def introduce_left_crossing(x: ETangleStrands, b1: int, b2: int,
crossing: int) -> StrandDiagram:
a1 = x.left_y_pos(b1)
a2 = x.left_y_pos(b2)
orange_strands = {}
orange_signs = {}
for orange in range(1, len(x.etangle.signs)):
if x.etangle.left_y_pos(orange):
# (left, middle, middle) breaks {0: 0, 1: 2} on a cup with (-1, 1)
orange_strands[orange] = (x.etangle.left_y_pos(orange),
x.etangle.left_y_pos(orange),
x.etangle.middle_y_pos(orange))
orange_signs[orange] = x.etangle.signs[orange]
black_strands = {}
for black in x.left_strands.values():
if black == b1:
black_strands[b1] = (a1, crossing + .1, b1)
elif black == b2:
black_strands[b2] = (a2, crossing + .2, b2)
else:
black_strands[black] = (x.left_y_pos(black), x.left_y_pos(black),
black)
return StrandDiagram(orange_strands, orange_signs, black_strands)
def d_mixed(module: Module, x: ETangleStrands) -> Module.TensorElement:
out = module.zero()
for b1 in x.right_strands.keys():
for b2 in x.right_strands.keys():
# if two black strands don't cross on the right
if b1 < b2 and x.right_y_pos(b1) < x.right_y_pos(b2):
out += d_mixed_case_1(module, x, b1, b2)
for b1 in x.left_strands.values():
for b2 in x.left_strands.values():
# if two black strands cross on the left
if b1 < b2 and x.left_y_pos(b1) > x.left_y_pos(b2):
out += d_mixed_case_2(module, x, b1, b2)
for b1 in x.right_strands.keys():
for b2 in x.left_strands.values():
if b1 < b2:
out += d_mixed_case_3(module, x, b1, b2)
for b1 in x.left_strands.values():
for b2 in x.right_strands.keys():
if b1 < b2:
out += d_mixed_case_4(module, x, b1, b2)
return out
def d_minus(module: Module, x: ETangleStrands) -> Module.TensorElement:
out = module.zero()
for black1, black2 in itertools.combinations(x.left_strands.values(), 2):
b1 = min(black1, black2)
b2 = max(black1, black2)
# if the black strands don't cross
if x.left_y_pos(b1) < x.left_y_pos(b2):
# introduce a crossing and add it to the output
out += introduce_left_crossing(module, x, b1, b2)
return out
def introduce_left_crossing(module: Module, x: ETangleStrands, b1: int,
b2: int) -> Module.TensorElement:
a1 = x.left_y_pos(b1)
a2 = x.left_y_pos(b2)
crossing_range_1 = range(a1, b2 + 1) if a1 < b2 else range(b2, a1 + 1)
crossing_range_2 = range(a2, b1 + 1) if a2 < b1 else range(b1, a2 + 1)
possible_crossings = list(set(crossing_range_1) & set(crossing_range_2))
sd = introduce_left_crossing(x, b1, b2, possible_crossings[0])
for crossing in possible_crossings[1:]:
new_sd = introduce_left_crossing(x, b1, b2, crossing)
if new_sd.num_orange_black_crossings() < sd.num_orange_black_crossings(
):
sd = new_sd
c = x.etangle.ring.one()
powers = sd.figure_7_relations()
if powers is None:
return module.zero()
for orange, power in powers.items():
c *= x.etangle.strand_index_to_variable(orange)**power
new_left_strands = swap_values(x.left_strands, a1, a2)
x_out = ETangleStrands(x.etangle, new_left_strands, x.right_strands)
return c * x_out.to_generator(module)
def d_mixed_case_2(module: Module, x: ETangleStrands, b1: int,
b2: int) -> Module.TensorElement:
c = x.etangle.ring.one()
powers = x.to_strand_diagram().figure_8_case_2b(b1, b2)
if powers is None:
return module.zero()
for orange, power in powers.items():
c *= x.etangle.strand_index_to_variable(orange)**power
a1 = x.left_y_pos(b1)
a2 = x.left_y_pos(b2)
x_out = ETangleStrands(x.etangle, swap_values(x.left_strands, a1, a2),
x.right_strands)
return c * x_out.to_generator(module)
def d_mixed_case_4(module: Module, x: ETangleStrands, b1: int,
b2: int) -> Module.TensorElement:
c = x.etangle.ring.one()
powers = x.to_strand_diagram().figure_8_case_4b(b1, b2)
if powers is None:
return module.zero()
for orange, power in powers.items():
c *= x.etangle.strand_index_to_variable(orange)**power
a1 = x.left_y_pos(b1)
a2 = x.right_y_pos(b2)
new_left_strands = dict(x.left_strands)
new_left_strands[a1] = b2
new_right_strands = dict(x.right_strands)
del new_right_strands[b2]
new_right_strands[b1] = a2
x_out = ETangleStrands(x.etangle, new_left_strands, new_right_strands)
return c * x_out.to_generator(module)