def test_m2():
# Figure 10 from "An introduction..."
cap2 = ETangle(ETangle.Type.CAP, (1, -1, -1), 1)
cap2_module = empty_da_module(cap2)
sd_cap2_1 = ETangleStrands(cap2, {0: 0, 1: 3}, {2: 0})
sd_cap2_1_out = ETangleStrands(cap2, {0: 0, 1: 3}, {2: 1}).to_generator(cap2_module)
algebra1 = AMinus((-1,))
elt1 = algebra1.generator({0: 1})
idem = algebra1.generator({0: 0})
assert m2(cap2_module, sd_cap2_1, elt1) == sd_cap2_1_out
assert m2(cap2_module, sd_cap2_1, idem) == sd_cap2_1.to_generator(cap2_module)
cup1 = ETangle(ETangle.Type.CUP, (1, -1), 1)
cup1_module = empty_da_module(cup1)
sd_cup1_1 = ETangleStrands(cup1, {}, {2: 1, 0: 0})
algebra2 = AMinus((1, -1))
idem2 = algebra2.generator({0: 0, 1: 1})
elt = algebra2.generator({1: 0, 2: 2})
sd_cup1_2 = ETangleStrands(cup1, {}, {2: 2, 0: 1})
sd_cup1_2_out = cup1_module.ring['U1'] * ETangleStrands(cup1, {}, {2: 2, 0: 0}).to_generator(cup1_module)
assert m2(cup1_module, sd_cup1_1, idem2) == sd_cup1_1.to_generator(cup1_module)
assert m2(cup1_module, sd_cup1_2, elt) == sd_cup1_2_out
cup2 = ETangle(ETangle.Type.CUP, (-1, 1), 1)
cup2_module = empty_da_module(cup2)
sd_cup2_1 = ETangleStrands(cup2, {}, {0: 0, 2: 1})
elt2 = cup2.right_algebra.generator({0: 0, 1: 2})
sd_cup2_1_out = cup2_module.ring['U1'] * ETangleStrands(cup2, {}, {0: 0, 2: 2}).to_generator(cup2_module)
assert m2(cup2_module, sd_cup2_1, elt2) == sd_cup2_1_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 test_delta_ell():
over1 = ETangle(ETangle.Type.OVER, (-1, -1, 1), 2)
over1_module = empty_da_module(over1)
x = ETangleStrands(over1, {1: 0, 2: 1}, {3: 0, 2: 2})
y = ETangleStrands(over1, {0: 0, 2: 1}, {3: 0, 2: 2})
c = over1.ring.one()
elt = over1.left_algebra.generator({0: 1, 3: 3}).to_element()
out = elt ** (c * y.to_generator(over1_module))
assert delta_ell(over1_module, x) == out
def test_delta_ell_case_4():
over1 = ETangle(ETangle.Type.OVER, (-1, -1), 1)
over1_module = empty_da_module(over1)
x = ETangleStrands(over1, {0: 1, 1: 0}, {2: 2})
a1 = 0
a2 = 2
y = ETangleStrands(over1, {2: 1, 1: 0}, {2: 2})
out = over1.left_algebra.generator({2: 0}).to_element() ** (over1.ring['U2'] * y.to_generator(over1_module))
assert delta_ell_case_4(over1_module, x, a1, a2) == out
def test_delta_ell_case_2():
over1 = ETangle(ETangle.Type.OVER, (1, 1), 1)
over1_module = empty_da_module(over1)
x = ETangleStrands(over1, {0: 1, 1: 0}, {2: 2})
a1 = 0
a2 = 1
y = ETangleStrands(over1, {0: 0, 1: 1}, {2: 2})
out = (over1.left_algebra.ring['U1'] * over1.left_algebra.generator({2: 2}).to_element()) ** \
y.to_generator(over1_module)
assert delta_ell_case_2(over1_module, x, a1, a2) == out
def d_mixed_case_1(module: Module, x: ETangleStrands, b1: int,
b2: int) -> Module.TensorElement:
c = x.etangle.ring.one()
powers = x.to_strand_diagram().figure_8_case_1b(b1, b2)
if powers is None:
return module.zero()
for orange, power in powers.items():
c *= x.etangle.strand_index_to_variable(orange)**power
x_out = ETangleStrands(x.etangle, x.left_strands,
swap_values(x.right_strands, b1, b2))
return c * x_out.to_generator(module)
def delta_ell_case_2(module: Module, x: ETangleStrands, a1: int,
a2: int) -> Module.TensorElement:
c1 = module.left_algebra.ring.one()
c2 = x.etangle.ring.one()
powers = x.idempotent_and_left_strands().figure_8_case_2a(a1, a2)
if powers is None:
return module.zero()
for orange, power in powers.items():
if x.etangle.strand_index_to_left_sign(orange) == 1:
c1 *= x.etangle.left_algebra_strand_index_to_variable(
orange)**power
else:
c2 *= x.etangle.strand_index_to_variable(orange)**power
elt = x.left_idempotent()
y = ETangleStrands(x.etangle, swap_values(x.left_strands, a1, a2),
x.right_strands)
return (c1 * elt)**(c2 * y.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)
def m2(module: Module, x: ETangleStrands,
a: AMinus.Generator) -> Module.TensorElement:
# if the sign sequences do not match, return 0
if x.etangle.right_sign_sequence() != a.algebra.ss:
return module.zero()
# if the strands cannot be merged, return 0
if set(x.right_strands.values()) != set(a.strands.keys()):
return module.zero()
orange_strands = {}
orange_signs = {}
for orange in range(1, len(x.etangle.signs)):
if x.etangle.right_y_pos(orange):
orange_strands[orange] = (x.etangle.middle_y_pos(orange),
x.etangle.right_y_pos(orange),
x.etangle.right_y_pos(orange))
orange_signs[orange] = x.etangle.signs[orange]
black_strands = {}
for black in x.right_strands.keys():
black_strands[black] = (black, x.right_strands[black],
a.strands[x.right_strands[black]])
c = x.etangle.ring.one()
sd = StrandDiagram(orange_strands, orange_signs, black_strands)
powers = sd.figure_6_relations()
if powers is None:
return module.zero()
for orange, power in powers.items():
for _ in range(power):
c *= x.etangle.strand_index_to_variable(orange)
new_right_strands = {
sd.black_left_pos(black): sd.black_right_pos(black)
for black in x.right_strands.keys()
}
x_out = ETangleStrands(x.etangle, x.left_strands, new_right_strands)
return c * x_out.to_generator(module)
def delta_ell_case_4(module: Module, x: ETangleStrands, a1: int,
a2: int) -> Module.TensorElement:
c1 = module.left_algebra.ring.one()
c2 = x.etangle.ring.one()
powers = x.idempotent_and_left_strands().figure_8_case_4a(a1, a2)
if powers is None:
return module.zero()
for orange, power in powers.items():
if x.etangle.strand_index_to_left_sign(orange) == 1:
c1 *= x.etangle.left_algebra_strand_index_to_variable(
orange)**power
else:
c2 *= x.etangle.strand_index_to_variable(orange)**power
b1 = x.left_strands[a1]
b2 = a2
elt_strands = x.left_idempotent_strands()
elt_strands[b2] = a1
new_left_strands = dict(x.left_strands)
del new_left_strands[a1]
new_left_strands[a2] = b1
elt = x.etangle.left_algebra.generator(elt_strands)
y = ETangleStrands(x.etangle, new_left_strands, x.right_strands)
return (c1 * elt)**(c2 * y.to_generator(module))
