def _LKB_matrix_(self, braid, variab):
"""
Compute the Lawrence-Krammer-Bigelow representation matrix.
The variables of the matrix must be given. This actual
computation is done in this helper method for caching
purposes.
INPUT:
- ``braid`` -- tuple of integers. The Tietze list of the
braid.
- ``variab`` -- string. the names of the variables that will
appear in the matrix. They must be given as a string,
separated by a comma
OUTPUT:
The LKB matrix of the braid, with respect to the variables.
TESTS::
sage: B=BraidGroup(3)
sage: B._LKB_matrix_((2, 1, 2), 'x, y')
[ 0 -x^4*y + x^3*y -x^4*y]
[ 0 -x^3*y 0]
[ -x^2*y x^3*y - x^2*y 0]
sage: B._LKB_matrix_((1, 2, 1), 'x, y')
[ 0 -x^4*y + x^3*y -x^4*y]
[ 0 -x^3*y 0]
[ -x^2*y x^3*y - x^2*y 0]
sage: B._LKB_matrix_((-1, -2, -1, 2, 1, 2), 'x, y')
[1 0 0]
[0 1 0]
[0 0 1]
"""
n = self.strands()
if len(braid)>1:
A = self._LKB_matrix_(braid[:1], variab)
for i in braid[1:]:
A = A*self._LKB_matrix_((i,), variab)
return A
l = list(Set(range(n)).subsets(2))
R = LaurentPolynomialRing(IntegerRing(), variab)
q = R.gens()[0]
t = R.gens()[1]
if len(braid)==0:
return identity_matrix(R, len(l), sparse=True)
A = matrix(R, len(l), sparse=True)
if braid[0]>0:
i = braid[0]-1
for m in range(len(l)):
j = min(l[m])
k = max(l[m])
if i==j-1:
A[l.index(Set([i, k])), m] = q
A[l.index(Set([i, j])), m] = q*q-q
A[l.index(Set([j, k])), m] = 1-q
elif i==j and not j==k-1:
A[l.index(Set([j, k])), m] = 0
A[l.index(Set([j+1, k])), m] = 1
elif k-1==i and not k-1==j:
A[l.index(Set([j, i])), m] = q
A[l.index(Set([j, k])), m] = 1-q
A[l.index(Set([i, k])), m] = (1-q)*q*t
elif i==k:
A[l.index(Set([j, k])), m] = 0
A[l.index(Set([j, k+1])), m] = 1
elif i==j and j==k-1:
A[l.index(Set([j, k])), m] = -t*q*q
else:
A[l.index(Set([j, k])), m] = 1
return A
else:
i = -braid[0]-1
for m in range(len(l)):
j = min(l[m])
k = max(l[m])
if i==j-1:
A[l.index(Set([j-1, k])), m] = 1
elif i==j and not j==k-1:
A[l.index(Set([j+1, k])), m] = q**(-1)
A[l.index(Set([j, k])), m] = 1-q**(-1)
A[l.index(Set([j, j+1])), m] = t**(-1)*q**(-1)-t**(-1)*q**(-2)
elif k-1==i and not k-1==j:
A[l.index(Set([j, k-1])), m] = 1
elif i==k:
A[l.index(Set([j, k+1])), m] = q**(-1)
A[l.index(Set([j, k])), m] = 1-q**(-1)
A[l.index(Set([k, k+1])), m] = -q**(-1)+q**(-2)
elif i==j and j==k-1:
A[l.index(Set([j, k])), m] = -t**(-1)*q**(-2)
else:
A[l.index(Set([j, k])), m] = 1
return A
def Omega_ge(a, exponents):
r"""
Return `\Omega_{\ge}` of the expression specified by the input.
To be more precise, calculate
.. MATH::
\Omega_{\ge} \frac{\mu^a}{
(1 - z_0 \mu^{e_0}) \dots (1 - z_{n-1} \mu^{e_{n-1}})}
and return its numerator and a factorization of its denominator.
Note that `z_0`, ..., `z_{n-1}` only appear in the output, but not in the
input.
INPUT:
- ``a`` -- an integer
- ``exponents`` -- a tuple of integers
OUTPUT:
A pair representing a quotient as follows: Its first component is the
numerator as a Laurent polynomial, its second component a factorization
of the denominator as a tuple of Laurent polynomials, where each
Laurent polynomial `z` represents a factor `1 - z`.
The parents of these Laurent polynomials is always a
Laurent polynomial ring in `z_0`, ..., `z_{n-1}` over `\ZZ`, where
`n` is the length of ``exponents``.
EXAMPLES::
sage: from sage.rings.polynomial.omega import Omega_ge
sage: Omega_ge(0, (1, -2))
(1, (z0, z0^2*z1))
sage: Omega_ge(0, (1, -3))
(1, (z0, z0^3*z1))
sage: Omega_ge(0, (1, -4))
(1, (z0, z0^4*z1))
sage: Omega_ge(0, (2, -1))
(z0*z1 + 1, (z0, z0*z1^2))
sage: Omega_ge(0, (3, -1))
(z0*z1^2 + z0*z1 + 1, (z0, z0*z1^3))
sage: Omega_ge(0, (4, -1))
(z0*z1^3 + z0*z1^2 + z0*z1 + 1, (z0, z0*z1^4))
sage: Omega_ge(0, (1, 1, -2))
(-z0^2*z1*z2 - z0*z1^2*z2 + z0*z1*z2 + 1, (z0, z1, z0^2*z2, z1^2*z2))
sage: Omega_ge(0, (2, -1, -1))
(z0*z1*z2 + z0*z1 + z0*z2 + 1, (z0, z0*z1^2, z0*z2^2))
sage: Omega_ge(0, (2, 1, -1))
(-z0*z1*z2^2 - z0*z1*z2 + z0*z2 + 1, (z0, z1, z0*z2^2, z1*z2))
::
sage: Omega_ge(0, (2, -2))
(-z0*z1 + 1, (z0, z0*z1, z0*z1))
sage: Omega_ge(0, (2, -3))
(z0^2*z1 + 1, (z0, z0^3*z1^2))
sage: Omega_ge(0, (3, 1, -3))
(-z0^3*z1^3*z2^3 + 2*z0^2*z1^3*z2^2 - z0*z1^3*z2
+ z0^2*z2^2 - 2*z0*z2 + 1,
(z0, z1, z0*z2, z0*z2, z0*z2, z1^3*z2))
::
sage: Omega_ge(0, (3, 6, -1))
(-z0*z1*z2^8 - z0*z1*z2^7 - z0*z1*z2^6 - z0*z1*z2^5 - z0*z1*z2^4 +
z1*z2^5 - z0*z1*z2^3 + z1*z2^4 - z0*z1*z2^2 + z1*z2^3 -
z0*z1*z2 + z0*z2^2 + z1*z2^2 + z0*z2 + z1*z2 + 1,
(z0, z1, z0*z2^3, z1*z2^6))
TESTS::
sage: Omega_ge(0, (2, 2, 1, 1, 1, 1, 1, -1, -1))[0].number_of_terms() # long time
27837
::
sage: Omega_ge(1, (2,))
(1, (z0,))
"""
import logging
logger = logging.getLogger(__name__)
logger.info('Omega_ge: a=%s, exponents=%s', a, exponents)
from sage.arith.misc import lcm
from sage.arith.srange import srange
from sage.misc.misc_c import prod
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
from sage.rings.number_field.number_field import CyclotomicField
if not exponents or any(e == 0 for e in exponents):
raise NotImplementedError
rou = sorted(set(abs(e) for e in exponents) - set([1]))
ellcm = lcm(rou)
B = CyclotomicField(ellcm, 'zeta')
zeta = B.gen()
z_names = tuple('z{}'.format(i) for i in range(len(exponents)))
L = LaurentPolynomialRing(B, ('t', ) + z_names, len(z_names) + 1)
t = L.gens()[0]
Z = LaurentPolynomialRing(ZZ, z_names, len(z_names))
powers = {i: L(zeta**(ellcm // i)) for i in rou}
powers[2] = L(-1)
powers[1] = L(1)
exponents_and_values = tuple(
(e, tuple(powers[abs(e)]**j * z for j in srange(abs(e))))
for z, e in zip(L.gens()[1:], exponents))
x = tuple(v for e, v in exponents_and_values if e > 0)
y = tuple(v for e, v in exponents_and_values if e < 0)
def subs_power(expression, var, exponent):
r"""
Substitute ``var^exponent`` by ``var`` in ``expression``.
It is assumed that ``var`` only occurs with exponents
divisible by ``exponent``.
"""
p = tuple(var.dict().popitem()[0]).index(
1) # var is the p-th generator
def subs_e(e):
e = list(e)
assert e[p] % exponent == 0
e[p] = e[p] // exponent
return tuple(e)
parent = expression.parent()
result = parent(
{subs_e(e): c
for e, c in iteritems(expression.dict())})
return result
def de_power(expression):
expression = Z(expression)
for e, var in zip(exponents, Z.gens()):
if abs(e) == 1:
continue
expression = subs_power(expression, var, abs(e))
return expression
logger.debug('Omega_ge: preparing denominator')
factors_denominator = tuple(
de_power(1 - factor) for factor in _Omega_factors_denominator_(x, y))
logger.debug('Omega_ge: preparing numerator')
numerator = de_power(_Omega_numerator_(a, x, y, t))
logger.info('Omega_ge: completed')
return numerator, factors_denominator
def Omega_ge(a, exponents):
r"""
Return `\Omega_{\ge}` of the expression specified by the input.
To be more precise, calculate
.. MATH::
\Omega_{\ge} \frac{\mu^a}{
(1 - z_0 \mu^{e_0}) \dots (1 - z_{n-1} \mu^{e_{n-1}})}
and return its numerator and a factorization of its denominator.
Note that `z_0`, ..., `z_{n-1}` only appear in the output, but not in the
input.
INPUT:
- ``a`` -- an integer
- ``exponents`` -- a tuple of integers
OUTPUT:
A pair representing a quotient as follows: Its first component is the
numerator as a Laurent polynomial, its second component a factorization
of the denominator as a tuple of Laurent polynomials, where each
Laurent polynomial `z` represents a factor `1 - z`.
The parents of these Laurent polynomials is always a
Laurent polynomial ring in `z_0`, ..., `z_{n-1}` over `\ZZ`, where
`n` is the length of ``exponents``.
EXAMPLES::
sage: from sage.rings.polynomial.omega import Omega_ge
sage: Omega_ge(0, (1, -2))
(1, (z0, z0^2*z1))
sage: Omega_ge(0, (1, -3))
(1, (z0, z0^3*z1))
sage: Omega_ge(0, (1, -4))
(1, (z0, z0^4*z1))
sage: Omega_ge(0, (2, -1))
(z0*z1 + 1, (z0, z0*z1^2))
sage: Omega_ge(0, (3, -1))
(z0*z1^2 + z0*z1 + 1, (z0, z0*z1^3))
sage: Omega_ge(0, (4, -1))
(z0*z1^3 + z0*z1^2 + z0*z1 + 1, (z0, z0*z1^4))
sage: Omega_ge(0, (1, 1, -2))
(-z0^2*z1*z2 - z0*z1^2*z2 + z0*z1*z2 + 1, (z0, z1, z0^2*z2, z1^2*z2))
sage: Omega_ge(0, (2, -1, -1))
(z0*z1*z2 + z0*z1 + z0*z2 + 1, (z0, z0*z1^2, z0*z2^2))
sage: Omega_ge(0, (2, 1, -1))
(-z0*z1*z2^2 - z0*z1*z2 + z0*z2 + 1, (z0, z1, z0*z2^2, z1*z2))
::
sage: Omega_ge(0, (2, -2))
(-z0*z1 + 1, (z0, z0*z1, z0*z1))
sage: Omega_ge(0, (2, -3))
(z0^2*z1 + 1, (z0, z0^3*z1^2))
sage: Omega_ge(0, (3, 1, -3))
(-z0^3*z1^3*z2^3 + 2*z0^2*z1^3*z2^2 - z0*z1^3*z2
+ z0^2*z2^2 - 2*z0*z2 + 1,
(z0, z1, z0*z2, z0*z2, z0*z2, z1^3*z2))
::
sage: Omega_ge(0, (3, 6, -1))
(-z0*z1*z2^8 - z0*z1*z2^7 - z0*z1*z2^6 - z0*z1*z2^5 - z0*z1*z2^4 +
z1*z2^5 - z0*z1*z2^3 + z1*z2^4 - z0*z1*z2^2 + z1*z2^3 -
z0*z1*z2 + z0*z2^2 + z1*z2^2 + z0*z2 + z1*z2 + 1,
(z0, z1, z0*z2^3, z1*z2^6))
TESTS::
sage: Omega_ge(0, (2, 2, 1, 1, 1, 1, 1, -1, -1))[0].number_of_terms() # long time
27837
::
sage: Omega_ge(1, (2,))
(1, (z0,))
"""
import logging
logger = logging.getLogger(__name__)
logger.info('Omega_ge: a=%s, exponents=%s', a, exponents)
from sage.arith.all import lcm, srange
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
from sage.rings.number_field.number_field import CyclotomicField
if not exponents or any(e == 0 for e in exponents):
raise NotImplementedError
rou = sorted(set(abs(e) for e in exponents) - set([1]))
ellcm = lcm(rou)
B = CyclotomicField(ellcm, 'zeta')
zeta = B.gen()
z_names = tuple('z{}'.format(i) for i in range(len(exponents)))
L = LaurentPolynomialRing(B, ('t',) + z_names, len(z_names) + 1)
t = L.gens()[0]
Z = LaurentPolynomialRing(ZZ, z_names, len(z_names))
powers = {i: L(zeta**(ellcm//i)) for i in rou}
powers[2] = L(-1)
powers[1] = L(1)
exponents_and_values = tuple(
(e, tuple(powers[abs(e)]**j * z for j in srange(abs(e))))
for z, e in zip(L.gens()[1:], exponents))
x = tuple(v for e, v in exponents_and_values if e > 0)
y = tuple(v for e, v in exponents_and_values if e < 0)
def subs_power(expression, var, exponent):
r"""
Substitute ``var^exponent`` by ``var`` in ``expression``.
It is assumed that ``var`` only occurs with exponents
divisible by ``exponent``.
"""
p = tuple(var.dict().popitem()[0]).index(1) # var is the p-th generator
def subs_e(e):
e = list(e)
assert e[p] % exponent == 0
e[p] = e[p] // exponent
return tuple(e)
parent = expression.parent()
result = parent({subs_e(e): c for e, c in iteritems(expression.dict())})
return result
def de_power(expression):
expression = Z(expression)
for e, var in zip(exponents, Z.gens()):
if abs(e) == 1:
continue
expression = subs_power(expression, var, abs(e))
return expression
logger.debug('Omega_ge: preparing denominator')
factors_denominator = tuple(de_power(1 - factor)
for factor in _Omega_factors_denominator_(x, y))
logger.debug('Omega_ge: preparing numerator')
numerator = de_power(_Omega_numerator_(a, x, y, t))
logger.info('Omega_ge: completed')
return numerator, factors_denominator
def _LKB_matrix_(self, braid, variab):
"""
Compute the Lawrence-Krammer-Bigelow representation matrix.
The variables of the matrix must be given. This actual
computation is done in this helper method for caching
purposes.
INPUT:
- ``braid`` -- tuple of integers. The Tietze list of the
braid.
- ``variab`` -- string. the names of the variables that will
appear in the matrix. They must be given as a string,
separated by a comma
OUTPUT:
The LKB matrix of the braid, with respect to the variables.
TESTS::
sage: B=BraidGroup(3)
sage: B._LKB_matrix_((2, 1, 2), 'x, y')
[ 0 -x^4*y + x^3*y -x^4*y]
[ 0 -x^3*y 0]
[ -x^2*y x^3*y - x^2*y 0]
sage: B._LKB_matrix_((1, 2, 1), 'x, y')
[ 0 -x^4*y + x^3*y -x^4*y]
[ 0 -x^3*y 0]
[ -x^2*y x^3*y - x^2*y 0]
sage: B._LKB_matrix_((-1, -2, -1, 2, 1, 2), 'x, y')
[1 0 0]
[0 1 0]
[0 0 1]
"""
n = self.strands()
if len(braid) > 1:
A = self._LKB_matrix_(braid[:1], variab)
for i in braid[1:]:
A = A * self._LKB_matrix_((i, ), variab)
return A
l = list(Set(range(n)).subsets(2))
R = LaurentPolynomialRing(IntegerRing(), variab)
q = R.gens()[0]
t = R.gens()[1]
if len(braid) == 0:
return identity_matrix(R, len(l), sparse=True)
A = matrix(R, len(l), sparse=True)
if braid[0] > 0:
i = braid[0] - 1
for m in range(len(l)):
j = min(l[m])
k = max(l[m])
if i == j - 1:
A[l.index(Set([i, k])), m] = q
A[l.index(Set([i, j])), m] = q * q - q
A[l.index(Set([j, k])), m] = 1 - q
elif i == j and not j == k - 1:
A[l.index(Set([j, k])), m] = 0
A[l.index(Set([j + 1, k])), m] = 1
elif k - 1 == i and not k - 1 == j:
A[l.index(Set([j, i])), m] = q
A[l.index(Set([j, k])), m] = 1 - q
A[l.index(Set([i, k])), m] = (1 - q) * q * t
elif i == k:
A[l.index(Set([j, k])), m] = 0
A[l.index(Set([j, k + 1])), m] = 1
elif i == j and j == k - 1:
A[l.index(Set([j, k])), m] = -t * q * q
else:
A[l.index(Set([j, k])), m] = 1
return A
else:
i = -braid[0] - 1
for m in range(len(l)):
j = min(l[m])
k = max(l[m])
if i == j - 1:
A[l.index(Set([j - 1, k])), m] = 1
elif i == j and not j == k - 1:
A[l.index(Set([j + 1, k])), m] = q**(-1)
A[l.index(Set([j, k])), m] = 1 - q**(-1)
A[l.index(Set([j, j + 1])),
m] = t**(-1) * q**(-1) - t**(-1) * q**(-2)
elif k - 1 == i and not k - 1 == j:
A[l.index(Set([j, k - 1])), m] = 1
elif i == k:
A[l.index(Set([j, k + 1])), m] = q**(-1)
A[l.index(Set([j, k])), m] = 1 - q**(-1)
A[l.index(Set([k, k + 1])), m] = -q**(-1) + q**(-2)
elif i == j and j == k - 1:
A[l.index(Set([j, k])), m] = -t**(-1) * q**(-2)
else:
A[l.index(Set([j, k])), m] = 1
return A
def alexander_polynomial(self,
multivar=True,
v='no',
method='default',
norm=True,
factored=False):
"""
Calculates the Alexander polynomial of the link. For links with one component,
can evaluate the alexander polynomial at v::
sage: K = Link('4_1')
sage: K.alexander_polynomial()
t^2 - 3*t + 1
sage: K.alexander_polynomial(v=[4])
5
sage: K = Link('L7n1')
sage: K.alexander_polynomial(norm=False)
t1^-1*t2^-1 + t1^-2*t2^-4
The default algorithm for *knots* is Bar-Natan's super-fast
tangle-based algorithm. For links, we apply Fox calculus to a
Wirtinger presentation for the link::
sage: L = Link('K13n123')
sage: L.alexander_polynomial() == L.alexander_polynomial(method='wirtinger')
True
"""
# sign normalization still missing, but when "norm=True" the
# leading coefficient with respect to the first variable is made
# positive.
if method == 'snappy':
try:
return self.exterior().alexander_polynomial()
except ImportError:
raise RuntimeError('this method for alexander_polynomial ' +
no_snappy_msg)
else:
comp = len(self.link_components)
if comp < 2:
multivar = False
# If single variable, use the super-fast method of Bar-Natan.
if comp == 1 and method == 'default' and norm:
p = alexander.alexander(self)
else: # Use a simple method based on the Wirtinger presentation.
if method not in ['default', 'wirtinger']:
raise ValueError(
"Available methods are 'default' and 'wirtinger'")
if (multivar):
L = LaurentPolynomialRing(
QQ, ['t%d' % (i + 1) for i in range(comp)])
t = list(L.gens())
else:
L = LaurentPolynomialRing(QQ, 't')
t = [L.gen()]
M = self.alexander_matrix(mv=multivar)
C = M[0]
m = C.nrows()
n = C.ncols()
if n > m:
k = m - 1
else:
k = n - 1
subMatrix = C[0:k, 0:k]
p = subMatrix.determinant()
if p == 0: return 0
if multivar:
t_i = M[1][-1]
p = (p.factor()) / (t_i - 1)
p = p.expand()
if (norm):
p = normalize_alex_poly(p, t)
if v != 'no':
return p(*v)
if multivar and factored: # it's easier to view this way
return p.factor()
else:
return p
