)
{
assert (!kd.crossings[prevc][previ]);
assert (placed % prevc);
unsigned c = prevc,
i = add_base1_mod4 (previ, dir);
for (;;)
{
if (c == prevc && i == previ)
return 0;
if (c == target && under == (i == 1 || i == 3) && kd.crossings[c][i] == 0)
return i;
unsigned p = kd.crossings[c][i];
if (p)
{
unsigned p2 = kd.edge_other_ept (p);
c = kd.ept_crossing[p2];
i = kd.ept_index[p2];
}
i = add_base1_mod4 (i, dir);
}
}
unsigned
dt_layout::comp_next_edge (unsigned e) const
{
for (unsigned comp = 1; comp <= n_components; comp ++)
{
if (comp_first_edge[comp] <= e
&& e <= comp_last_edge[comp])
{
if (e == comp_last_edge[comp])
return comp_first_edge[comp];
else
return e + 1;
}
}
abort ();
}
bool
dt_layout::layout_2 (unsigned e,
unsigned prevc, unsigned previ,
unsigned c, unsigned i)
{
assert (placed % prevc);
assert (placed % c);
unsigned from = kd.edge_from_ept (e),
to = kd.edge_to_ept (e);
assert (kd.crossings[prevc][previ] == 0);
kd.crossings[prevc][previ] = from;
kd.ept_crossing[from] = prevc;
kd.ept_index[from] = previ;
assert (kd.crossings[c][i] == 0);
kd.crossings[c][i] = to;
kd.ept_crossing[to] = c;
kd.ept_index[to] = i;
if (layout_1 (comp_next_edge (e), c, add_base1_mod4 (i, 2)))
return 1;
kd.crossings[prevc][previ] = 0;
kd.crossings[c][i] = 0;
kd.ept_crossing[from] = kd.ept_crossing[to] = 0;
kd.ept_index[from] = kd.ept_index[to] = 0;
return 0;
}
bool
dt_layout::layout_1 (unsigned e, unsigned prevc, unsigned previ)
{
if (kd.crossings[prevc][previ])
{
unsigned from = kd.edge_from_ept (e);
assert (kd.crossings[prevc][previ] == from);
assert (kd.ept_crossing[from] == prevc);
assert (kd.ept_index[from] == previ);
for (unsigned e = 1; e <= kd.num_edges (); e ++)
{
unsigned to = kd.edge_to_ept (e);
unsigned prevc = edge_to_crossing[e];
if (placed % prevc
&& !kd.ept_crossing[to])
{
unsigned previ1 = edge_to_under % e ? 3 : 4;
if (layout_1 (comp_next_edge (e), prevc, previ1))
return 1;
unsigned previ2 = edge_to_under % e ? 1 : 2;
return layout_1 (comp_next_edge (e), prevc, previ2);
}
}
/* done! */
return 1;
}
unsigned c = edge_to_crossing[e];
if (placed % c)
{
unsigned i = find_crossing (prevc, previ, c, edge_to_under % e, 1);
if (i && layout_2 (e, prevc, previ, c, i))
return 1;
unsigned i2 = find_crossing (prevc, previ, c, edge_to_under % e, 3);
return i2 && layout_2 (e, prevc, previ, c, i2); // return i && ... ??
}
else
{
unsigned i = (edge_to_under % e) ? 1 : 2;
placed.push (c);
if (!layout_2 (e, prevc, previ, c, i))
{
placed -= c;
return 0;
}
else
return 1;
}
}
void
dt_layout::layout ()
{
unsigned last = comp_last_edge[1];
unsigned prevc = edge_to_crossing[last];
unsigned previ = (edge_to_under % last) ? 3 : 4;
placed.push (prevc);
bool b = layout_1 (comp_first_edge[1], prevc, previ);
assert (b);
assert (placed.card () == kd.n_crossings);
}
knot_diagram::knot_diagram (const dt_code &dt)
: name(dt.name),
n_crossings(dt.num_crossings ()),
marked_edge(0),
crossings(n_crossings),
ept_crossing(num_epts ()),
ept_index(num_epts ()),
nminus(0), nplus(0)
{
for (unsigned i = 1; i <= n_crossings; i ++)
{
basedvector<unsigned, 1> v (4);
v[1] = v[2] = v[3] = v[4] = 0;
crossings[i] = v;
}
dt_layout layout (dt, *this);
layout.layout ();
calculate_nminus_nplus ();
calculate_smoothing_orientation ();
}
knot_diagram::knot_diagram (sublink,
smallbitset c,
const knot_diagram &kd)
: name(kd.name),
n_crossings(0),
marked_edge(0),
nminus(0),
nplus(0)
{
// ??? assert (!kd.marked_edge);
assert (c.card () > 0); // no empty diagrams
// edge x component
unionfind<1> u (kd.num_edges ());
for (unsigned i = 1; i <= kd.n_crossings; i ++)
{
u.join (kd.ept_edge (kd.crossings[i][1]),
kd.ept_edge (kd.crossings[i][3]));
u.join (kd.ept_edge (kd.crossings[i][2]),
kd.ept_edge (kd.crossings[i][4]));
}
assert (u.num_sets () == c.size ());
ullmanset<1> u_sets (kd.num_edges ());
for (unsigned i = 1; i <= kd.num_edges (); i ++)
u_sets += u.find (i);
ullmanset<1> sub_crossings (kd.n_crossings),
sub_edges (kd.num_edges ());
for (unsigned i = 1; i <= kd.num_edges (); i ++)
{
if (c % (u_sets.position (u.find (i)) + 1))
sub_edges.push (i);
}
// sub edge x sublink edge
unionfind<1> subu (sub_edges.card ());
set<unsigned> active_comps;
for (unsigned i = 1; i <= kd.n_crossings; i ++)
{
unsigned c1 = u_sets.position (u.find (kd.ept_edge (kd.crossings[i][1]))) + 1,
c2 = u_sets.position (u.find (kd.ept_edge (kd.crossings[i][2]))) + 1;
if (c % c1
&& c % c2)
{
sub_crossings.push (i);
active_comps += c1;
active_comps += c2;
}
else
{
if (c % (u_sets.position (u.find (kd.ept_edge (kd.crossings[i][1]))) + 1))
{
subu.join (sub_edges.position (kd.ept_edge (kd.crossings[i][1])) + 1,
sub_edges.position (kd.ept_edge (kd.crossings[i][3])) + 1);
}
if (c % (u_sets.position (u.find (kd.ept_edge (kd.crossings[i][2]))) + 1))
{
subu.join (sub_edges.position (kd.ept_edge (kd.crossings[i][2])) + 1,
sub_edges.position (kd.ept_edge (kd.crossings[i][4])) + 1);
}
}
}
n_crossings = (sub_crossings.card ()
+ (c.card () - active_comps.card ()));
assert (n_crossings > 0);
ullmanset<1> subu_sets (sub_edges.card ());
for (unsigned i = 1; i <= sub_edges.card (); i ++)
subu_sets += subu.find (i);
crossings = basedvector<basedvector<unsigned, 1>, 1> (n_crossings);
for (unsigned i = 1; i <= n_crossings; i ++)
crossings[i] = basedvector<unsigned, 1> (4);
for (ullmanset_const_iter<1> i = sub_crossings; i; i ++)
{
unsigned c = i.val (),
new_c = i.pos () + 1;
unsigned e1 = (subu_sets.position
(subu.find (sub_edges.position (kd.ept_edge (kd.crossings[c][1])) + 1)) + 1),
e2 = (subu_sets.position
(subu.find (sub_edges.position (kd.ept_edge (kd.crossings[c][2])) + 1)) + 1),
e3 = (subu_sets.position
(subu.find (sub_edges.position (kd.ept_edge (kd.crossings[c][3])) + 1)) + 1),
e4 = (subu_sets.position
(subu.find (sub_edges.position (kd.ept_edge (kd.crossings[c][4])) + 1)) + 1);
if (kd.is_from_ept (kd.crossings[c][1]))
crossings[new_c][1] = edge_from_ept (e1);
else
crossings[new_c][1] = edge_to_ept (e1);
if (kd.is_from_ept (kd.crossings[c][2]))
crossings[new_c][2] = edge_from_ept (e2);
else
crossings[new_c][2] = edge_to_ept (e2);
if (kd.is_from_ept (kd.crossings[c][3]))
crossings[new_c][3] = edge_from_ept (e3);
else
crossings[new_c][3] = edge_to_ept (e3);
if (kd.is_from_ept (kd.crossings[c][4]))
crossings[new_c][4] = edge_from_ept (e4);
else
crossings[new_c][4] = edge_to_ept (e4);
}
unsigned e = subu.num_sets ();
unsigned new_c = sub_crossings.card ();
for (smallbitset_const_iter i = c; i; i ++)
{
if (active_comps % i.val ())
continue;
unsigned e1 = (subu_sets.position
(subu.find (sub_edges.position (u_sets.nth (i.val () - 1)) + 1)) + 1);
unsigned e2 = ++ e;
unsigned c = ++ new_c;
crossings[c][1] = edge_from_ept (e1);
crossings[c][2] = edge_to_ept (e1);
crossings[c][3] = edge_to_ept (e2);
crossings[c][4] = edge_from_ept (e2);
}
assert (e == num_edges ());
assert (new_c == n_crossings);
// ?? break this out into aux function
ept_crossing = basedvector<unsigned, 1> (num_epts ());
ept_index = basedvector<unsigned, 1> (num_epts ());
for (unsigned i = 1; i <= n_crossings; i ++)
{
for (unsigned j = 1; j <= 4; j ++)
{
unsigned p = crossings[i][j];
ept_crossing[p] = i;
ept_index[p] = j;
}
}
#ifndef NDEBUG
check_crossings ();
#endif
calculate_smoothing_orientation ();
calculate_nminus_nplus ();
}
knot_diagram::knot_diagram (disjoint_union,
const knot_diagram &kd1,
const knot_diagram &kd2)
: name(kd1.name + "+" + kd2.name),
n_crossings(kd1.n_crossings + kd2.n_crossings),
marked_edge(0),
crossings(n_crossings),
nminus(kd1.nminus + kd2.nminus),
nplus(kd1.nplus + kd2.nplus)
{
assert (kd1.marked_edge == 0);
assert (kd2.marked_edge == 0);
for (unsigned i = 1; i <= n_crossings; i ++)
crossings[i] = basedvector<unsigned, 1> (4);
for (unsigned i = 1; i <= kd1.n_crossings; i ++)
for (unsigned j = 1; j <= 4; j ++)
crossings[i][j] = kd1.crossings[i][j];
for (unsigned e = 1; e <= kd1.num_edges (); e ++)
{
if (kd1.edge_smoothing_oriented % e)
edge_smoothing_oriented.push (e);
}
for (unsigned i = 1; i <= kd2.n_crossings; i ++)
for (unsigned j = 1; j <= 4; j ++)
crossings[kd1.n_crossings + i][j] = kd1.num_epts () + kd2.crossings[i][j];
for (unsigned e = 1; e <= kd2.num_edges (); e ++)
{
if (kd2.edge_smoothing_oriented % e)
edge_smoothing_oriented.push (kd1.num_edges () + e);
}
// ?? break this out into aux function
ept_crossing = basedvector<unsigned, 1> (num_epts ());
ept_index = basedvector<unsigned, 1> (num_epts ());
for (unsigned i = 1; i <= n_crossings; i ++)
{
for (unsigned j = 1; j <= 4; j ++)
{
unsigned p = crossings[i][j];
ept_crossing[p] = i;
ept_index[p] = j;
}
}
#ifndef NDEBUG
check_crossings ();
#endif
}
knot_diagram::knot_diagram (mirror, const knot_diagram &kd)
: name("mirror(" + kd.name + ")"),
n_crossings(kd.n_crossings),
marked_edge(0),
crossings(n_crossings),
ept_crossing(num_epts ()),
ept_index(num_epts ()),
nminus(0), nplus(0)
{
for (unsigned i = 1; i <= n_crossings; i ++)
{
basedvector<unsigned, 1> v (4);
v[1] = kd.crossings[i][1];
v[2] = kd.crossings[i][4];
v[3] = kd.crossings[i][3];
v[4] = kd.crossings[i][2];
crossings[i] = v;
}
for (unsigned i = 1; i <= n_crossings; i ++)
{
for (unsigned j = 1; j <= 4; j ++)
{
unsigned c = crossings[i][j];
ept_crossing[c] = i;
ept_index[c] = j;
}
}
calculate_smoothing_orientation ();
calculate_nminus_nplus ();
assert (nminus == kd.nplus);
assert (nplus == kd.nminus);
}
void
knot_diagram::check_crossings ()
{
for (unsigned i = 1; i <= n_crossings; i ++)
{
for (unsigned j = 1; j <= 4; j ++)
{
unsigned p = crossings[i][j];
assert (ept_crossing[p] == i);
assert (ept_index[p] == j);
}
}
for (unsigned i = 1; i <= num_edges (); i ++)
{
unsigned to = edge_to_ept (i);
assert (is_from_ept (crossings[ept_crossing[to]][add_base1_mod4 (ept_index[to], 2)]));
}
}
void
knot_diagram::rotate_crossing (unsigned c)
{
std::swap (crossings[c][1], crossings[c][3]);
std::swap (crossings[c][2], crossings[c][4]);
for (unsigned j = 1; j <= 4; j ++)
{
unsigned p = crossings[c][j];
ept_index[p] = j;
}
}
unsigned
knot_diagram::total_linking_number () const
{
unionfind<1> u (num_edges ());
for (unsigned i = 1; i <= n_crossings; i ++)
{
u.join (ept_edge (crossings[i][1]),
ept_edge (crossings[i][3]));
u.join (ept_edge (crossings[i][2]),
ept_edge (crossings[i][4]));
}
unsigned n = u.num_sets ();
map<unsigned, unsigned> root_comp;
unsigned t = 0;
for (unsigned i = 1; i <= num_edges (); i ++)
{
if (u.find (i) == i)
{
++ t;
root_comp.push (i, t);
}
}
assert (t == n);
unsigned total_lk = 0;
for (unsigned i = 1; i <= n; i ++)
for (unsigned j = i + 1; j <= n; j ++)
{
if (i == j)
continue;
int lk = 0;
for (unsigned x = 1; x <= n_crossings; x ++)
{
unsigned r1 = root_comp(u.find (ept_edge (crossings[x][1]))),
r2 = root_comp(u.find (ept_edge (crossings[x][2])));
if (((r1 == i) && (r2 == j))
|| ((r2 == i) && (r1 == j)))
{
if (is_to_ept (crossings[x][1]) == is_to_ept (crossings[x][4]))
lk ++;
else
lk --;
}
}
assert (is_even (lk));
lk /= 2;
total_lk += abs (lk);
}
return total_lk;
}
knot_diagram::knot_diagram (connect_sum,
const knot_diagram &d1,
const knot_diagram &d2)
: name(d1.name + "\\#" + d2.name),
n_crossings(d1.n_crossings + d2.n_crossings),
marked_edge(0),
crossings(n_crossings),
ept_crossing(num_epts ()),
ept_index(num_epts ()),
nminus(0), nplus(0)
{
for (unsigned i = 1; i <= d1.n_crossings; i ++)
crossings[i] = basedvector<unsigned, 1> (COPY, d1.crossings[i]);
for (unsigned i = 1; i <= d2.n_crossings; i ++)
{
basedvector<unsigned, 1> v (4);
for (unsigned j = 1; j <= 4; j ++)
v[j] = d1.num_epts () + d2.crossings[i][j];
crossings[d1.n_crossings + i] = v;
}
unsigned p1 = d1.edge_from_ept (1);
crossings[d1.ept_crossing[p1]][d1.ept_index[p1]]
= edge_from_ept (d1.num_edges () + 1);
unsigned p2 = d2.edge_from_ept (1);
crossings[d1.n_crossings + d2.ept_crossing[p2]][d2.ept_index[p2]]
= d1.edge_from_ept (1);
for (unsigned i = 1; i <= n_crossings; i ++)
{
for (unsigned j = 1; j <= 4; j ++)
{
unsigned p = crossings[i][j];
ept_crossing[p] = i;
ept_index[p] = j;
}
}
#ifndef NDEBUG
check_crossings ();
#endif
orient ();
calculate_smoothing_orientation ();
calculate_nminus_nplus ();
}
knot_diagram::knot_diagram (const std::string &name_, unsigned n_crossings_, unsigned crossings_ar[][4])
: name(name_),
n_crossings(n_crossings_),
marked_edge(0),
crossings(n_crossings),
ept_crossing(num_epts ()),
ept_index(num_epts ()),
nminus(0), nplus(0)
{
for (unsigned i = 1; i <= n_crossings; i ++)
{
basedvector<unsigned, 1> v (4);
v[1] = crossings_ar[i - 1][0];
v[2] = crossings_ar[i - 1][1];
v[3] = crossings_ar[i - 1][2];
v[4] = crossings_ar[i - 1][3];
crossings[i] = v;
for (unsigned j = 1; j <= 4; j ++)
{
unsigned p = crossings[i][j];
ept_crossing[p] = i;
ept_index[p] = j;
}
}
calculate_smoothing_orientation ();
calculate_nminus_nplus ();
}
knot_diagram::knot_diagram (const std::string &name_, const basedvector<basedvector<unsigned, 1>, 1> &crossings_)
: name(name_),
n_crossings(crossings_.size ()),
marked_edge(0),
crossings(crossings_),
ept_crossing(num_epts ()),
ept_index(num_epts ()),
nminus(0), nplus(0)
{
for (unsigned i = 1; i <= n_crossings; i ++)
{
for (unsigned j = 1; j <= 4; j ++)
{
unsigned p = crossings[i][j];
ept_crossing[p] = i;
ept_index[p] = j;
}
}
calculate_smoothing_orientation ();
calculate_nminus_nplus ();
}
void
knot_diagram::orient ()
{
bitset done (num_edges ());
for (unsigned i = 1; i <= num_edges (); i ++)
{
if (done % i)
continue;
unsigned p = edge_to_ept (i);
for (;;)
{
unsigned e = ept_edge (p);
done.push (e);
if (is_from_ept (p))
{
unsigned e_from = edge_from_ept (e),
e_to = edge_to_ept (e),
c_from = ept_crossing[e_from],
c_to = ept_crossing[e_to],
j_from = ept_index[e_from],
j_to = ept_index[e_to];
ept_crossing[e_to] = c_from;
ept_index[e_to] = j_from;
ept_crossing[e_from] = c_to;
ept_index[e_from] = j_to;
crossings[c_from][j_from] = e_to;
crossings[c_to][j_to] = e_from;
p = e_to;
}
p = crossings[ept_crossing[p]][add_base1_mod4 (ept_index[p], 2)];
if (ept_edge (p) == i)
break;
p = edge_other_ept (p);
}
}
assert (done.card () == num_edges ());
#ifndef NDEBUG
check_crossings ();
#endif
}
void
knot_diagram::calculate_nminus_nplus ()
{
assert (nplus == 0 && nminus == 0);
for (unsigned i = 1; i <= n_crossings; i ++)
{
if (is_to_ept (crossings[i][1]) == is_to_ept (crossings[i][4]))
nplus ++;
else
nminus ++;
}
}
void
knot_diagram::calculate_smoothing_orientation ()
{
edge_smoothing_oriented = set<unsigned> ();
set<unsigned> done;
vector<unsigned> q;
for (unsigned i = 1; i <= num_edges (); i ++)
{
if (done % i)
continue;
edge_smoothing_oriented.push (i);
done.push (i);
q.append (i);
while (q.size () != 0)
{
unsigned e = q.pop ();
assert (done % e);
unsigned p = edge_smoothing_to_ept (e);
unsigned r = resolve_next_ept (p, 0);
unsigned f = ept_edge (r);
if (done % f)
assert (is_smoothing_from_ept (r));
else
{
if (is_from_ept (r))
edge_smoothing_oriented.push (f);
done.push (f);
q.append (f);
}
unsigned r2 = resolve_next_ept (p, 1);
unsigned f2 = ept_edge (r2);
if (done % f2)
assert (is_smoothing_from_ept (r2));
else
{
if (is_from_ept (r2))
edge_smoothing_oriented.push (f2);
done.push (f2);
q.append (f2);
}
}
}
assert (done.card () == num_edges ());
}
unsigned
knot_diagram::resolve_next_ept (unsigned p, bool resolution) const
{
static unsigned lookup[2][4] = { { 2, 1, 4, 3 }, { 4, 3, 2, 1 } };
unsigned c = ept_crossing[p],
i = ept_index[p];
assert (i >= 1 && i <= 4);
i = lookup[(int)resolution][i - 1];
return crossings[c][i];
}
static unsigned crossing_index_corner (unsigned c, unsigned i)
{
return (c - 1) * 4 + i;
}
static unsigned corner_crossing (unsigned x)
{
return (x - 1) / 4 + 1;
}
static unsigned corner_index (unsigned x)
{
return ((x - 1) % 4) + 1;
}
unsigned
knot_diagram::num_components () const
{
unionfind<1> u (num_edges ());
for (unsigned i = 1; i <= n_crossings; i ++)
{
u.join (ept_edge (crossings[i][1]),
ept_edge (crossings[i][3]));
u.join (ept_edge (crossings[i][2]),
ept_edge (crossings[i][4]));
}
return u.num_sets ();
}
directed_multigraph
knot_diagram::black_graph (basedvector<unsigned, 1> &bg_edge_height) const
{
unsigned n_corners = n_crossings * 4;
#if 1
for (unsigned c = 1; c <= n_crossings; c ++)
for (unsigned j = 1; j <= 4; j ++)
{
unsigned x = crossing_index_corner (c, j);
assert (x >= 1 && x <= n_corners);
assert (corner_crossing (x) == c);
assert (corner_index (x) == j);
}
for (unsigned x = 1; x <= n_corners; x ++)
{
unsigned c = corner_crossing (x);
assert (c >= 1 && c <= n_crossings);
unsigned j = corner_index (x);
assert (j >= 1 && j <= 4);
assert (crossing_index_corner (c, j) == x);
}
#endif
unionfind<1> u (n_corners);
unsigned n_edges = num_edges ();
for (unsigned i = 1; i <= n_edges; i ++)
{
unsigned i_from = edge_from_ept (i),
i_to = edge_to_ept (i);
u.join (crossing_index_corner (ept_crossing (i_from),
ept_index (i_from)),
crossing_index_corner (ept_crossing (i_to),
add_base1_mod4 (ept_index (i_to), 3)));
u.join (crossing_index_corner (ept_crossing (i_from),
add_base1_mod4 (ept_index (i_from), 3)),
crossing_index_corner (ept_crossing (i_to),
ept_index (i_to)));
}
ullmanset<1> present (n_corners);
for (unsigned i = 1; i <= n_corners; i ++)
present += u.find (i);
basedvector<unsigned, 1> edge_from, edge_to;
basedvector<unsigned, 1> edge_height;
for (unsigned i = 1; i <= n_crossings; i ++)
{
edge_from.append (present.position (u.find (crossing_index_corner (i, 1))) + 1);
edge_to.append (present.position (u.find (crossing_index_corner (i, 3))) + 1);
edge_height.append (1);
edge_from.append (present.position (u.find (crossing_index_corner (i, 2))) + 1);
edge_to.append (present.position (u.find (crossing_index_corner (i, 4))) + 1);
edge_height.append (0);
}
assert (u.num_sets () == present.card ());
directed_multigraph bwg (u.num_sets (), edge_from, edge_to);
// display ("bwg:\n", bwg);
assert (bwg.num_components () == 2);
basedvector<unsigned, 1> edge_inj,
vertex_inj;
directed_multigraph bg = bwg.component (1, edge_inj, vertex_inj);
bg_edge_height = basedvector<unsigned, 1> (bg.num_edges ());
for (unsigned i = 1; i <= bg.num_edges (); i ++)
bg_edge_height[i] = edge_height[edge_inj[i]];
return bg;
}
basedvector<basedvector<int, 1>, 1>
knot_diagram::planar_diagram_crossings () const
{
unsigned n_edges = num_edges ();
ullmanset<1> edges (n_edges);
for (unsigned e = 1; e <= n_edges; e ++)
{
if (edges % e)
continue;
for (unsigned i = e;;)
{
edges.push (i);
unsigned c = ept_crossing[edge_to_ept (i)],
j = ept_index[edge_to_ept (i)];
unsigned j2 = add_base1_mod4 (j, 2);
assert (is_from_ept (crossings[c][j2]));
i = ept_edge (crossings[c][j2]);
if (i == e)
break;
}
}
assert (edges.card () == n_edges);
basedvector<basedvector<int, 1>, 1> r (n_crossings);