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pmc.py
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pmc.py
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"""Pointed matched circle and its algebras."""
import itertools
from algebra import E0
from fractions import Fraction
from algebra import DGAlgebra, Element, Generator, Tensor, TensorGenerator
from grading import BigGradingElement, BigGradingGroup, SmallGradingElement, \
SmallGradingGroup
from grading import DEFAULT_REFINEMENT
from utility import memorize, memorizeHash
from utility import BIG_GRADING, DEFAULT_GRADING, F2, MULT_ONE, ZZ
class PMC(object):
"""Represents a pointed matched circle."""
def __init__(self, matching):
"""Creates a pointed matched circle from a list of matched pairs. The
indices start at 0.
"""
self.n = len(matching) * 2
self.num_pair = self.n // 2
self.genus = self.n // 4
# otherp[i] is the point paired to i (0 <= i < n)
self.otherp = [0] * self.n
for p, q in matching:
self.otherp[p] = q
self.otherp[q] = p
# pairid[i] is the ID of the pair containing i (0 <= pairid[i] < n/2)
self.pairid = [-1] * self.n
# pairs[i] is the pair of points with ID i (0 <= i < n/2)
self.pairs = []
pairCount = 0
for pos in range(self.n):
if self.pairid[pos] == -1:
self.pairid[pos] = self.pairid[self.otherp[pos]] = pairCount
self.pairs.append((pos, self.otherp[pos]))
pairCount += 1
def __eq__(self, other):
return self.otherp == other.otherp
def __ne__(self, other):
return not (self == other)
@memorizeHash
def __hash__(self):
return hash(tuple(self.otherp))
def __str__(self):
return str(self.pairs)
def __repr__(self):
return "PMC(%s)" % str(self)
@memorize
def opp(self):
"""Returns a new PMC that represents the opposite PMC."""
return PMC([(self.n-1-p, self.n-1-q) for p, q in self.pairs])
def sd(self, data, mult_one = MULT_ONE):
"""Simple way to obtain a strand diagram for this PMC. Each element of
data is either an integer or a pair. An integer specifies a double
horizontal at this position (and its paired position). A pair (p, q)
specifies a strand from p to q.
"""
idem_size = len(data)
parent = StrandAlgebra(F2, self, idem_size, mult_one)
left_idem = []
strands = []
for d in data:
if isinstance(d, int):
left_idem.append(self.pairid[d])
else:
left_idem.append(self.pairid[d[0]])
strands.append(d)
return StrandDiagram(parent, left_idem, strands)
def idem(self, data):
"""Simple way to obtain an idempotent for this PMC. Each element of
data specifies a double horizontal at this position (and its paired
position).
"""
idem_data = [self.pairid[d] for d in data]
return Idempotent(self, idem_data)
def big_gr(self, maslov, spinc):
"""Simple way to obtain an element of the big grading group for this
PMC.
"""
grading_group = BigGradingGroup(self)
return BigGradingElement(grading_group, maslov, spinc)
def small_gr(self, maslov, spinc):
"""Simple way to obtain an element of the small grading group for this
PMC.
"""
grading_group = SmallGradingGroup(self)
return SmallGradingElement(grading_group, maslov, spinc)
def getAlgebra(self, ring = F2, idem_size = None, mult_one = MULT_ONE):
"""Returns the algebra with a given size of idempotent (the default
value, with size half the number of pairs, is most used).
"""
if idem_size == None: idem_size = self.genus
return StrandAlgebra(ring, self, idem_size, mult_one)
def getIdempotents(self, idem_size = None):
"""Get the list of all idempotents."""
if idem_size == None: idem_size = self.genus
return [Idempotent(self, data) for data in
itertools.combinations(list(range(self.num_pair)), idem_size)]
def getStrandDiagrams(self, algebra):
"""Get the list of generators of the strand algebra. algebra should be
of type StrandAlgebra.
"""
result = []
idem_size = algebra.idem_size
def helper(l_idem, r_idem, strands, pos):
# Both l_idem and r_idem are lists of pair ID's. The first
# 'pos' of them are already uesd to generate strands or double
# horizontals. 'strands' keep track of strands generated.
if pos == idem_size:
result.append(StrandDiagram(algebra, l_idem, strands))
return
for i in range(pos, idem_size):
r_idem[i], r_idem[pos] = r_idem[pos], r_idem[i]
if l_idem[pos] == r_idem[pos]:
helper(l_idem, r_idem, strands, pos+1)
for p in self.pairs[l_idem[pos]]:
for q in self.pairs[r_idem[pos]]:
if p < q:
helper(l_idem, r_idem, strands + [(p, q)], pos+1)
r_idem[i], r_idem[pos] = r_idem[pos], r_idem[i]
idems = self.getIdempotents(idem_size)
for l_idem in idems:
for r_idem in idems:
helper(list(l_idem), list(r_idem), [], 0)
# If mult_one is True, filter the generators
if algebra.mult_one is True:
result = [sd for sd in result
if all([x <= 1 for x in sd.multiplicity])]
return result
def splitPMC(genus):
"""Returns the split pmc with a given genus."""
return PMC(sum([[(4*i, 4*i+2), (4*i+1, 4*i+3)] for i in range(0,genus)],[]))
def linearPMC(genus):
"""Returns the linear pmc with a given genus."""
matching = [(0, 2),(4*genus-3, 4*genus-1)]
matching += [(2*i-1, 2*i+2) for i in range(1, 2*genus-1)]
return PMC(matching)
def antipodalPMC(genus):
"""Returns the antipodal pmc with a given genus."""
return PMC([(i, 2*genus+i) for i in range(2*genus)])
def connectSumPMC(pmc1, pmc2):
"""Return the connect sum of two PMC's."""
pairs2 = [(p+pmc1.n, q+pmc1.n) for p, q in pmc2.pairs]
return PMC(pmc1.pairs + pairs2)
@memorize
def unconnectSumPMC(pmc, genus1):
"""Returns a pair (pmc1, pmc2) such that pmc1 has genus1 and
pmc1 # pmc2 = pmc.
"""
cut_point = 4 * genus1
for p, q in pmc.pairs:
assert (p < cut_point and q < cut_point) or \
(p >= cut_point and q >= cut_point)
pmc1 = PMC([(p, q) for p, q in pmc.pairs if p < cut_point])
pmc2 = PMC([(p-cut_point, q-cut_point)
for p, q in pmc.pairs if p >= cut_point])
return (pmc1, pmc2)
class Idempotent(tuple):
"""Represents an idempotent in a certain PMC. Stored as a tuple of pairid
of occupied pairs.
"""
def __new__(cls, pmc, data):
return tuple.__new__(cls, tuple(sorted(data)))
def __init__(self, pmc, data):
self.pmc = pmc
def __eq__(self, other):
if isinstance(other, Idempotent):
return self.pmc == other.pmc and tuple.__eq__(self, other)
else:
return False
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return hash((self.pmc, tuple(self), "Idempotent"))
def __str__(self):
return repr(self)
def __repr__(self):
return "(%s)" % ",".join(str(self.pmc.pairs[i]) for i in self)
@memorize
def opp(self):
"""Get the same idempotent in the opposite PMC."""
pmc, pmcopp = self.pmc, self.pmc.opp()
return Idempotent(pmcopp, [pmcopp.pairid[pmc.n-1-pmc.pairs[i][0]]
for i in self])
def comp(self):
"""Get the complementary idempotent in the same PMC."""
return Idempotent(self.pmc,
set(range(self.pmc.num_pair))-set(self))
def toAlgElt(self, parent):
"""Get the strand diagram corresponding to this idempotent, in the
specified strand algebra.
"""
return StrandDiagram(parent, self, [])
def unconnectSumIdem(idem, genus1):
"""Returns the pair of idempotents (idem1, idem2) in (pmc1, pmc2), where
(pmc1, pmc2) is the pair returned by unconnectSumPMC(idem.pmc, genus1).
"""
cut_pair = 2 * genus1
pmc1, pmc2 = unconnectSumPMC(idem.pmc, genus1)
return (Idempotent(pmc1, [pair for pair in idem if pair < cut_pair]),
Idempotent(pmc2,
[pair-cut_pair for pair in idem if pair >= cut_pair]))
class Strands(tuple):
"""Represents a (fixed) list of strands in a certain PMC. Stored as a tuple
of pairs.
"""
def __new__(cls, pmc, data):
return tuple.__new__(cls, tuple(sorted(data)))
def __init__(self, pmc, data):
self.pmc = pmc
# Compute multiplicity at each interval
self.multiplicity = [0] * (self.pmc.n - 1)
for st in self:
assert len(st) == 2 and st[0] < st[1]
for pos in range(st[0], st[1]):
self.multiplicity[pos] += 1
def __eq__(self, other):
if isinstance(other, Strands):
return self.pmc == other.pmc and tuple.__eq__(self, other)
else:
return False
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return hash((self.pmc, tuple(self), "Strands"))
def __str__(self):
return "(%s)" % ",".join("%d->%d" % (s, t) for s, t in self)
def __repr__(self):
return str(self)
def opp(self):
"""Returns the same strands (with direction reversed) in the opposite
PMC.
"""
n = self.pmc.n
return Strands(self.pmc.opp(), [(n-1-q, n-1-p) for p, q in self])
def leftCompatible(self, idem):
"""Tests whether this set of strands is compatible with a given left
idempotent.
"""
return self.propagateRight(idem) is not None
def rightCompatible(self, idem):
"""Tests whether this set of strands is compatible with a given right
idempotent.
"""
return self.propagateLeft(idem) is not None
def idemCompatible(self, left_idem, right_idem):
"""Tests whether this set of strands is compatible with the given
idempotents on the two sides. Note this is not the same as left and
right compatible.
"""
return self.leftCompatible(left_idem) and \
self.propagateRight(left_idem) == right_idem
def propagateRight(self, left_idem):
"""Find the right_idem given left_idem and strand info. If not
compatible, return None.
"""
idemCount = [0] * self.pmc.num_pair
for pair in left_idem:
idemCount[pair] += 1
for st in self:
if idemCount[self.pmc.pairid[st[0]]] == 0: return None
idemCount[self.pmc.pairid[st[0]]] -= 1
for st in self:
if idemCount[self.pmc.pairid[st[1]]] == 1: return None
idemCount[self.pmc.pairid[st[1]]] += 1
right_idem = [i for i in range(self.pmc.n//2) if idemCount[i] == 1]
return Idempotent(self.pmc, right_idem)
def propagateLeft(self, right_idem):
"""Find the left_idem given right_idem and strand info. If not
compatible, return None.
"""
idemCount = [0] * self.pmc.num_pair
for pair in right_idem:
idemCount[pair] += 1
for st in self:
if idemCount[self.pmc.pairid[st[1]]] == 0: return None
idemCount[self.pmc.pairid[st[1]]] -= 1
for st in self:
if idemCount[self.pmc.pairid[st[0]]] == 1: return None
idemCount[self.pmc.pairid[st[0]]] += 1
left_idem = [i for i in range(self.pmc.n//2) if idemCount[i] == 1]
return Idempotent(self.pmc, left_idem)
def isMultOne(self):
"""Tests whether this set of strands have total multiplicity <= 1
everywhere.
"""
return all([n <= 1 for n in self.multiplicity])
def unconnectSumStrands(strands, genus1):
"""Returns pairs of strands (strand1, strand2) in (pmc1, pmc2), where
(pmc1, pmc2) is the pair returned by unconnectSumPMC(strands.pmc, genus1).
"""
cut_pos = 4 * genus1
pmc1, pmc2 = unconnectSumPMC(strands.pmc, genus1)
for p, q in strands:
assert q < cut_pos or p >= cut_pos
return (Strands(pmc1, [(p, q) for p, q in strands if q < cut_pos]),
Strands(pmc2, [(p-cut_pos, q-cut_pos)
for p, q in strands if p >= cut_pos]))
class StrandDiagram(Generator):
"""Represents a strand diagram, or a generator of the strand algebra."""
def __init__(self, parent, left_idem, strands, right_idem = None):
"""Specifies PMC, left idempotent and right idempotent as list of pair
ID's, and strands as a list of pairs (start, end).
For example, in the split PMC of genus 2, the strand diagram with
double horizontal at (1,3) and strand from 2 to 5 would be encoded as:
left_idem = [1,2], right_idem = [1,3], strands = [(2,5)], since pair
(1,3) has index 1, pair (2,4) has index 2, and pair (5,7) has index 3.
"""
Generator.__init__(self, parent)
self.pmc = parent.pmc
self.mult_one = parent.mult_one
self.strands = strands
if not isinstance(self.strands, Strands):
self.strands = Strands(self.pmc, self.strands)
# Calculate left idempotent if necessary
if left_idem is None:
assert right_idem is not None
left_idem = self.strands.propagateLeft(right_idem)
self.left_idem = left_idem
if not isinstance(self.left_idem, Idempotent):
self.left_idem = Idempotent(self.pmc, self.left_idem)
# Calculate right idempotent if necessary
if right_idem is None:
right_idem = self.strands.propagateRight(self.left_idem)
assert right_idem is not None, \
"Invalid init data for strand diagram: cannot propagate to right."
self.right_idem = right_idem
if not isinstance(self.right_idem, Idempotent):
self.right_idem = Idempotent(self.pmc, self.right_idem)
# Enumerate double horizontals
self.double_hor = list(self.left_idem)
for st in self.strands:
self.double_hor.remove(self.pmc.pairid[st[0]])
self.double_hor = tuple(self.double_hor)
# Get multiplicity from strands
self.multiplicity = self.strands.multiplicity
@memorize
def getBigGrading(self):
return self.pmc.big_gr(self.maslov(), self.multiplicity)
@memorize
def getSmallGrading(self, refinement = DEFAULT_REFINEMENT):
refine_data = refinement(self.pmc, len(self.left_idem))
p_l, p_r = [refine_data[i] for i in (self.left_idem, self.right_idem)]
return (p_l * self.getBigGrading() * p_r.inverse()).toSmallGrading()
def getGrading(self):
if DEFAULT_GRADING == BIG_GRADING:
return self.getBigGrading()
else: # DEFAULT_GRADING == SMALL_GRADING
return self.getSmallGrading()
def __eq__(self, other):
return self.parent == other.parent \
and self.left_idem == other.left_idem \
and self.strands == other.strands
def __ne__(self, other):
return not (self == other)
@memorizeHash
def __hash__(self):
return hash((self.parent, self.left_idem, self.strands))
def __str__(self):
return "[%s]" % \
",".join([str(self.pmc.pairs[i]) for i in self.double_hor] +
["%s->%s" % (p, q) for (p, q) in self.strands])
def __repr__(self):
return str(self)
def isIdempotent(self):
"""Tests whether this generator is an idempotent."""
return len(self.strands) == 0
@memorize
def opp(self):
"""Returns the opposite strand diagram in the opposite strand
algebra.
"""
return StrandDiagram(self.parent.opp(), self.right_idem.opp(),
self.strands.opp(), self.left_idem.opp())
def numCrossing(self):
"""Returns the number of crossings between moving strands."""
return sum(1 for (s1, t1) in self.strands for (s2, t2) in self.strands
if s1 < s2 and t1 > t2)
def maslov(self):
"""Returns the Maslov index, defined as i(a) = inv(a) - m([a],S)."""
maslov = Fraction()
for s, t in self.strands:
maslov -= Fraction(self.multiplicity[s], 2)
if s != 0:
maslov -= Fraction(self.multiplicity[s-1], 2)
maslov += self.numCrossing()
return maslov
def getLeftIdem(self):
"""Return the left idempotent."""
return self.left_idem
def getRightIdem(self):
"""Return the right idempotent."""
return self.right_idem
def unconnectSumStrandDiagram(sd, genus1):
"""Returns a pair of strand diagrams (sd1, sd2) in the algebra of
(pmc1, pmc2), where (pmc1, pmc2) is the pair returned by
unconnectSumPMC(sd.pmc, genus1).
"""
pmc1, pmc2 = unconnectSumPMC(sd.pmc, genus1)
left_idem1, left_idem2 = unconnectSumIdem(sd.left_idem, genus1)
strands1, strands2 = unconnectSumStrands(sd.strands, genus1)
alg1 = StrandAlgebra(F2, pmc1, len(left_idem1), sd.mult_one)
alg2 = StrandAlgebra(F2, pmc2, len(left_idem2), sd.mult_one)
return (StrandDiagram(alg1, left_idem1, strands1),
StrandDiagram(alg2, left_idem2, strands2))
class StrandAlgebra(DGAlgebra):
"""Represents the strand algebra of a PMC."""
def __init__(self, ring, pmc, idem_size, mult_one = MULT_ONE):
"""Specifies the PMC, size of idempotent, and whether this is a
multiplicity one algebra.
"""
DGAlgebra.__init__(self, ring)
self.pmc = pmc
self.idem_size = idem_size
self.mult_one = mult_one
def __str__(self):
return "Strand algebra over %s with idem_size = %d and mult_one = %r" \
% (str(self.pmc), self.idem_size, self.mult_one)
def __eq__(self, other):
if not isinstance(other, StrandAlgebra):
return False
return self.pmc == other.pmc and self.idem_size == other.idem_size \
and self.mult_one == other.mult_one
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return hash((self.pmc, self.idem_size, self.mult_one))
@memorize
def getStrandDiagram(self, left_idem, strands):
"""Memorized version of creating new strand diagrams."""
return StrandDiagram(self, left_idem, strands)
def opp(self):
"""Returns the opposite algebra, as the strand algebra associated to
the opposite PMC.
"""
return StrandAlgebra(self.ring, self.pmc.opp(), self.idem_size,
self.mult_one)
@memorize
def diffRaw(self, gen):
"""Returns a list of elements of the form ((s1, s2), diff_term), where
s1 < s2 are starting points of strands in gen that crosses, and
diff_term is a generator in gen.diff() obtained by uncrossing these two
strands. Together they specify all terms in gen.diff().
"""
target_maslov = gen.maslov() - 1
cur_strands = gen.strands
result = []
def appendCandidate(new_strands, s1, s2):
# Same info except strands, then check grading
assert s1 < s2
diff_term = self.getStrandDiagram(
tuple(gen.left_idem), new_strands)
if self.mult_one or diff_term.maslov() == target_maslov:
result.append(((s1, s2), diff_term))
# Uncross two moving strands
for s1, t1 in cur_strands:
for s2, t2 in cur_strands:
if s1 < s2 and t1 > t2:
new_strands = list(cur_strands)
new_strands.remove((s1, t1))
new_strands.remove((s2, t2))
new_strands.extend([(s1, t2), (s2, t1)])
appendCandidate(tuple(sorted(new_strands)), s1, s2)
# Uncross a moving strand with a double horizontal
for st_id in range(len(cur_strands)):
s, t = cur_strands[st_id]
for i in gen.double_hor:
for p in gen.pmc.pairs[i]:
if s <= p and p <= t:
# Automatically sorted.
new_strands = cur_strands[:st_id] + \
((s, p), (p, t)) + cur_strands[st_id+1:]
appendCandidate(new_strands, s, p)
return result
@memorize
def diff(self, gen):
result = E0
if self.ring is F2:
for (s1, s2), dgen_term in self.diffRaw(gen):
result += dgen_term.elt()
else:
return NotImplemented
return result
@memorize
def getGenerators(self):
return self.pmc.getStrandDiagrams(self)
@memorize
def getGeneratorsForIdem(self, left_idem = None, right_idem = None):
"""Returns the list of generators with the specified left and right
idempotents. Giving None as input means no constraints there.
"""
return [gen for gen in self.getGenerators() if
(left_idem is None or gen.left_idem == left_idem) and
(right_idem is None or gen.right_idem == right_idem)]
@memorize
def getIdempotents(self):
"""Returns the set of idempotents. Use corresponding function in PMC.
"""
return self.pmc.getIdempotents()
def _multiplyRaw(self, gen1, gen2):
"""If gen1 and gen2 can be multiplied, return the generator that is
their product. Otherwise, return None.
"""
pmc = gen1.pmc
new_strands = []
# Keep track of which strands at right are not yet used.
strands_right = list(gen2.strands)
for sd in gen1.strands:
mid_idem = pmc.pairid[sd[1]]
possible_match = [sd2 for sd2 in strands_right
if pmc.pairid[sd2[0]] == mid_idem]
if len(possible_match) == 0:
new_strands.append(sd)
else: # len(possible_match) == 1
sd2 = possible_match[0]
if sd2[0] != sd[1]:
return None
else:
new_strands.append((sd[0], sd2[1]))
strands_right.remove(sd2)
new_strands.extend(strands_right)
new_strands = sorted(new_strands)
mult_term = self.getStrandDiagram(
tuple(gen1.left_idem), tuple(new_strands))
if self.mult_one or mult_term.getBigGrading() == \
gen1.getBigGrading() * gen2.getBigGrading():
return mult_term
else:
return None
def multiply(self, gen1, gen2):
if not isinstance(gen1, StrandDiagram):
return NotImplemented
if not isinstance(gen2, StrandDiagram):
return NotImplemented
assert gen1.parent == self and gen2.parent == self, \
"Algebra not compatible."
if gen1.right_idem != gen2.left_idem:
return E0
if self.mult_one:
# Enforce the multiplicity one condition
if not all(x <= 1 for x in [
m1 + m2 for m1, m2 in zip(gen1.multiplicity,
gen2.multiplicity)]):
return E0
prod_raw = self._multiplyRaw(gen1, gen2)
if prod_raw is None:
return E0
if self.ring is F2:
return prod_raw.elt()
else:
return NotImplemented
class StrandAlgebraElement(Element):
"""An element of strand algebra."""
def isIdempotent(self):
"""Tests whether this element is an idempotent."""
for sd, coeff in list(self.items()):
if not sd.isIdempotent():
return False
return True
def invertible(self):
"""Tests whether this element is invertible."""
return self != 0 and self.isIdempotent()
def inverse(self):
"""Returns the inverse of this element, if invertible. Undefined
behavior if the element is not invertible.
"""
return self
StrandDiagram.ELT_CLASS = StrandAlgebraElement