def testIdentityDDLocal(self):
splitting = PMCSplitting(linearPMC(2), [(1, 4)])
local_pmc = splitting.local_pmc
id_local = identityDALocal(local_pmc)
extended_id = ExtendedDAStructure(id_local, splitting, splitting)
id_dd = extended_id.tensorDD(identityDD(linearPMC(2)))
self.assertTrue(id_dd.compareDDStructures(identityDD(linearPMC(2))))
def __init__(self, genus, c_pair):
"""Specifies genus of the starting pmc and the id of the pair of
anti-braid resolution.
"""
self.genus = genus
self.c_pair = c_pair
self.start_pmc = linearPMC(genus)
self.end_pmc = self.start_pmc
self.n = 4 * genus
self.c1, self.c2 = self.start_pmc.pairs[c_pair]
if self.c2 == self.c1 + 3:
self.is_degenerate = False
else:
assert self.c2 == self.c1 + 2
assert self.c1 == 0 or self.c2 == self.n - 1
self.is_degenerate = True
if self.is_degenerate:
# One position between c1 and c2, called p
self.p = self.c1 + 1
self.p_pair = self.start_pmc.pairid[self.p]
else:
# Two positions between c1 and c2, for (d)own and (u)p
self.d = self.c1 + 1
self.u = self.c1 + 2
self.d_pair = self.start_pmc.pairid[self.d]
self.u_pair = self.start_pmc.pairid[self.u]
def testAbsGrading(self):
def testOneAlgebra(alg, test_op = True):
abs_gr = AbsZ2Grading(alg)
for gen in alg.getGenerators():
# Test asserts in getAbsGrading
abs_gr.getAbsGrading(gen)
if not test_op:
return
# Test differential and multiplication
for a in alg.getGenerators():
for term in a.diff():
a_gr, da_gr = [abs_gr.getAbsGrading(gen) for gen in a, term]
assert (a_gr - 1) % 2 == da_gr
for a in alg.getGenerators():
for b in alg.getGenerators():
if a * b != 0:
a_gr, b_gr, ab_gr = [abs_gr.getAbsGrading(gen)
for gen in a, b, (a*b).getElt()]
assert (a_gr + b_gr) % 2 == ab_gr
for pmc in [splitPMC(1), splitPMC(2), linearPMC(2)]:
testOneAlgebra(pmc.getAlgebra())
for pmc in [antipodalPMC(2), splitPMC(3)]:
testOneAlgebra(pmc.getAlgebra(), test_op = False)
for (pmc, idem_size) in [(splitPMC(1), 1), (splitPMC(1), 2),
(splitPMC(2), 1), (splitPMC(2), 2)]:
testOneAlgebra(PreStrandAlgebra(F2, pmc, idem_size))
for (pmc, idem_size) in [(splitPMC(2), 3), (splitPMC(2), 4),
(splitPMC(3), 2)]:
testOneAlgebra(PreStrandAlgebra(F2, pmc, idem_size),
test_op = False)
def testGetIntervalOrdering(self):
tests = [(splitPMC(1), [0,1,2]),
(splitPMC(2), [4,5,6,3,0,1,2]),
(antipodalPMC(2), [2,5,0,3,6,1,4]),
(linearPMC(2), [4,0,1,3,5,6,2])]
for pmc, order in tests:
self.assertEqual(_getIntervalOrdering(pmc), order)
def __init__(self, genus, c_pair, orientation):
"""Specifies genus of the starting pmc, id of the pair of Dehn twist,
and orientation of the twist (POS or NEG).
"""
self.genus = genus
self.orientation = orientation
self.start_pmc = linearPMC(genus)
self.end_pmc = self.start_pmc
self.n = 4 * genus
self.c1, self.c2 = self.start_pmc.pairs[c_pair]
self.c_pair = c_pair
if self.c2 == self.c1 + 3:
self.is_degenerate = False
else:
assert self.c2 == self.c1 + 2
assert self.c1 == 0 or self.c2 == self.n - 1
self.is_degenerate = True
if not self.is_degenerate:
# Two positions between c1 and c2, for (d)own and (u)p
self.d = self.c1 + 1
self.u = self.c1 + 2
self.splitting = PMCSplitting(self.start_pmc, [(self.c1, self.c2)])
self.local_pmc = self.splitting.local_pmc
def testGeneralOverslideDown(self):
slides_to_test = [
Arcslide(splitPMC(1), 3, 2),
Arcslide(splitPMC(2), 3, 2),
Arcslide(splitPMC(2), 4, 3),
Arcslide(splitPMC(2), 7, 6),
Arcslide(linearPMC(2), 3, 2),
Arcslide(linearPMC(2), 5, 4),
Arcslide(linearPMC(2), 7, 6),
Arcslide(antipodalPMC(2), 5, 4),
Arcslide(antipodalPMC(2), 6, 5),
Arcslide(antipodalPMC(2), 7, 6),
]
for slide in slides_to_test:
print slide
dastr = ArcslideDA(slide).toSimpleDAStructure()
self.assertTrue(dastr.testDelta())
def testGrading(self):
slides_to_test = [
Arcslide(splitPMC(1), 1, 0),
Arcslide(splitPMC(2), 4, 3),
Arcslide(linearPMC(2), 1, 0),
]
for slide in slides_to_test:
dastr = ArcslideDA(slide)
dastr.toSimpleDAStructure().checkGrading()
def testToDStructure(self):
ddstr = identityDD(linearPMC(1))
dstr = ddstr.toDStructure()
hochchild = dstr.morToD(dstr)
self.assertEqual(len(hochchild), 18)
hochchild.simplify(find_homology_basis = True)
self.assertEqual(len(hochchild), 4)
meaning_len = [len(gen.prev_meaning)
for gen in hochchild.getGenerators()]
self.assertEqual(sorted(meaning_len), [1,1,1,2])
def testShortUnderslideDown(self):
slides_to_test = [
Arcslide(splitPMC(1), 1, 0),
Arcslide(splitPMC(1), 2, 1),
Arcslide(splitPMC(2), 1, 0),
Arcslide(splitPMC(2), 6, 5),
Arcslide(linearPMC(2), 1, 0),
Arcslide(linearPMC(2), 6, 5),
Arcslide(PMC([(0, 2), (1, 6), (3, 5), (4, 7)]), 1, 0),
Arcslide(PMC([(0, 3), (1, 6), (2, 4), (5, 7)]), 6, 5),
Arcslide(splitPMC(2), 2, 1),
Arcslide(splitPMC(2), 5, 4),
Arcslide(PMC([(0, 2), (1, 6), (3, 5), (4, 7)]), 4, 3),
Arcslide(PMC([(0, 3), (1, 6), (2, 4), (5, 7)]), 3, 2),
]
for slide in slides_to_test:
print slide
dastr = ArcslideDA(slide).toSimpleDAStructure()
self.assertTrue(dastr.testDelta())
def testIdentityDiagram(self):
pmc_to_test = [splitPMC(1), splitPMC(2), linearPMC(2), antipodalPMC(2)]
pmc_to_test.append(PMC([(0, 2), (1, 6), (3, 5), (4, 7)]))
pmc_to_test += [splitPMC(3), antipodalPMC(4)]
genus_to_size = [2, 6, 20, 70]
for pmc in pmc_to_test:
diagram = getIdentityDiagram(pmc)
self.assertEqual(diagram.getPMCs(), [pmc.opp(), pmc])
expected_size = genus_to_size[pmc.genus - 1]
self.assertEqual(len(diagram.getHFGenerators()), expected_size)
self.assertEqual(len(diagram.getPeriodicDomains()), pmc.genus * 2)
def testOverslideAgreesWithDD(self):
# This is not guaranteed (since there is choice involved in type DD for
# overslides, but appears to work
slides_to_test = [
# General overslides down
Arcslide(splitPMC(1), 3, 2),
Arcslide(splitPMC(2), 3, 2),
Arcslide(splitPMC(2), 4, 3),
Arcslide(splitPMC(2), 7, 6),
Arcslide(linearPMC(2), 3, 2),
Arcslide(linearPMC(2), 5, 4),
Arcslide(linearPMC(2), 7, 6),
Arcslide(antipodalPMC(2), 5, 4),
Arcslide(antipodalPMC(2), 6, 5),
Arcslide(antipodalPMC(2), 7, 6),
# General overslides up
Arcslide(splitPMC(1), 0, 1),
Arcslide(splitPMC(2), 0, 1),
Arcslide(splitPMC(2), 3, 4),
Arcslide(splitPMC(2), 4, 5),
Arcslide(linearPMC(2), 0, 1),
Arcslide(linearPMC(2), 2, 3),
Arcslide(linearPMC(2), 4, 5),
Arcslide(antipodalPMC(2), 0, 1),
Arcslide(antipodalPMC(2), 1, 2),
Arcslide(antipodalPMC(2), 2, 3),
]
for slide in slides_to_test:
dastr = ArcslideDA(slide)
ddstr = dastr.tensorDD(identityDD(slide.end_pmc))
ori_ddstr = slide.getDDStructure()
self.assertTrue(ddstr.compareDDStructures(ori_ddstr))
def testUnderslideAgreesWithDD(self):
slides_to_test = [
# Short underslides down
Arcslide(splitPMC(1), 1, 0),
Arcslide(linearPMC(2), 6, 5),
Arcslide(PMC([(0, 2), (1, 6), (3, 5), (4, 7)]), 1, 0),
# General underslides down
Arcslide(antipodalPMC(2), 1, 0),
Arcslide(PMC([(0, 2), (1, 4), (3, 6), (5, 7)]), 4, 3),
# Short underslides up
Arcslide(splitPMC(1), 1, 2),
Arcslide(linearPMC(2), 1, 2),
Arcslide(PMC([(0, 2), (1, 6), (3, 5), (4, 7)]), 4, 5),
# General underslides up
Arcslide(antipodalPMC(2), 6, 7),
Arcslide(PMC([(0, 3), (1, 6), (2, 4), (5, 7)]), 2, 3),
]
for slide in slides_to_test:
dastr = ArcslideDA(slide)
ddstr = dastr.tensorDD(identityDD(slide.end_pmc))
ori_ddstr = slide.getDDStructure()
self.assertTrue(ddstr.compareDDStructures(ori_ddstr))
def setUp(self):
self.pmc = linearPMC(2)
self.splitting = PMCSplitting(self.pmc, [(1, 4)])
# Correspondence between points in pmc and local / outer pmc:
# pmc - 0 1 2 3 4 5 6 7
# local_pmc - 0* 1 2 3 4 5*
# outer_pmc - 0 1* 2* 3 4 5
local_pmc = self.splitting.local_pmc
outer_pmc = self.splitting.outer_pmc
# Construct the local generators. In local_pmc, idempotent 0 is (1, 4),
# idempotent 1 is (2,), idempotent 2 is (3,).
local_da = LocalDAStructure(
F2, local_pmc.getAlgebra(), local_pmc.getAlgebra(), single_idems1=[1, 2], single_idems2=[1, 2]
)
idems = {"x1": ([0], [1]), "x2": ([0], [2]), "x3": ([0, 2], [1, 2]), "x4": ([0, 1], [1, 2])}
gens = {}
for name, (l_idem, r_idem) in idems.items():
gens[name] = SimpleDAGenerator(
local_da, LocalIdempotent(local_pmc, l_idem), LocalIdempotent(local_pmc, r_idem), name
)
local_da.addGenerator(gens[name])
local_da.auto_u_map()
for name_from, name_to, algs_a, alg_d in [
# Some examples of local DA actions
("x4", "x3", [], [1, (2, 3)]),
("x1", "x2", [[(2, 3)]], [1]),
]:
local_da.addDelta(
gens[name_from], gens[name_to], local_pmc.sd(alg_d), [local_pmc.sd(alg_a) for alg_a in algs_a], 1
)
self.extended_da = ExtendedDAStructure(local_da, self.splitting, self.splitting)
mod_gens = self.extended_da.getGenerators()
# Set up short names for the extended generators. Here y_i is the unique
# extension of x_i.
extended_idems = {
"y1": ([1, 3], [0, 3]),
"y2": ([1, 3], [2, 3]),
"y3": ([1, 2], [0, 2]),
"y4": ([0, 1], [0, 2]),
}
self.extended_gens = {}
for name, (l_idem, r_idem) in extended_idems.items():
for gen in mod_gens:
if gen.idem1 == Idempotent(self.pmc, l_idem) and gen.idem2 == Idempotent(self.pmc, r_idem):
self.extended_gens[name] = gen
def platTypeD(genus):
"""Returns the type D structure for the plat handlebody of a given
genus.
"""
pmc = linearPMC(genus)
algebra = pmc.getAlgebra()
dstr = SimpleDStructure(F2, algebra)
idem = pmc.idem([4*i+1 for i in range(genus-1)]+[4*genus-3])
genx = SimpleDGenerator(dstr, idem, "x")
dstr.addGenerator(genx)
strands = [(4*i+1,4*i+4) for i in range(genus-1)]+[(4*genus-3, 4*genus-1)]
for st in strands:
sd = StrandDiagram(algebra, idem, [st])
dstr.addDelta(genx, genx, sd, 1)
dstr.registerHDiagram(getPlatDiagram(genus), genx)
return dstr
def __init__(self, genus, c_pair, orientation):
"""Specifies genus of the starting pmc, id of the pair of Dehn twist,
and orientation of the twist (POS or NEG).
"""
self.genus = genus
self.orientation = orientation
self.start_pmc = linearPMC(genus)
self.end_pmc = self.start_pmc
self.n = 4 * genus
self.c1, self.c2 = self.start_pmc.pairs[c_pair]
self.c_pair = c_pair
assert self.c2 == self.c1 + 3
# Two positions between c1 and c2, for (d)own and (u)p
self.d = self.c1 + 1
self.u = self.c1 + 2
self.d_pair = self.start_pmc.pairid[self.d]
self.u_pair = self.start_pmc.pairid[self.u]
def __init__(self, genus, c_pair):
"""Specifies genus of the starting PMC and the ID of the pair of
anti-braid resolution.
"""
self.genus = genus
self.c_pair = c_pair
self.pmc = linearPMC(genus)
self.n = 4 * genus
self.c1, self.c2 = self.pmc.pairs[c_pair]
if self.c2 == self.c1 + 3:
self.is_degenerate = False
else:
assert self.c2 == self.c1 + 2
assert self.c1 == 0 or self.c2 == self.n - 1
self.is_degenerate = True
if self.is_degenerate:
# One position between c1 and c2, called p
self.p = self.c1 + 1
self.p_pair = self.pmc.pairid[self.p]
else:
# Two positions between c1 and c2, for (d)own and (u)p
self.d = self.c1 + 1
self.u = self.c1 + 2
self.d_pair = self.pmc.pairid[self.d]
self.u_pair = self.pmc.pairid[self.u]
# Necessary to get local DA structure.
self.splitting = PMCSplitting(self.pmc, [(self.c1, self.c2)])
self.local_pmc = self.splitting.local_pmc
self.mapping = self.splitting.local_mapping
# Local DA Structure
self.local_da = self.getLocalDAStructure()
### Uncomment to use autocompleteda to construct arrows from seeds.
# autoCompleteDA(self.local_da, ([]))
# Initiate the ExtendedDAStructure
ExtendedDAStructure.__init__(self, self.local_da,
self.splitting, self.splitting)
def testHochschildCohomology(self):
# Check HH^*(A)
# Rank 4 is right for HH^* of A in genus 1 case
pmc = splitPMC(1)
ddstr = identityDD(pmc)
cx = ddstr.hochschildCochains()
cx.simplify()
self.assertEqual(len(cx), 4)
# Rank 1 in the extremal strands grading
pmc = splitPMC(2)
ddstr = identityDD(pmc, 0)
cx = ddstr.hochschildCochains()
cx.simplify()
self.assertEqual(len(cx), 1)
# Rank should be indepdendent of PMC (equals 16)
for pmc in [splitPMC(2), linearPMC(2), PMC([(0,2),(1,6),(3,5),(4,7)])]:
ddstr = identityDD(pmc)
cx = ddstr.hochschildCochains()
cx.simplify()
self.assertEqual(len(cx), 16)
def __init__(self, cob):
"""Specifies the cobordism to use. cob should be of type Cobordism,
with cob.side == RIGHT.
"""
assert cob.side == RIGHT
self.n, self.genus, self.c_pair = cob.n, cob.genus, cob.c_pair
start_pmc = linearPMC(self.genus-1)
# Divide into four cases like in CobordismDALeft
if self.c_pair == 0:
# Bottom case:
self.start_da = SimpleCobordismDA(start_pmc, 1)
self.slides = [(0, 1)]
elif self.c_pair == 2*self.genus-2:
# Next to top case:
self.start_da = SimpleCobordismDA(start_pmc, self.n-5)
self.slides = [(self.n-1, self.n-2)]
elif self.c_pair == 2*self.genus-1:
# Top case:
self.start_da = SimpleCobordismDA(start_pmc, self.n-5)
self.slides = [(self.n-2, self.n-3), (self.n-1, self.n-2)]
else:
# Middle case
self.c1 = cob.c1
self.start_da = SimpleCobordismDA(start_pmc, self.c1)
self.slides = [(self.c1+4, self.c1+3), (self.c1+1, self.c1),
(self.c1+2, self.c1+1), (self.c1+5, self.c1+4)]
self.da_list = [self.start_da]
for b1, c1 in self.slides:
self.da_list.append(
ArcslideDA(Arcslide(self.da_list[-1].pmc2, b1, c1)))
ComposedDAStructure.__init__(self, self.da_list)
def __init__(self, num_strands):
"""Specifies the number of strands in the braid."""
self.num_strands = num_strands
self.genus = (num_strands - 2)/2
self.pmc = linearPMC(self.genus)
def __init__(self, genus, c_pair, side):
"""Specifies the genus of the larger (linear PMC), the c-pair at which
the cobordism occurred, and the side of the larger PMC (LEFT or RIGHT).
"""
self.genus = genus # genus of larger PMC
self.n = 4 * self.genus # number of points in the larger PMC
self.c_pair = c_pair
self.side = side # LEFT or RIGHT
if c_pair == 0 or c_pair == 2*self.genus-1:
self.is_degenerate = True
else:
self.is_degenerate = False
self.large_pmc = linearPMC(self.genus)
self.small_pmc = linearPMC(self.genus-1)
# Some special points and pairs
self.c1, self.c2 = self.large_pmc.pairs[self.c_pair]
if self.is_degenerate:
assert self.c2 == self.c1 + 2
self.p = self.c1 + 1
self.p_pair = self.large_pmc.pairid[self.p]
else:
assert self.c2 == self.c1 + 3
self.d, self.u = self.c1 + 1, self.c1 + 2
self.d_pair = self.large_pmc.pairid[self.d]
self.u_pair = self.large_pmc.pairid[self.u]
# Construct the to_s dictionary. Keys are points on the large PMC that
# match points on the small PMC. Value is the point that it matches.
self.to_s = dict()
cur_pt = 0
for i in range(self.n):
if self.is_degenerate:
pair_i = self.large_pmc.pairid[i]
if pair_i == self.c_pair or pair_i == self.p_pair:
continue
else:
if self.c1 <= i <= self.c2:
continue
self.to_s[i] = cur_pt
cur_pt += 1
# The pair_to_s dictionary is similar, but for pairs. The c-pair does
# not match to anything. In the non-degenerate case, the u and d pairs
# both match the (u',d') pair on the left. In the degenerate case, the
# p-pair also does not match anything.
self.pair_to_s = dict()
for i in range(self.n/2):
for p in self.large_pmc.pairs[i]:
if p in self.to_s:
self.pair_to_s[i] = self.small_pmc.pairid[self.to_s[p]]
if not self.is_degenerate:
# Special pair on the small PMC
self.du_pair = self.pair_to_s[self.d_pair]
assert self.du_pair == self.pair_to_s[self.u_pair]
if self.side == LEFT:
self.start_pmc, self.end_pmc = self.large_pmc, self.small_pmc
else:
self.start_pmc, self.end_pmc = self.small_pmc, self.large_pmc