def d_plus_raw(gen): #checked
''' d_+ map that smooths a black-black crossing of a generator `gen`
of CT-(T_i)) contained in right halfs ( i- 1/2, i)
Returns a list of elements of the form ((s1, s2), d_plus_raw_term), where
s1 < s2 are pairs of strands that d_plus is applied, and
diff_term is a generator in d_plus() obtained by uncrossing these two
strands. Together they specify all terms in gen.d_plus(). '''
# applying it to line Bj
lst = []
l_strands = list(gen.strands.left_converted.items())
r_strands = list(gen.strands.right_converted.items())
types = mod_helper(l_strands, r_strands)
t_r = list(gen.tangle.r_pairs_wc.items())
r_crossings = types[4] # type 4 : right crossings
for s1, s2 in r_crossings:
r_s_pairs = r_strands.copy()
r_s_pairs.remove(s1)
r_s_pairs.remove(s1)
is_crossed = False
for strands in r_s_pairs: # check double black - black crossing
if mod_between(strands, (s1[0], s2[0]), (s1[1], s2[1]),
False) == 0:
is_crossed = True
break
for tangle in t_r: # tangle double cross a strand
if mod_between(tangle, (s1[0], s2[0]), (s1[1], s2[1]), False) == 0:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2, is_crossed))
return lst
def d_minus_raw(gen): # checked
'''d_- map that introduces a black-black crossing to a generator 'gen`
of CT-(T_i) contained in left halfs, (i, i+1/2).
Returns a list of elements of the form ((s1, s2), d_minus_raw_term), where
s1 < s2 are pairs of strands that d_minus is applied, and
diff_term is a generator in d_minus() obtained by crossing these two
strands. Together they specify all terms in gen.d_minus(). '''
lst = [] # applying it to line Bj
l_strands = list(gen.strands.left_converted.items())
r_strands = list(gen.strands.right_converted.items())
types = mod_helper(l_strands, r_strands)
t_l = list(gen.tangle.l_pairs_wc.items())
# type 1 : left horizontals
horizontals = types[1]
for s1, s2 in horizontals:
l_s_pairs = l_strands.copy()
is_crossed = False
l_s_pairs.remove(s1)
l_s_pairs.remove(s2)
for strands in l_s_pairs: # check double strand-strand crossing
if mod_between(strands, (s1[0], s2[0]), (s1[1], s2[1]), True) == 0:
is_crossed = True
break
for tangle in t_l: # tangle double cross a strand
if mod_between(tangle, (s1[0], s2[0]), (s1[1], s2[1]), True) == 0:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
return lst
def d_plus_raw(gen): #checked
''' d_+ map that smooths a black-black crossing of a generator `gen`
of CT-(T_i)) contained in right halfs ( i- 1/2, i)
Returns a list of elements of the form ((s1, s2), d_plus_raw_term), where
s1 < s2 are pairs of strands that d_plus is applied, and
diff_term is a generator in d_plus() obtained by uncrossing these two
strands. Together they specify all terms in gen.d_plus(). '''
# applying it to line Bj
lst = []
l_strands = list(gen.strands.left_converted.items())
r_strands = list(gen.strands.right_converted.items())
types = mod_helper(l_strands, r_strands)
t_r = list(gen.tangle.r_pairs_wc.items())
r_crossings = types[4] # type 4 : right crossings
for s1, s2 in r_crossings:
r_s_pairs = r_strands.copy()
r_s_pairs.remove(s1)
r_s_pairs.remove(s1)
is_crossed = False
for strands in r_s_pairs:# check double black - black crossing
if mod_between(strands, (s1[0],s2[0]),(s1[1],s2[1]), False) == 0:
is_crossed = True
break
for tangle in t_r:# tangle double cross a strand
if mod_between(tangle, (s1[0],s2[0]),(s1[1],s2[1]), False) == 0:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2, is_crossed))
return lst
def d_minus_raw(gen): # checked
'''d_- map that introduces a black-black crossing to a generator 'gen`
of CT-(T_i) contained in left halfs, (i, i+1/2).
Returns a list of elements of the form ((s1, s2), d_minus_raw_term), where
s1 < s2 are pairs of strands that d_minus is applied, and
diff_term is a generator in d_minus() obtained by crossing these two
strands. Together they specify all terms in gen.d_minus(). '''
lst = [] # applying it to line Bj
l_strands = list(gen.strands.left_converted.items())
r_strands = list(gen.strands.right_converted.items())
types = mod_helper(l_strands, r_strands)
t_l = list(gen.tangle.l_pairs_wc.items())
# type 1 : left horizontals
horizontals = types[1]
for s1, s2 in horizontals:
l_s_pairs = l_strands.copy()
is_crossed = False
l_s_pairs.remove(s1)
l_s_pairs.remove(s2)
for strands in l_s_pairs: # check double strand-strand crossing
if mod_between(strands, (s1[0],s2[0]),(s1[1],s2[1]),True)==0:
is_crossed = True
break
for tangle in t_l:# tangle double cross a strand
if mod_between(tangle, (s1[0],s2[0]),(s1[1],s2[1]),True) == 0:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
return lst
def helper(tang_left, tang_right, pair_1, pair_2, pair_3, is_left, option):
''' Helper methods for diff methods. Assumes compatibility. Given a generator x, taking the is_left side,
exchanges pairs = (a,b), it seems whether the
whether it violates `option`. Refer to options.jpg
for options.
* tang_left is dictionary object of pairs
* tang_right is dictionary object of pairs
* Pair_1 is the coordinate of two pairs of strand on the left
* Pair_2 is the coordinate of two pairs of strand in the middle
* Pair_3 is the coordinate of two pairs of strand in the right.'''
mod = False
if is_left: # only check mod relations 1,2,3
for k, v in tang_left.items():
if option == 1:
if mod_helper((k, v), pair_1, pair_2, is_left) == 1:
mod = True
elif option == 2:
if mod_helper((k, v), pair_1, pair_2, is_left) == 0:
mod = True
elif option == 3:
if mod_helper((k, v), pair_1, pair_2, is_left) == -1:
mod = True
else:
raise AssertionError(
"Something is wrong -- options arent either 1, 2 or 3 ")
else: # only check mod relations 4,5,6
for k, v in tang_right.items():
if option == 4:
if mod_helper((k, v), pair_2, pair_3, is_left) == 1:
mod = True
elif option == 5:
if mod_helper((k, v), pair_2, pair_3, is_left) == 0:
mod = True
elif option == 6:
if mod_helper((k, v), pair_2, pair_3, is_left) == -1:
mod = True
else:
raise AssertionError(
"Something is wrong about `is_left`-- options arent either 4, 5, 6"
)
return mod
def helper(tang_left, tang_right, pair_1, pair_2, pair_3, is_left, option):
''' Helper methods for diff methods. Assumes compatibility. Given a generator x, taking the is_left side,
exchanges pairs = (a,b), it seems whether the
whether it violates `option`. Refer to options.jpg
for options.
* tang_left is dictionary object of pairs
* tang_right is dictionary object of pairs
* Pair_1 is the coordinate of two pairs of strand on the left
* Pair_2 is the coordinate of two pairs of strand in the middle
* Pair_3 is the coordinate of two pairs of strand in the middle.'''
mod = False
if is_left: # only check mod relations 1,2,3
for k,v in tang_left.items():
if option ==1:
if mod_helper((k,v), pair_1, pair_2, is_left) == 1:
mod = True
elif option == 2:
if mod_helper((k,v), pair_1, pair_2, is_left) == 0:
mod = True
elif option == 3:
if mod_helper((k,v), pair_1, pair_2, is_left) == -1:
mod = True
else:
raise AssertionError("Something is wrong -- options arent either 1, 2 or 3 ")
else: # only check mod relations 4,5,6
for k,v in tang_right.items():
if option == 4:
if mod_helper((k,v), pair_2, pair_3, is_left) == 1:
mod = True
elif option ==5:
if mod_helper((k,v), pair_2, pair_3, is_left) == 0:
mod = True
elif option == 6:
if mod_helper((k,v), pair_2, pair_3, is_left) == -1:
mod = True
else:
raise AssertionError("Something is wrong about `is_left`-- options arent either 4, 5, 6" )
return mod
def d_m_raw_B(gen): # checked
''' dm map that picks two pair of points along Bj.
Exchanges two ends of the corresponding pair of black strands of generator
`gen` in CT(Ti)
Returns a list of elements of the form ((s1, s2), d_m_raw_term), where
s1 < s2 are pairs of strands that d_m is applied, and
diff_term is a generator in d_plus() obtained by uncrossing or crossing
these two strands. Together they specify all terms in d_m(). '''
lst = []
l_strands = list(gen.strands.left_converted.items())
r_strands = list(gen.strands.right_converted.items())
types = mod_helper(l_strands, r_strands)
t_l = list(gen.tangle.l_pairs_wc.items())
t_r = list(gen.tangle.r_pairs_wc.items())
# right no cross strands
r_horizontals = types[3]
for s1, s2 in r_horizontals:
is_crossed = False
for tangle in t_r: # right half tangles
if abs(mod_between(tangle, (s1[0], s2[0]), (s1[1], s2[1]),
False)) == 1:
is_crossed = True
break
for tangle in t_l: #left half tangles
x = (tuple(np.subtract(s1[1],
(1, 0))), tuple(np.subtract(s2[1], (1, 0))))
if mod_between(tangle, x, (s1[0], s2[0]), True) != -2:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
# left cross strands
l_cross = types[2]
for s1, s2 in l_cross:
is_crossed = False
for tangle in t_l: #left half tangles
if abs(mod_between(tangle, (s1[0], s2[0]), (s1[1], s2[1]),
True)) == 1:
is_crossed = True
break
for tangle in t_r: # right half tangles
x = (tuple(np.add(s1[0], (1, 0))), tuple(np.add(s2[0], (1, 0))))
if mod_between(tangle, (s1[1], s2[1]), x, False) != -2:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
# left above right strands
l_above_r = types[6]
for s1, s2 in l_above_r:
is_crossed = False
left_pair = (tuple(np.subtract(s2[0], (0.5, 0))), s1[0])
mid_pair = (s2[0], s1[1])
right_pair = (tuple(np.add(s1[1], (0.5, 0))), s2[1])
for tangle in t_l: # left half tangles
if -2 < mod_between(tangle, left_pair, mid_pair, True) < 1:
is_crossed = True
break
for tangle in t_r: # right_half tangles
if mod_between(tangle, mid_pair, right_pair, False) == -1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
# left below right strands
l_below_r = types[5]
for s1, s2 in l_below_r: #s1 on the left, s2 on the right
is_crossed = False
left_pair = (tuple(np.subtract(s2[0], (0.5, 0))), s1[0])
mid_pair = (s2[0], s1[1])
right_pair = (tuple(np.add(s1[1], (0.5, 0))), s2[1])
for tangle in t_l: # left half tangles
if mod_between(tangle, left_pair, mid_pair, True) > -1:
is_crossed = True
break
for tangle in t_r:
if mod_between(tangle, mid_pair, right_pair, False) == 1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
return lst
def d_m_raw_A(left_idem, gen):
'''dm map that picks two pair of points along A_j, between x and algebra element
E_L_D(x), a left idempotent of generator x.
Exchanges two ends of the corresponding pair of black strands.
if `alg_left` == True, A(-dLT_i) * CT(T_ii)
Returns a list of elements of the form ((s1, s2), new_alg_element, new_gen), where
s1 < s2 are pairs of strands that d_m is applied, and
diff_term is a generator in d_plus() obtained by uncrossing or crossing
these two strands. Together they specify all terms in d_m()'''
# So far only implemented for A(-dLT_i) * CT(T_ii)
if not isinstance(left_idem, Simple_Strand):
raise TypeError(
"The left element, must be an idempotent in the form of Strand Algebra Element"
)
lst = [] # CB
l_strands = list(left_idem.strands.right_converted.items())
r_strands = list(gen.strands.left_converted.items())
t_l = list(left_idem.tangle.r_pairs_wc.items())
t_r = list(gen.tangle.l_pairs_wc.items())
types = mod_helper(l_strands, r_strands)
# left no cross strands
l_horizontals = types[1]
for s1, s2 in l_horizontals:
is_crossed = False
for tangle in t_l: # left half tangles
if abs(mod_between(tangle, (s1[0], s2[0]), (s1[1], s2[1]),
True)) == 1:
is_crossed = True
break
for tangle in t_r: # right half tangles
x = (tuple(np.add(s1[0], (1, 0))), tuple(np.add(s2[0], (1, 0))))
if mod_between(tangle, (s1[1], s2[1]), x, False) != -2:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
# right cross strands
r_cross = types[4]
for s1, s2 in r_cross:
is_crossed = False
for tangle in t_l: # left half tangles
x = (tuple(np.subtract(s1[1],
(1, 0))), tuple(np.subtract(s2[1], (1, 0))))
if mod_between(tangle, x, (s1[0], s2[0]), True) != -2:
is_crossed = True
break
for tangle in t_r: # right half tangles
if abs(mod_between(tangle, (s1[0], s2[0]), (s1[1], s2[1]),
False)) == 1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
# left below right
l_below_r = types[5]
for s1, s2 in l_below_r:
is_crossed = False
left_pair = (tuple(np.subtract(s2[0], (0.5, 0))), s1[0])
mid_pair = (s2[0], s1[1])
right_pair = (tuple(np.add(s1[1], (0.5, 0))), s2[1])
for tangle in t_l: #left half tangles
if mod_between(tangle, left_pair, mid_pair, True) == -1:
is_crossed = True
break
for tangle in t_r: # right half tangles
if -2 < mod_between(tangle, mid_pair, right_pair, False) < 1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
# left above right strands
l_above_r = types[6]
for s1, s2 in l_above_r:
is_crossed = False
left_pair = (tuple(np.subtract(s2[0], (0.5, 0))), s1[0])
mid_pair = (s2[0], s1[1])
right_pair = (tuple(np.add(s1[1], (0.5, 0))), s2[1])
for tangle in t_l: # left half tangle
if mod_between(tangle, left_pair, mid_pair, True) == 1:
is_crossed = True
break
for tangle in t_r: # right half tangle
if mod_between(tangle, mid_pair, right_pair, False) > -1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
return lst
def d_m_raw_B(gen): # checked
''' dm map that picks two pair of points along Bj.
Exchanges two ends of the corresponding pair of black strands of generator
`gen` in CT(Ti)
Returns a list of elements of the form ((s1, s2), d_m_raw_term), where
s1 < s2 are pairs of strands that d_m is applied, and
diff_term is a generator in d_plus() obtained by uncrossing or crossing
these two strands. Together they specify all terms in d_m(). '''
lst = []
l_strands = list(gen.strands.left_converted.items())
r_strands = list(gen.strands.right_converted.items())
types = mod_helper(l_strands, r_strands)
t_l = list(gen.tangle.l_pairs_wc.items())
t_r = list(gen.tangle.r_pairs_wc.items())
# right no cross strands
r_horizontals = types[3]
print("\n Right no cross strands:{0} \n".format(r_horizontals))
for s1,s2 in r_horizontals:
is_crossed = False
for tangle in t_r: # right half tangles
if abs(mod_between(tangle, (s1[0],s2[0]),(s1[1],s2[1]),False)) == 1:
is_crossed = True
break
for tangle in t_l: #left half tangles
x = (tuple(np.subtract(s1[1],(1,0))), tuple(np.subtract(s2[1],(1,0))))
if mod_between(tangle, x,(s1[0],s2[0]), True) != -2:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
# left cross strands
l_cross = types[2]
print("\n left cross strands:{0} \n".format(l_cross))
for s1,s2 in l_cross:
is_crossed = False
for tangle in t_l: #left half tangles
if abs(mod_between(tangle, (s1[0],s2[0]),(s1[1],s2[1]),True)) == 1:
is_crossed = True
break
for tangle in t_r: # right half tangles
x = (tuple(np.add(s1[0],(1,0))), tuple(np.add(s2[0],(1,0))))
if mod_between(tangle,(s1[1],s2[1]),x, False) != -2:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
# left above right strands
l_above_r = types[6]
print("\n left above right strands:{0} \n".format(l_above_r))
print(len(l_above_r))
for s1, s2 in l_above_r:
is_crossed = False
left_pair = (tuple(np.subtract(s2[0],(0.5,0))), s1[0])
mid_pair = (s2[0],s1[1])
right_pair = (tuple(np.add(s1[1],(0.5,0))),s2[1])
for tangle in t_l: # left half tangles
if -2 < mod_between(tangle, left_pair, mid_pair, True) < 1:
is_crossed = True
break
for tangle in t_r: # right_half tangles
if mod_between(tangle, mid_pair, right_pair, False) == -1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
# left below right strands
l_below_r = types[5]
print("\n left below right strands:{0} \n".format(l_below_r))
print(len(l_below_r))
for s1,s2 in l_below_r: #s1 on the left, s2 on the right
is_crossed = False
left_pair = (tuple(np.subtract(s2[0],(0.5,0))), s1[0])
mid_pair = (s2[0],s1[1])
right_pair = (tuple(np.add(s1[1],(0.5,0))),s2[1])
for tangle in t_l: # left half tangles
if mod_between(tangle, left_pair, mid_pair, True) > -1 :
is_crossed = True
break
for tangle in t_r:
if mod_between(tangle, mid_pair, right_pair, False) == 1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_2(gen, s1, s2))
return lst
def d_m_raw_A(left_idem, gen):
'''dm map that picks two pair of points along A_j, between x and algebra element
E_L_D(x), a left idempotent of generator x.
Exchanges two ends of the corresponding pair of black strands.
if `alg_left` == True, A(-dLT_i) * CT(T_ii)
Returns a list of elements of the form ((s1, s2), new_alg_element, new_gen), where
s1 < s2 are pairs of strands that d_m is applied, and
diff_term is a generator in d_plus() obtained by uncrossing or crossing
these two strands. Together they specify all terms in d_m()'''
# So far only implemented for A(-dLT_i) * CT(T_ii)
if not isinstance(left_idem, Simple_Strand):
raise TypeError("The left element, must be an idempotent in the form of Strand Algebra Element")
lst = [] # CB
l_strands = list(left_idem.strands.right_converted.items())
r_strands = list(gen.strands.left_converted.items())
t_l = list(left_idem.tangle.r_pairs_wc.items())
t_r = list(gen.tangle.l_pairs_wc.items())
types = mod_helper(l_strands, r_strands)
# left no cross strands
l_horizontals = types[1]
for s1, s2 in l_horizontals:
is_crossed = False
for tangle in t_l: # left half tangles
if abs(mod_between(tangle, (s1[0],s2[0]),(s1[1],s2[1]), True)) == 1:
is_crossed = True
break
for tangle in t_r: # right half tangles
x = (tuple(np.add(s1[0],(1,0))), tuple(np.add(s2[0],(1,0))))
if mod_between(tangle, (s1[1], s2[1]),x, False ) != -2:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
# right cross strands
r_cross = types[4]
for s1, s2 in r_cross:
is_crossed = False
for tangle in t_l: # left half tangles
x = (tuple(np.subtract(s1[1],(1,0))), tuple(np.subtract(s2[1],(1,0))))
if mod_between(x,(s1[0],s2[0]),True) != -2:
is_crossed = True
break
for tangle in t_r: # right half tangles
if abs(mod_between(tangle, (s1[0],s2[0]),(s1[1],s2[1]),False)) == 1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
# left below right
l_below_r = types[5]
for s1, s2 in l_below_r:
is_crossed = False
left_pair = (tuple(np.subtract(s2[0],(0.5,0))), s1[0])
mid_pair = (s2[0],s1[1])
right_pair = (tuple(np.add(s1[1],(0.5,0))),s2[1])
for tangle in t_l: #left half tangles
if mod_between(tangle, left_pair, mid_pair, True) == -1:
is_crossed = True
break
for tangle in t_r: # right half tangles
if -2 < mod_between(tangle, mid_pair, right_pair, False) < 1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
# left above right strands
l_above_r = types[6]
for s1, s2 in l_above_r:
is_crossed = False
left_pair = (tuple(np.subtract(s2[0],(0.5,0))), s1[0])
mid_pair = (s2[0],s1[1])
right_pair = (tuple(np.add(s1[1],(0.5,0))),s2[1])
for tangle in t_l: # left half tangle
if mod_between(tangle, left_pair, mid_pair, True) == 1:
is_crossed = True
break
for tangle in t_r: # right half tangle
if mod_between(tangle, mid_pair, right_pair, False) > -1:
is_crossed = True
break
if not is_crossed:
lst.append(mod_helper_3(left_idem, gen, s1, s2))
return lst