marchwicka/periodicity_of_links
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correction in Murasugi's criterion

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Maria Marchwicka 2020-01-20 05:38:45 +01:00
parent 61e0669b3d
commit 737a556b8b
1 changed files with 59 additions and 34 deletions
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93
verbose_periodicity.sage
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@ -36,7 +36,9 @@ class MySettings(object):
self.f_results_out = os.path.join(os.getcwd(), "results.out") self.f_results_out = os.path.join(os.getcwd(), "results.out")
self.f_old_results = os.path.join(os.getcwd(), "old_results.input") self.f_old_results = os.path.join(os.getcwd(), "old_results.input")
self.periods = [3, 5, 7, 9, 11] self.periods = [9]
# self.periods = [3, 5, 7, 9, 11]
self.set_to_check = self.get_set() self.set_to_check = self.get_set()
# check only knots from defined set # check only knots from defined set
@ -44,7 +46,7 @@ class MySettings(object):
self.only_chosen = False self.only_chosen = False
self.debugging = True self.debugging = True
self.debugging = False # self.debugging = False
# only if debugging # only if debugging
self.print_matrices = True self.print_matrices = True
@ -70,7 +72,7 @@ class MySettings(object):
# self.input_file_with_homflypt = False # self.input_file_with_homflypt = False
self.check_old_results = True self.check_old_results = True
self.check_old_results = False # self.check_old_results = False
if self.input_file_with_homflypt: if self.input_file_with_homflypt:
if not os.path.isfile(self.f_homfly_lm_in): if not os.path.isfile(self.f_homfly_lm_in):
@ -83,7 +85,7 @@ class MySettings(object):
periodic_burde = set(["3_1", "5_1", "7_1", "8_19", "9_1", periodic_burde = set(["3_1", "5_1", "7_1", "8_19", "9_1",
"9_35", "9_40", "9_41", "9_47", "9_49", "9_35", "9_40", "9_41", "9_47", "9_49",
"10_3", "10_123", "10_124"]) "10_3", "10_123", "10_124"])
# set_to_check |= periodic_burde set_to_check |= periodic_burde
# knots that fail Borodzik criterion # knots that fail Borodzik criterion
self.fails_dict = { self.fails_dict = {
@ -353,7 +355,7 @@ class MySettings(object):
} }
set_to_check |= set(self.success_dict.keys()) set_to_check |= set(self.success_dict.keys())
# knots that are known to be periodic # knots that are known to be (not) periodic
self.periods_dict = { self.periods_dict = {
"3_1": [3], "3_1": [3],
"5_1": [5], "5_1": [5],
@ -392,7 +394,7 @@ class MySettings(object):
"15a40549": [-5], "15a40549": [-5],
"15a53966": [-5] "15a53966": [-5]
} }
# set_to_check |= set(self.periods_dict.keys()) set_to_check |= set(self.periods_dict.keys())
# knots that have Alexander polynomial = 1 # knots that have Alexander polynomial = 1
self.alexander_1 = set(["11n34", self.alexander_1 = set(["11n34",
@ -596,9 +598,11 @@ class MySettings(object):
"15n163337", "15n163337",
"15n165398", "15n165398",
]) ])
# set_to_check |= self.alexander_1 set_to_check |= self.alexander_1
set_to_check = set(["10_123"]) set_to_check |= set(["10_123"])
set_to_check |= set(["15n166130"])
set_to_check = set(self.fails_dict.keys())
return set_to_check return set_to_check
@ -723,16 +727,17 @@ class PeriodicityTester(object):
def check_murasugi(self, q): def check_murasugi(self, q):
''' '''
Select these delta factors and natural number r such that: Select these delta factors and natural number r such that:
delta = delta_prime^q * (1 + t^1 + ... + t^(r-1))^(q-1) mod q delta = delta_prime^q * (1 + t^1 + ... + t^(r-1))^(q-1) mod q_factor
where "delta_prime" is a delta factor. where "delta_prime" is a delta factor.
''' '''
quotient_delta = self.delta.change_ring(GF(q)) q_factor = factor(q)[0][0]
quotient_delta = self.delta.change_ring(GF(q_factor))
# Underlying polynomial of quotient_delta: # Underlying polynomial of quotient_delta:
quotient_delta = quotient_delta.polynomial_construction()[0] quotient_delta = quotient_delta.polynomial_construction()[0]
delta_degree = quotient_delta.degree() delta_degree = quotient_delta.degree()
for candidate in self.delta_factors: for candidate in self.delta_factors:
quotient_candidate = candidate.change_ring(GF(q)) quotient_candidate = candidate.change_ring(GF(q_factor))
power_candidate = quotient_candidate^q power_candidate = quotient_candidate^q
power_candidate = power_candidate.polynomial_construction()[0] power_candidate = power_candidate.polynomial_construction()[0]
# (r - 1) - possible t-polynomial degree # (r - 1) - possible t-polynomial degree
@ -754,18 +759,19 @@ class PeriodicityTester(object):
t_delta_factors = [f for f in t_delta_dict.keys() t_delta_factors = [f for f in t_delta_dict.keys()
if f != 2 and gcd(q, f) == 1] if f != 2 and gcd(q, f) == 1]
for f in t_delta_factors: for f in t_delta_factors:
f_q = naik_number_dict.setdefault((f, q), get_naik_number(f, q)) q_f = naik_number_dict.setdefault((q, f), get_naik_number(q, f))
if not (t_delta_dict[f] / (2 * f_q)).is_integer(): if not (t_delta_dict[f] / (2 * q_f)).is_integer():
return None return None
return t_delta_factors return t_delta_factors
def check_naik_1(self, q): def check_naik_1(self, q):
''' '''
For each delta' find a set P of prime numbers p such that: For each delta' find a set P of prime numbers p such that:
gcd(p, q) == 1, p != 2 and p| t_delta, t_delta = delta(-1)/delta'(-1). gcd(q, p) == 1, p != 2 and p| t_delta, t_delta = delta(-1)/delta'(-1).
Check if all p factors of t_delta has multiplicity divisible by 2*[p|q]. Check if all p factors of t_delta has multiplicity divisible by 2*[q|p].
If it holds for at least one delta' candidate, set naik_1 = True. If it holds for at least one delta' candidate, set naik_1 = True.
''' '''
# Proposition 2.7.
delta_evaluated = self.delta(-1) delta_evaluated = self.delta(-1)
for delta_prime, _ in self.murasugi_fulfilling: for delta_prime, _ in self.murasugi_fulfilling:
@ -780,7 +786,7 @@ class PeriodicityTester(object):
delta_prime_bases = [] delta_prime_bases = []
maximum_in_diagonal = self.get_maximum_in_diagonal() maximum_in_diagonal = self.get_maximum_in_diagonal()
for p in p_list: for p in p_list:
p_q = naik_number_dict[(p, q)] q_p = naik_number_dict[(q, p)]
bases_for_p_torsion = [] bases_for_p_torsion = []
factor_power = p factor_power = p
# find all p^k torsion parts # find all p^k torsion parts
@ -795,7 +801,7 @@ class PeriodicityTester(object):
basis_for_p_k_part.append(0) basis_for_p_k_part.append(0)
len_non_zero = sum(x != 0 for x in basis_for_p_k_part) len_non_zero = sum(x != 0 for x in basis_for_p_k_part)
# check if dimension is multiple of 2 * naik_number # check if dimension is multiple of 2 * naik_number
if not (len_non_zero / (2 * p_q)).is_integer(): if not (len_non_zero / (2 * q_p)).is_integer():
return None return None
factor_power *= p factor_power *= p
bases_for_p_torsion.append(basis_for_p_k_part) bases_for_p_torsion.append(basis_for_p_k_part)
@ -812,6 +818,7 @@ class PeriodicityTester(object):
In particular naik_2 is set to be -1 if the criterion passes, In particular naik_2 is set to be -1 if the criterion passes,
but only in cases where P is an empty set. but only in cases where P is an empty set.
''' '''
# Proposition 2.8.
for delta_prime, p_list in self.naik_1_fulfilling: for delta_prime, p_list in self.naik_1_fulfilling:
delta_prime_factors = set([d[0] for d in factor(delta_prime(-1))]) delta_prime_factors = set([d[0] for d in factor(delta_prime(-1))])
p_list = [p for p in p_list if p not in delta_prime_factors] p_list = [p for p in p_list if p not in delta_prime_factors]
@ -860,7 +867,7 @@ class PeriodicityTester(object):
episilon_1 = 1, else: episilon_1 = -1. episilon_1 = 1, else: episilon_1 = -1.
If p == 3 mod(4) and a rank of p^k torsion part n == 2 mod(4), then: If p == 3 mod(4) and a rank of p^k torsion part n == 2 mod(4), then:
epsilon_2 = -1, else: epsilon_2 = 1. epsilon_2 = -1, else: epsilon_2 = 1.
eta = naik_sign ^ d, where d = n / (2 * [p, q]). eta = naik_sign^d, where d = n / (2 * [p, q]).
If p^([p, q]) % q == 1, then: naik_sign = 1, else: naik_sign = -1. If p^([p, q]) % q == 1, then: naik_sign = 1, else: naik_sign = -1.
''' '''
for k, p_k_basis in enumerate(bases): for k, p_k_basis in enumerate(bases):
@ -883,10 +890,10 @@ class PeriodicityTester(object):
if not mod(P_det, p).is_square(): if not mod(P_det, p).is_square():
epsilon *= -1 # epsilon = epsilon_1 * epsilon_2 epsilon *= -1 # epsilon = epsilon_1 * epsilon_2
p_q = naik_number_dict[(p, q)] q_p = naik_number_dict[(q, p)]
d = n / (2 * p_q) d = n / (2 * q_p)
# sign(p_q) - whether rest is -1 or 1 # sign(q_p) - whether rest is -1 or 1
if sign(p_q)^d != epsilon: if sign(q_p)^d != epsilon:
return False return False
return True return True
@ -1070,7 +1077,7 @@ class PeriodicityTester(object):
if not p_list: if not p_list:
print "List of factors was empty." print "List of factors was empty."
for p in p_list: for p in p_list:
g = abs(naik_number_dict[(p, q)]) g = abs(naik_number_dict[(q, p)])
print "factor of del/del'(-1): " + str(p) print "factor of del/del'(-1): " + str(p)
print "Naik number: " + str(g) print "Naik number: " + str(g)
print "2 * Naik number:\t" + str(2 * g) print "2 * Naik number:\t" + str(2 * g)
@ -1117,7 +1124,7 @@ class PeriodicityTester(object):
for p, bases_for_p in delta_prime_bases: for p, bases_for_p in delta_prime_bases:
print "\nfactor p for delta prime:\t\t\t" + str(p) print "\nfactor p for delta prime:\t\t\t" + str(p)
g = abs(naik_number_dict[(p, q)]) g = abs(naik_number_dict[(q, p)])
print "Naik number:\t\t" + str(g) print "Naik number:\t\t" + str(g)
print "2 * Naik number:\t" + str(2 * g) print "2 * Naik number:\t" + str(2 * g)
test_naik_number = p^g % q test_naik_number = p^g % q
@ -1210,11 +1217,11 @@ class PeriodicityTester(object):
# epsilon and eta # epsilon and eta
print "epsilon = epsilon_1 * epsilon_2 = " + str(epsilon_1 * epsilon_2) print "epsilon = epsilon_1 * epsilon_2 = " + str(epsilon_1 * epsilon_2)
p_q = naik_number_dict[(p, q)] q_p = naik_number_dict[(q, p)]
d = n / (2 * abs(p_q)) d = n / (2 * abs(q_p))
print "\nnaik_sign = " + str(sign(p_q)) print "\nnaik_sign = " + str(sign(q_p))
print "eta = naik_sign^d = " + str(sign(p_q)^d) print "eta = naik_sign^d = " + str(sign(q_p)^d)
if sign(p_q)^d == epsilon_1 * epsilon_2: if sign(q_p)^d == epsilon_1 * epsilon_2:
print "eta == epsilon\n" print "eta == epsilon\n"
else: else:
print "eta != epsilon\n" print "eta != epsilon\n"
@ -1306,9 +1313,9 @@ def check_criteria(name, pd_code, f_homfly_in=None):
return tester return tester
def get_naik_number(p, q): def get_naik_number(q, p):
''' '''
Calculate the smallest integer i such that p^i == +/-1 mod q. Calculate the smallest integer i = [q, p] such that p^i == +/-1 mod q.
Signum of i shows whether rest is -1 or 1 Signum of i shows whether rest is -1 or 1
''' '''
if gcd(q, p) > 1: if gcd(q, p) > 1:
@ -1389,10 +1396,28 @@ if __name__ == '__main__':
R.<t> = LaurentPolynomialRing(ZZ) R.<t> = LaurentPolynomialRing(ZZ)
prime_numbers = Primes() prime_numbers = Primes()
naik_number_dict = {} naik_number_dict = {}
if not os.path.isfile(settings.f_old_results):
f = open(settings.f_old_results, 'w+') if not os.path.isfile(settings.f_old_results) \
or not settings.check_old_results:
settings.check_old_results = False settings.check_old_results = False
f.close() if settings.save_homfly and settings.input_file_with_homflypt:
with open(settings.f_results_out, 'w') as f_out,\
open(settings.f_homfly_lm_out, 'w') as f_homfly_out,\
open(settings.f_homfly_lm_in, 'r') as f_homfly_in:
test_all(f_out, f_homfly_out, f_homfly_in)
elif settings.save_homfly:
with open(settings.f_results_out, 'w') as f_out,\
open(settings.f_homfly_lm_out, 'w') as f_homfly_out:
test_all(f_out, f_homfly_out)
elif settings.input_file_with_homflypt:
with open(settings.f_results_out, 'w') as f_out,\
open(settings.f_homfly_lm_in, 'r') as f_homfly_in:
test_all(f_out, None, f_homfly_in)
else:
with open(settings.f_results_out, 'w') as f_out:
test_all(f_out)
sys.exit()
with open(settings.f_old_results, 'r') as f_old_results: with open(settings.f_old_results, 'r') as f_old_results:
if settings.save_homfly and settings.input_file_with_homflypt: if settings.save_homfly and settings.input_file_with_homflypt:
with open(settings.f_results_out, 'w') as f_out,\ with open(settings.f_results_out, 'w') as f_out,\