Strengthened b'_lk to use Khovanov homology. Set up to compute sp
bounds in paralllel.
This commit is contained in:
Cotton Seed
parent
e19a274c8e
commit
f3d7e22f51
@ -454,6 +454,60 @@ knot_diagram::knot_diagram (sublink,
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calculate_nminus_nplus ();
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}
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knot_diagram::knot_diagram (disjoint_union,
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const knot_diagram &kd1,
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const knot_diagram &kd2)
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: name(kd1.name + "+" + kd2.name),
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n_crossings(kd1.n_crossings + kd2.n_crossings),
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marked_edge(0),
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crossings(n_crossings),
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nminus(kd1.nminus + kd2.nminus),
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nplus(kd1.nplus + kd2.nplus)
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{
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assert (kd1.marked_edge == 0);
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assert (kd2.marked_edge == 0);
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for (unsigned i = 1; i <= n_crossings; i ++)
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crossings[i] = basedvector<unsigned, 1> (4);
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for (unsigned i = 1; i <= kd1.n_crossings; i ++)
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for (unsigned j = 1; j <= 4; j ++)
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crossings[i][j] = kd1.crossings[i][j];
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for (unsigned e = 1; e <= kd1.num_edges (); e ++)
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{
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if (kd1.edge_smoothing_oriented % e)
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edge_smoothing_oriented.push (e);
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}
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for (unsigned i = 1; i <= kd2.n_crossings; i ++)
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for (unsigned j = 1; j <= 4; j ++)
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crossings[kd1.n_crossings + i][j] = kd1.num_epts () + kd2.crossings[i][j];
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for (unsigned e = 1; e <= kd2.num_edges (); e ++)
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{
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if (kd2.edge_smoothing_oriented % e)
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edge_smoothing_oriented.push (kd1.num_edges () + e);
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}
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// ?? break this out into aux function
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ept_crossing = basedvector<unsigned, 1> (num_epts ());
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ept_index = basedvector<unsigned, 1> (num_epts ());
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for (unsigned i = 1; i <= n_crossings; i ++)
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{
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for (unsigned j = 1; j <= 4; j ++)
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{
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unsigned p = crossings[i][j];
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ept_crossing[p] = i;
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ept_index[p] = j;
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}
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}
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#ifndef NDEBUG
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check_crossings ();
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#endif
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}
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knot_diagram::knot_diagram (mirror, const knot_diagram &kd)
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: name("mirror(" + kd.name + ")"),
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n_crossings(kd.n_crossings),
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@ -11,6 +11,7 @@ add_base1_mod4 (unsigned x, unsigned y)
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enum mirror { MIRROR };
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enum connect_sum { CONNECT_SUM };
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enum sublink { SUBLINK };
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enum disjoint_union { DISJOINT_UNION };
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class knot_diagram
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{
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@ -122,6 +123,9 @@ class knot_diagram
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knot_diagram (sublink,
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smallbitset c,
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const knot_diagram &kd);
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knot_diagram (disjoint_union,
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const knot_diagram &kd1,
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const knot_diagram &kd2);
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knot_diagram (const std::string &name_, unsigned n_crossings_, unsigned crossings_ar[][4]);
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knot_diagram (const std::string &name_, const basedvector<basedvector<unsigned, 1>, 1> &crossings_);
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126
main.cpp
126
main.cpp
@ -2262,6 +2262,106 @@ permutations (unsigned n)
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return permutations (v);
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}
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template<class R> multivariate_laurentpoly<Z>
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Kh_poincare_polynomial (knot_diagram &kd)
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{
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cube<R> c (kd);
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mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
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chain_complex_simplifier<R> s (c.khC, d, 0);
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assert (s.new_d == 0);
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return s.new_C->free_poincare_polynomial ();
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}
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unsigned
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compute_b_lk_weak (knot_diagram &kd)
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{
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unsigned m = kd.num_components ();
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assert (m > 1);
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if (m == 2)
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{
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unsigned total_lk = kd.total_linking_number ();
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return total_lk == 0 ? 2 : total_lk;
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}
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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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assert (m == 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 <= 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 == m);
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unsigned b_lk_weak = 0;
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for (unsigned i = 1; i <= m; i ++)
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for (unsigned j = i + 1; j <= m; j ++)
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{
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assert (i < j);
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int lk = 0;
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for (unsigned x = 1; x <= kd.n_crossings; x ++)
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{
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unsigned r1 = root_comp(u.find (kd.ept_edge (kd.crossings[x][1]))),
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r2 = root_comp(u.find (kd.ept_edge (kd.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 (kd.is_to_ept (kd.crossings[x][1]) == kd.is_to_ept (kd.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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if (lk == 0)
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{
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smallbitset ci (m);
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ci.push (i);
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smallbitset cj (m);
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cj.push (j);
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smallbitset c (m);
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c.push (i);
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c.push (j);
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knot_diagram Lij (SUBLINK, c, kd);
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multivariate_laurentpoly<Z> P_Lij = Kh_poincare_polynomial<Z2> (Lij);
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knot_diagram Li_join_Lj (DISJOINT_UNION,
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knot_diagram (SUBLINK, ci, kd),
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knot_diagram (SUBLINK, cj, kd));
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multivariate_laurentpoly<Z> P_Li_join_Lj = Kh_poincare_polynomial<Z2> (Li_join_Lj);
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if (P_Lij != P_Li_join_Lj)
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lk = 2; // non-split
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}
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b_lk_weak += abs (lk);
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}
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return b_lk_weak == 0 ? 2 : b_lk_weak;
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}
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void
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compute_splitting_bounds ()
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{
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@ -2319,25 +2419,18 @@ compute_splitting_bounds ()
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printf (" b = %d\n", b);
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unsigned total_lk = kd.total_linking_number ();
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unsigned b_lk_weak = total_lk == 0 ? 2 : total_lk;
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unsigned b_lk_weak = compute_b_lk_weak (kd);
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unsigned b_lk_weaker = total_lk == 0 ? 2 : total_lk;
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printf (" b_lk_weaker = %d\n", b_lk_weaker);
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printf (" b_lk_weak = %d\n", b_lk_weak);
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basedvector<basedvector<unsigned, 1>, 1> ps = permutations (m);
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#if 0
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printf ("ps, |ps| = %d, m = %d:\n", ps.size (), m);
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for (unsigned i = 1; i <= ps.size (); i ++)
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{
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basedvector<unsigned, 1> p = ps[i];
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assert (p.size () == m);
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printf (" % 3d: ", i);
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for (unsigned j = 1; j <= m; j ++)
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printf (" %d", p[j]);
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newline ();
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}
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#endif
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assert (b_lk_weaker <= b_lk_weak);
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if (b_lk_weaker < b_lk_weak)
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printf (" > STRICTLY WEAKER\n");
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basedvector<basedvector<unsigned, 1>, 1> ps = permutations (m);
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unsigned r = kd.n_crossings;
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for (unsigned i = 1; i <= ps.size (); i ++)
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{
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@ -2362,6 +2455,7 @@ compute_splitting_bounds ()
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}
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printf (" r = %d\n", r);
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assert (b_lk_weak <= r);
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assert (b <= r);
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// non-trivial link, sp at least 1.
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@ -2383,12 +2477,10 @@ compute_splitting_bounds ()
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printf (" > sp = %d (b + b_lk_weak)\n", b);
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else
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printf (" > %d <= sp <= %d (b + b_lk_weak)\n", b, r);
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}
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else if (b_lk_weak == r)
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{
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assert (b < b_lk_weak);
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printf (" > sp = %d (b_lk_weak)\n", b_lk_weak);
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}
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}
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227
mpimain.cpp
227
mpimain.cpp
@ -14,10 +14,10 @@ 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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for (unsigned i = 1; i <= 10; i ++)
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for (unsigned j = 1; j <= mt_links (i); j ++)
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{
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knot_diagram kd (mt_links (i, 0, j));
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knot_diagram kd (mt_links (i, 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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@ -116,6 +116,7 @@ file_exists (const char *file)
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return 1;
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}
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#if 0
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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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@ -291,6 +292,215 @@ compute_forgetful (int rank, knot_desc desc, const char *buf)
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write (w, desc);
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write (w, pages);
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}
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#endif
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unsigned
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compute_b_lk_weak (knot_diagram &kd)
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{
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unsigned m = kd.num_components ();
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assert (m > 1);
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if (m == 2)
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{
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unsigned total_lk = kd.total_linking_number ();
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return total_lk == 0 ? 2 : total_lk;
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}
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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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assert (m == 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 <= 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 == m);
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unsigned b_lk_weak = 0;
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for (unsigned i = 1; i <= m; i ++)
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for (unsigned j = i + 1; j <= m; j ++)
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{
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assert (i < j);
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int lk = 0;
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for (unsigned x = 1; x <= kd.n_crossings; x ++)
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{
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unsigned r1 = root_comp(u.find (kd.ept_edge (kd.crossings[x][1]))),
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r2 = root_comp(u.find (kd.ept_edge (kd.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 (kd.is_to_ept (kd.crossings[x][1]) == kd.is_to_ept (kd.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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if (lk == 0)
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{
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smallbitset ci (m);
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ci.push (i);
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smallbitset cj (m);
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cj.push (j);
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smallbitset c (m);
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c.push (i);
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c.push (j);
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knot_diagram Lij (SUBLINK, c, kd);
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multivariate_laurentpoly<Z> P_Lij = Kh_poincare_polynomial<Z2> (Lij);
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knot_diagram Li_join_Lj (DISJOINT_UNION,
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knot_diagram (SUBLINK, ci, kd),
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knot_diagram (SUBLINK, cj, kd));
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multivariate_laurentpoly<Z> P_Li_join_Lj = Kh_poincare_polynomial<Z2> (Li_join_Lj);
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if (P_Lij != P_Li_join_Lj)
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lk = 2; // non-split
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}
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b_lk_weak += abs (lk);
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}
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return b_lk_weak == 0 ? 2 : b_lk_weak;
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}
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void
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compute_splitting_bounds (knot_diagram &kd)
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{
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int rank = self_rank ();
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typedef fraction_field<polynomial<Z2> > Z2x;
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unsigned m = kd.num_components ();
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assert (m > 1);
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printf ("[% 2d] ", rank); show (kd); newline ();
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printf ("[% 2d] m = %d\n", rank, m);
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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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assert (u.num_sets () == m);
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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 == m);
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basedvector<Q, 1> comp_weightQ (m);
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for (unsigned i = 1; i <= m; i ++)
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comp_weightQ[i] = Q (i);
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unsigned bQ = splitting_bound<Q> (kd, comp_weightQ);
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basedvector<Z2x, 1> comp_weightZ2x (m);
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for (unsigned i = 1; i <= m; i ++)
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comp_weightZ2x[i] = Z2x (polynomial<Z2> (Z2 (1), i));
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unsigned bZ2x = splitting_bound<Z2x> (kd, comp_weightZ2x);
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// lower bound
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unsigned b = std::max (bQ, bZ2x);
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printf ("[% 2d] bQ = %d\n", rank, bQ);
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printf ("[% 2d] bZ2x = %d\n", rank, bZ2x);
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printf ("[% 2d] b = %d\n", rank, b);
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unsigned total_lk = kd.total_linking_number ();
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unsigned b_lk_weaker = total_lk == 0 ? 2 : total_lk;
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unsigned b_lk_weak = compute_b_lk_weak (kd);
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assert (b_lk_weaker <= b_lk_weak);
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printf ("[% 2d] b_lk_weaker = %d\n", rank, b_lk_weaker);
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printf ("[% 2d] b_lk_weak = %d\n", rank, b_lk_weak);
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if (b_lk_weaker < b_lk_weak)
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printf ("[% 2d] > STRICTLY WEAKER\n", rank);
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basedvector<basedvector<unsigned, 1>, 1> ps = permutations (m);
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unsigned r = kd.n_crossings;
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for (unsigned i = 1; i <= ps.size (); i ++)
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{
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basedvector<unsigned, 1> p = ps[i];
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unsigned ri = 0;
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for (unsigned j = 1; j <= kd.n_crossings; j ++)
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{
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unsigned upper_e = kd.ept_edge (kd.crossings[j][2]),
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lower_e = kd.ept_edge (kd.crossings[j][1]);
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unsigned upper_c = root_comp(u.find (upper_e)),
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lower_c = root_comp(u.find (lower_e));
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if (upper_c != lower_c
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&& p[upper_c] < p[lower_c])
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ri ++;
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}
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if (ri < r)
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r = ri;
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}
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printf ("[% 2d] r = %d\n", rank, r);
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assert (b_lk_weak <= r);
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assert (b <= r);
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// non-trivial link, sp at least 1.
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unsigned best = std::max (b, b_lk_weak);
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if (best == r)
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printf ("[% 2d] > sp = %d", rank, r);
|
||||
else
|
||||
printf ("[% 2d] > %d <= sp <= %d", rank, best, r);
|
||||
|
||||
printf (" ");
|
||||
|
||||
if (b == best
|
||||
&& b_lk_weak == best)
|
||||
printf ("(b + b_lk_weak)");
|
||||
else if (b == best)
|
||||
printf ("(b)");
|
||||
else
|
||||
{
|
||||
assert (b_lk_weak == best);
|
||||
printf ("(b_lk_weak)");
|
||||
}
|
||||
|
||||
fflush (stdout);
|
||||
}
|
||||
|
||||
void
|
||||
slave ()
|
||||
@ -311,15 +521,8 @@ 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/forgetful/L%d_%d.dat.gz",
|
||||
desc.i, desc.j);
|
||||
|
||||
if (! file_exits ())
|
||||
compute_forgetful (rank, desc, buf);
|
||||
else
|
||||
printf ("skip %s: exists.\n", buf);
|
||||
knot_diagram kd = desc.diagram ();
|
||||
compute_splitting_bounds (kd);
|
||||
|
||||
send_int (0, 0);
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user