def tensorDoubleDD(self, dd_graph1, dd_graph2): """Compute the type DD structure DD1 * CFAA(Id) * DD2.""" assert dd_graph1.tensor_side == 2 and dd_graph2.tensor_side == 1 assert dd_graph1.algebra2.opp() == self.pmc_alg assert dd_graph2.algebra1 == self.pmc_alg ddstr = SimpleDDStructure(F2, dd_graph1.algebra1, dd_graph2.algebra2) # Generators of the type DD structure: for ddgen1, node1 in dd_graph1.ddgen_node.items(): for ddgen2, node2 in dd_graph2.ddgen_node.items(): if node1.idem2 == node2.idem1.opp().comp(): cur_gen = DATensorDDGenerator(ddstr, ddgen1, ddgen2) ddstr.addGenerator(cur_gen) # Search the graphs for type DD operations for gen_start in ddstr.getGenerators(): ddgen1, ddgen2 = gen_start d1_pos = dd_graph1.ddgen_node[ddgen1] d2_pos = dd_graph2.ddgen_node[ddgen2] aa_pos = self.homology_node[ddgen1.idem2.opp()] pos = [(d1_pos, d2_pos, aa_pos)] end_states = self._searchDoubleD(dd_graph1, dd_graph2, pos)[0] for d1_end, d2_end, aa_end in end_states: gen_end = DATensorDDGenerator(ddstr, d1_end.ddgen, d2_end.ddgen) ddstr.addDelta(gen_start, gen_end, d1_end.sd, d2_end.sd, 1) return ddstr
def tensorDD(self, ddstr): """Compute the box tensor product DA * DD of this bimodule with the given type DD structure. Returns the resulting type DD structure. Uses delta() and deltaPrefix() functions of this type DA structure. """ ddstr_result = SimpleDDStructure(F2, self.algebra1, ddstr.algebra2) # Compute list of generators in the box tensor product for gen_left in self.getGenerators(): for gen_right in ddstr.getGenerators(): if gen_left.idem2 == gen_right.idem1: ddstr_result.addGenerator(DATensorDDGenerator( ddstr_result, gen_left, gen_right)) def search(start_gen, cur_ddgen, cur_algd, cur_coeffs_a): """Searching for an arrow in the box tensor product. - start_gen: starting generator in the box tensor product. The resulting arrow will start from here. - cur_ddgen: current location in the type DD structure. - cur_algd: current product algebra outputs on the right side of the DD structure. - cur_coeffs_a: current list of A-side inputs to the type DA structure (or alternatively, list of algebra outputs on the left side of the DD structure). """ start_dagen, start_dgen = start_gen cur_delta = self.delta(start_dagen, cur_coeffs_a) for (coeff_d, gen_to), ring_coeff in cur_delta.items(): ddstr_result.addDelta(start_gen, DATensorDDGenerator( ddstr_result, gen_to, cur_ddgen), coeff_d, cur_algd, 1) if self.deltaPrefix(start_dagen, cur_coeffs_a): for (coeff_out1, coeff_out2, dgen_to), ring_coeff in \ ddstr.delta(cur_ddgen).items(): new_algd = cur_algd * coeff_out2 if new_algd != E0: search(start_gen, dgen_to, new_algd.getElt(), cur_coeffs_a + (coeff_out1,)) for x in ddstr_result.getGenerators(): dagen, ddgen = x search(x, ddgen, ddgen.idem2.toAlgElt(ddstr.algebra2), ()) # Add arrows coming from idempotent output on the left DD-side for (coeff_out1, coeff_out2, dgen_to), ring_coeff in \ ddstr.delta(ddgen).items(): if coeff_out1.isIdempotent(): ddstr_result.addDelta( x, DATensorDDGenerator(ddstr_result, dagen, dgen_to), dagen.idem1.toAlgElt(self.algebra1), coeff_out2, 1) # Grading is omitted. return ddstr_result
def getAdmissibleDDStructure(self): """Returns the type DD structure corresponding to the Heegaard diagram created by a finger move of the beta circle to the right. """ alg1 = self.start_pmc.getAlgebra(mult_one = True) alg2 = alg1 ddstr = SimpleDDStructure(F2, alg1, alg2) # Add generators for the non-admissible case - that is, those generators # that do not contain the two intersections created by the finger move. original_idems = self._getIdems() for i in range(len(original_idems)): left_idem, right_idem = original_idems[i] ddstr.addGenerator( SimpleDDGenerator(ddstr, left_idem, right_idem, "0_%d" % i)) # Now add the new generators. These just correspond to the complementary # idempotents with c_pair on the left, repeated twice. left_idems = [idem for idem in self.start_pmc.getIdempotents() if self.c_pair in idem] for i in range(len(left_idems)): left_idem = left_idems[i] right_idem = left_idem.opp().comp() ddstr.addGenerator( SimpleDDGenerator(ddstr, left_idem, right_idem, "1_%d" % i)) ddstr.addGenerator( SimpleDDGenerator(ddstr, left_idem, right_idem, "2_%d" % i)) gen_set = [] for i in range(3): gen_set.append([gen for gen in ddstr.getGenerators() if gen.name[:1] == "%d" % i]) # Enumerate the non-special chords (those that do not dependent on the # idempotent. See the functions themselves for the format of all_chords. if self.is_degenerate: all_chords = self._getAdmissibleNonSpecialChordsDegenerate() else: all_chords = self._getAdmissibleNonSpecialChords() for i, j in itertools.product(range(3), range(3)): all_chords[i][j] = [self._StrandsFromChords(chord1, chord2) for chord1, chord2 in all_chords[i][j]] # Now we emulate the logic in ddstructure.DDStrFromChords, except we # distinguish between ''classes'' of generators, by the first character # of the name of the generator. for i, j in itertools.product(range(3), range(3)): for x, y in itertools.product(gen_set[i], gen_set[j]): for l_chord, r_chord in all_chords[i][j]: if l_chord.idemCompatible(x.idem1, y.idem1) and \ r_chord.idemCompatible(x.idem2, y.idem2): ddstr.addDelta(x, y, StrandDiagram(alg1, x.idem1, l_chord), StrandDiagram(alg2, x.idem2, r_chord), 1) # Special handling for these. From class 2 to class 1, add only if the # c-pair is occupied on the left side (and not on the right). # Non-degenerate cases only. sp_chords = [] if not self.is_degenerate: for x in range(0, self.c1): for y in range(self.c2+1, self.n): sp_chords.append(([(x, y)], [(x, self.u), (self.u, y)])) sp_chords.append(([(x, y)], [(x, self.d), (self.d, y)])) sp_chords.append(([(x, self.d), (self.u, y)], [(x, self.d), (self.u, y)])) sp_chords = [self._StrandsFromChords(chord1, chord2) for chord1, chord2 in sp_chords] for x, y in itertools.product(gen_set[2], gen_set[1]): for l_chord, r_chord in sp_chords: if self.c_pair in x.idem1 and \ l_chord.idemCompatible(x.idem1, y.idem1) and \ r_chord.idemCompatible(x.idem2, y.idem2): assert self.c_pair not in x.idem2.opp() and \ self.c_pair in y.idem1 and \ self.c_pair not in y.idem2.opp() ddstr.addDelta(x, y, StrandDiagram(alg1, x.idem1, l_chord), StrandDiagram(alg2, x.idem2, r_chord), 1) assert ddstr.testDelta() return ddstr