Added compute_forgetful_signs to main. Set up mpimain to compute
forgetful spectral sequence over Z/97 for all links with 14 or fewer crossings.
This commit is contained in:
Cotton Seed
parent
177b37bb57
commit
f2375a2d9e
17
cube_impl.h
17
cube_impl.h
@ -365,6 +365,13 @@ cube<R>::compute_dinv (unsigned c)
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if (!unsigned_bittest (i, c))
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continue;
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int sign = 1;
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for (unsigned j = 1; j < c; j ++)
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{
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if (unsigned_bittest (i, j))
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sign *= -1;
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}
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smoothing from_s (kd, smallbitset (n_crossings, i));
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unsigned i2 = unsigned_bitclear (i, c);
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@ -398,12 +405,12 @@ cube<R>::compute_dinv (unsigned c)
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j2 = unsigned_bitset (j2, x);
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j2 = unsigned_bitclear (j2, y);
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p[generator (i, j)].muladd (1, generator (i2, j2));
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p[generator (i, j)].muladd (sign, generator (i2, j2));
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j2 = unsigned_bitclear (j2, x);
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j2 = unsigned_bitset (j2, y);
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p[generator (i, j)].muladd (1, generator (i2, j2));
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p[generator (i, j)].muladd (sign, generator (i2, j2));
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}
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else
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{
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@ -411,7 +418,7 @@ cube<R>::compute_dinv (unsigned c)
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j2 = unsigned_bitclear (j2, x);
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j2 = unsigned_bitclear (j2, y);
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p[generator (i, j)].muladd (1, generator (i2, j2));
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p[generator (i, j)].muladd (sign, generator (i2, j2));
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}
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}
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else
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@ -425,7 +432,7 @@ cube<R>::compute_dinv (unsigned c)
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// 1 -> 1
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j2 = unsigned_bitset (j2, x);
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p[generator (i, j)].muladd (1, generator (i2, j2));
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p[generator (i, j)].muladd (sign, generator (i2, j2));
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}
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else if ((unsigned_bittest (j, a)
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&& !unsigned_bittest (j, b))
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@ -435,7 +442,7 @@ cube<R>::compute_dinv (unsigned c)
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// a, b -> x
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j2 = unsigned_bitclear (j2, x);
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p[generator (i, j)].muladd (1, generator (i2, j2));
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p[generator (i, j)].muladd (sign, generator (i2, j2));
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}
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}
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}
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@ -22,7 +22,7 @@ knot_diagram::knot_diagram (const planar_diagram &pd)
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crossings[c] = basedvector<unsigned, 1> (4);
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}
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smallbitset done (num_edges ());
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set<unsigned> done;
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for (unsigned c0 = 1; c0 <= n_crossings; c0 ++)
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{
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for (unsigned i0 = 1; i0 <= 2; i0 ++)
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@ -521,6 +521,63 @@ knot_diagram::rotate_crossing (unsigned c)
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}
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}
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unsigned
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knot_diagram::total_linking_number () const
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{
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unionfind<1> u (num_edges ());
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for (unsigned i = 1; i <= n_crossings; i ++)
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{
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u.join (ept_edge (crossings[i][1]),
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ept_edge (crossings[i][3]));
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u.join (ept_edge (crossings[i][2]),
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ept_edge (crossings[i][4]));
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}
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unsigned n = u.num_sets ();
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map<unsigned, unsigned> root_comp;
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unsigned t = 0;
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for (unsigned i = 1; i <= num_edges (); i ++)
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{
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if (u.find (i) == i)
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{
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++ t;
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root_comp.push (i, t);
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}
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}
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assert (t == n);
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unsigned total_lk = 0;
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for (unsigned i = 1; i <= n; i ++)
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for (unsigned j = i + 1; j <= n; j ++)
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{
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if (i == j)
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continue;
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int lk = 0;
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for (unsigned x = 1; x <= n_crossings; x ++)
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{
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unsigned r1 = root_comp(u.find (ept_edge (crossings[x][1]))),
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r2 = root_comp(u.find (ept_edge (crossings[x][2])));
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if (((r1 == i) && (r2 == j))
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|| ((r2 == i) && (r1 == j)))
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{
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if (is_to_ept (crossings[x][1]) == is_to_ept (crossings[x][4]))
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lk ++;
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else
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lk --;
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}
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}
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assert (is_even (lk));
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lk /= 2;
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total_lk += std::abs (lk);
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}
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return total_lk;
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}
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knot_diagram::knot_diagram (connect_sum,
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const knot_diagram &d1,
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const knot_diagram &d2)
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@ -157,6 +157,8 @@ class knot_diagram
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int writhe () const { return (int)nplus - (int)nminus; }
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unsigned total_linking_number () const;
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basedvector<basedvector<int, 1>, 1> planar_diagram_crossings () const;
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hash_t hash_self () const;
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@ -1,7 +1,7 @@
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#include <knotkit.h>
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#define HOME "/u/cseed/src/knotkit/"
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#define HOME "/Users/cotton/src/knotkit/"
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bool verbose = 0;
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180
main.cpp
180
main.cpp
@ -849,6 +849,183 @@ test_forgetful_ss ()
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}
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}
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void
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test_forgetful_signs ()
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{
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typedef Zp<97> R;
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#if 0
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for (unsigned i = 1; i <= 14; i ++)
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for (unsigned j = 1; j <= mt_links (i, 0); j ++)
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{
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knot_diagram kd (mt_link (i, 0, j));
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}
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#endif
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#if 1
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for (unsigned i = 2; i <= 16; i ++)
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for (unsigned j = i; j <= 16; j ++)
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{
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knot_diagram kd (torus_knot (i, j));
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assert (kd.n_crossings == (i - 1) * j
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&& kd.n_crossings <= i * (j - 1));
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if (kd.n_crossings > 13)
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continue;
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#endif
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unsigned n = kd.num_components ();
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if (n < 2)
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continue;
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assert (n < 97);
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show (kd); newline ();
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unionfind<1> u (kd.num_edges ());
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for (unsigned i = 1; i <= kd.n_crossings; i ++)
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{
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u.join (kd.ept_edge (kd.crossings[i][1]),
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kd.ept_edge (kd.crossings[i][3]));
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u.join (kd.ept_edge (kd.crossings[i][2]),
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kd.ept_edge (kd.crossings[i][4]));
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}
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map<unsigned, unsigned> root_comp;
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unsigned t = 0;
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for (unsigned i = 1; i <= kd.num_edges (); i ++)
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{
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if (u.find (i) == i)
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{
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++ t;
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root_comp.push (i, t);
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}
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}
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assert (t == n);
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basedvector<R, 1> comp_weight (n);
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for (unsigned i = 1; i <= n; i ++)
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comp_weight[i] = R ((int)i);
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map<unsigned, R> crossing_over_sign;
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// crossings
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set<unsigned> pending;
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set<unsigned> finished;
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crossing_over_sign.push (1, R (1));
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pending.push (1);
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while (pending.card () > 0)
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{
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unsigned x = pending.pop ();
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finished.push (x);
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R s = crossing_over_sign(x);
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for (unsigned j = 1; j <= 4; j ++)
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{
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unsigned p = kd.crossings[x][j];
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R t = kd.is_over_ept (p) ? s : -s; // sign of (x, p)
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unsigned q = kd.edge_other_ept (p);
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unsigned x2 = kd.ept_crossing[q];
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R u = kd.is_over_ept (q) ? -t : t;
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if (crossing_over_sign % x2)
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assert (crossing_over_sign(x2) == u);
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else
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crossing_over_sign.push (x2, u);
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if (! (finished % x2))
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pending += x2;
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}
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}
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assert (finished.card () == kd.n_crossings);
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#if 0
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printf ("crossing_over_sign:\n");
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for (unsigned i = 1; i <= kd.n_crossings; i ++)
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{
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printf (" % 2d: ", i);
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display (crossing_over_sign(i));
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}
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#endif
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cube<R> c (kd);
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#if 0
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for (unsigned i = 0; i < c.n_resolutions; i ++)
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{
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smallbitset state (kd.n_crossings, i);
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smoothing s (kd, state);
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s.show_self (kd, state);
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newline ();
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}
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#endif
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mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
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assert (untwisted_d.compose (untwisted_d) == 0);
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mod_map<R> d = untwisted_d;
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for (unsigned x = 1; x <= kd.n_crossings; x ++)
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{
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unsigned p1 = kd.crossings[x][1],
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p2 = kd.crossings[x][2];
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assert (kd.is_over_ept (p2));
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unsigned r1 = u.find (kd.ept_edge (p1)),
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r2 = u.find (kd.ept_edge (p2));
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unsigned c1 = root_comp(r1),
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c2 = root_comp(r2);
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if (c1 != c2)
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{
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R s = crossing_over_sign(x);
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R w_under = comp_weight[c1];
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R w_over = comp_weight[c2];
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d = d + c.compute_dinv (x)*(s*(w_over - w_under))
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;
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}
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}
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// display ("d:\n", d);
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// mod_map<Z> h = d.compose (d);
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// display ("h:\n", h);
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assert (d.compose (d) == 0);
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ptr<const module<R> > Ek = c.khC;
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mod_map<R> dk = d;
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unsigned kinf;
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for (int dq = 0;; dq -= 2)
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{
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chain_complex_simplifier<R> s (Ek, dk, dq);
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Ek = s.new_C;
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dk = s.new_d;
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printf ("|E%d| = %d\n", ((- dq) / 2) + 1, Ek->dim ());
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if (dk == 0)
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{
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kinf = ((- dq) / 2);
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break;
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}
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}
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unsigned total_lk = kd.total_linking_number ();
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unsigned trivial_bound = total_lk == 0 ? 2 : total_lk;
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printf ("kinf = %d, trivial bound = %d (total_lk = %d)\n",
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kinf, trivial_bound, total_lk);
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if (trivial_bound < kinf)
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printf (" > BETTER\n");
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}
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}
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/* Kh homotopy type:
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- check for CP^2,
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- mutation invariance
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@ -1578,6 +1755,9 @@ convert_15 ()
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int
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main ()
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{
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test_forgetful_signs ();
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return 0;
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find_width4_in_h0 ();
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// convert_15 ();
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// test_knot_sq ();
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240
mpimain.cpp
240
mpimain.cpp
@ -14,11 +14,24 @@ master ()
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{
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basedvector<knot_desc, 1> work;
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for (unsigned i = 1; i <= 14; i ++)
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for (unsigned j = 1; j <= mt_links (i, 0); j ++)
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{
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knot_diagram kd (mt_links (i, 0, j));
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unsigned n = kd.num_components ();
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if (n < 2)
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continue;
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work.append (knot_desc (knot_desc::MT, i, j));
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}
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#if 0
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basedvector<basedvector<unsigned, 1>, 1> groups = mutant_knot_groups (15);
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for (unsigned i = 1; i <= groups.size (); i ++)
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for (unsigned j = 1; j <= groups[i].size (); j ++)
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work.append (knot_desc (knot_desc::HTW, 15, groups[i][j]));
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#endif
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#if 0
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for (unsigned i = 14; i >= 1; i --)
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@ -87,6 +100,198 @@ master ()
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}
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}
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bool
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file_exists (const char *file)
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{
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struct stat stat_buf;
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if (stat (file, &stat_buf) != 0)
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{
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if (errno == ENOENT)
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return 0;
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stderror ("stat: %s", file);
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exit (EXIT_FAILURE);
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}
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return 1;
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}
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void
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compute_forgetful (int rank, knot_desc desc, const char *buf)
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{
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knot_diagram kd = desc.diagram ();
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typedef Zp<97> R;
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unsigned n = kd.num_components ();
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assert (n >= 2);
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assert (n < 97);
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printf ("[% 2d] ", rank);
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show (kd);
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newline ();
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unionfind<1> u (kd.num_edges ());
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for (unsigned i = 1; i <= kd.n_crossings; i ++)
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{
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u.join (kd.ept_edge (kd.crossings[i][1]),
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kd.ept_edge (kd.crossings[i][3]));
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u.join (kd.ept_edge (kd.crossings[i][2]),
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kd.ept_edge (kd.crossings[i][4]));
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}
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map<unsigned, unsigned> root_comp;
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unsigned t = 0;
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for (unsigned i = 1; i <= kd.num_edges (); i ++)
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{
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if (u.find (i) == i)
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{
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++ t;
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root_comp.push (i, t);
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}
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}
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assert (t == n);
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basedvector<R, 1> comp_weight (n);
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for (unsigned i = 1; i <= n; i ++)
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comp_weight[i] = R ((int)i);
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map<unsigned, R> crossing_over_sign;
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// crossings
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set<unsigned> pending;
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set<unsigned> finished;
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crossing_over_sign.push (1, R (1));
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pending.push (1);
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while (pending.card () > 0)
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{
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unsigned x = pending.pop ();
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finished.push (x);
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R s = crossing_over_sign(x);
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for (unsigned j = 1; j <= 4; j ++)
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{
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unsigned p = kd.crossings[x][j];
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R t = kd.is_over_ept (p) ? s : -s; // sign of (x, p)
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unsigned q = kd.edge_other_ept (p);
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unsigned x2 = kd.ept_crossing[q];
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R u = kd.is_over_ept (q) ? -t : t;
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if (crossing_over_sign % x2)
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assert (crossing_over_sign(x2) == u);
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else
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crossing_over_sign.push (x2, u);
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if (! (finished % x2))
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pending += x2;
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}
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}
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assert (finished.card () == kd.n_crossings);
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#if 0
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printf ("crossing_over_sign:\n");
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for (unsigned i = 1; i <= kd.n_crossings; i ++)
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{
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printf (" % 2d: ", i);
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display (crossing_over_sign(i));
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}
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#endif
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cube<R> c (kd);
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#if 0
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for (unsigned i = 0; i < c.n_resolutions; i ++)
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{
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smallbitset state (kd.n_crossings, i);
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smoothing s (kd, state);
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s.show_self (kd, state);
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newline ();
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}
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#endif
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mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
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assert (untwisted_d.compose (untwisted_d) == 0);
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mod_map<R> d = untwisted_d;
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for (unsigned x = 1; x <= kd.n_crossings; x ++)
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{
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unsigned p1 = kd.crossings[x][1],
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p2 = kd.crossings[x][2];
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assert (kd.is_over_ept (p2));
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unsigned r1 = u.find (kd.ept_edge (p1)),
|
||||
r2 = u.find (kd.ept_edge (p2));
|
||||
|
||||
unsigned c1 = root_comp(r1),
|
||||
c2 = root_comp(r2);
|
||||
|
||||
if (c1 != c2)
|
||||
{
|
||||
R s = crossing_over_sign(x);
|
||||
|
||||
R w_under = comp_weight[c1];
|
||||
R w_over = comp_weight[c2];
|
||||
|
||||
d = d + c.compute_dinv (x)*(s*(w_over - w_under))
|
||||
;
|
||||
}
|
||||
}
|
||||
|
||||
// display ("d:\n", d);
|
||||
|
||||
// mod_map<Z> h = d.compose (d);
|
||||
// display ("h:\n", h);
|
||||
|
||||
assert (d.compose (d) == 0);
|
||||
|
||||
ptr<const module<R> > Ek = c.khC;
|
||||
mod_map<R> dk = d;
|
||||
|
||||
basedvector<multivariate_laurentpoly<Z>, 1> pages;
|
||||
|
||||
unsigned kinf;
|
||||
for (int dq = 0;; dq -= 2)
|
||||
{
|
||||
chain_complex_simplifier<R> s (Ek, dk, dq);
|
||||
Ek = s.new_C;
|
||||
dk = s.new_d;
|
||||
|
||||
pages.append (Ek->free_poincare_polynomial ());
|
||||
|
||||
printf ("[% 2d] ", rank);
|
||||
printf ("|E%d| = %d\n", ((- dq) / 2) + 1, Ek->dim ());
|
||||
if (dk == 0)
|
||||
{
|
||||
kinf = ((- dq) / 2);
|
||||
break;
|
||||
}
|
||||
}
|
||||
|
||||
unsigned total_lk = kd.total_linking_number ();
|
||||
|
||||
unsigned trivial_bound = total_lk == 0 ? 2 : total_lk;
|
||||
|
||||
printf ("[% 2d] ", rank);
|
||||
printf ("kinf = %d, trivial bound = %d (total_lk = %d)\n",
|
||||
kinf, trivial_bound, total_lk);
|
||||
|
||||
if (trivial_bound < kinf)
|
||||
{
|
||||
printf ("[% 2d] ", rank);
|
||||
printf (" > BETTER\n");
|
||||
}
|
||||
fflush (stdout);
|
||||
|
||||
assert (desc.t == knot_desc::MT);
|
||||
gzfile_writer w (buf);
|
||||
write (w, desc);
|
||||
write (w, pages);
|
||||
}
|
||||
|
||||
void
|
||||
slave ()
|
||||
{
|
||||
@ -106,40 +311,13 @@ slave ()
|
||||
|
||||
printf ("[% 2d] CMD_DO %s\n", rank, desc.name ().c_str ());
|
||||
|
||||
assert (desc.t == knot_desc::MT);
|
||||
char buf[1000];
|
||||
sprintf (buf, "/scratch/network/cseed/incoming/K%d_%d.dat.gz",
|
||||
sprintf (buf, "/scratch/network/cseed/forgetful/L%d_%d.dat.gz",
|
||||
desc.i, desc.j);
|
||||
|
||||
struct stat stat_buf;
|
||||
if (stat (buf, &stat_buf) != 0)
|
||||
{
|
||||
if (errno == ENOENT)
|
||||
{
|
||||
knot_diagram kd = desc.diagram ();
|
||||
|
||||
cube<Z2> c (kd);
|
||||
mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
|
||||
|
||||
chain_complex_simplifier<Z2> s (c.khC, d, 1);
|
||||
assert (s.new_d == 0);
|
||||
|
||||
steenrod_square sq (c, d, s);
|
||||
mod_map<Z2> sq1 = sq.sq1 ();
|
||||
mod_map<Z2> sq2 = sq.sq2 ();
|
||||
|
||||
assert (sq1.compose (sq1) == 0);
|
||||
assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
|
||||
|
||||
gzfile_writer w (buf);
|
||||
write (w, sq1);
|
||||
write (w, sq2);
|
||||
}
|
||||
else
|
||||
{
|
||||
stderror ("stat: %s", buf);
|
||||
exit (EXIT_FAILURE);
|
||||
}
|
||||
}
|
||||
if (! file_exits ())
|
||||
compute_forgetful (rank, desc, buf);
|
||||
else
|
||||
printf ("skip %s: exists.\n", buf);
|
||||
|
||||
|
||||
@ -63,7 +63,7 @@ class chain_complex_simplifier
|
||||
public:
|
||||
chain_complex_simplifier (ptr<const module<R> > C_,
|
||||
const mod_map<R> &d_,
|
||||
int dh);
|
||||
int dq);
|
||||
|
||||
};
|
||||
|
||||
@ -145,7 +145,7 @@ chain_complex_simplifier<R>::cancel (unsigned i, R b, unsigned j)
|
||||
template<class R>
|
||||
chain_complex_simplifier<R>::chain_complex_simplifier (ptr<const module<R> > C_,
|
||||
const mod_map<R> &d_,
|
||||
int dh)
|
||||
int dq)
|
||||
: C(C_), n(C_->dim ()), d(d_),
|
||||
new_d_columns(n),
|
||||
preim(n),
|
||||
@ -172,9 +172,8 @@ chain_complex_simplifier<R>::chain_complex_simplifier (ptr<const module<R> > C_,
|
||||
for (linear_combination_const_iter<R> j = new_d_columns[i]; j; j ++)
|
||||
{
|
||||
grading jgr = C->generator_grading (j.key ());
|
||||
// assert (jgr.h >= igr.h);
|
||||
if (j.val ().is_unit ()
|
||||
// && jgr.h - igr.h == dh
|
||||
&& jgr.q - igr.q == dq
|
||||
)
|
||||
{
|
||||
cancel (i, j.val (), j.key ());
|
||||
|
||||
Loading…
Reference in New Issue
Block a user