#!/usr/bin/env python
# Copyright (c) 2018: Maria Marchwicka, Wojciech Politarczyk.
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see
import sys
import os
import numpy as np
import warnings
class MySettings(object):
def __init__(self):
self.f_pd_knot_11_15 = os.path.join(os.getcwd(), "knots_11_15.txt")
self.f_knot_up_to_10 = os.path.join(os.getcwd(), "knots_3_10.txt")
self.f_homfly_lm_out = os.path.join(os.getcwd(), "homflypt.out")
self.f_homfly_lm_in = os.path.join(os.getcwd(), "homflypt.input")
self.f_results_out = os.path.join(os.getcwd(), "results.out")
self.f_old_results = os.path.join(os.getcwd(), "old_results.input")
self.periods = [9]
# self.periods = [3, 5, 7, 9, 11]
self.set_to_check = self.get_set()
# check only knots from defined set
self.only_chosen = True
self.only_chosen = False
self.debugging = True
# self.debugging = False
# only if debugging
self.print_matrices = True
self.print_matrices = False
# only if only_chosen
self.only_periods_where_borodzik = True
self.only_periods_where_borodzik = False
# only if only_chosen
self.only_periods = True
self.only_periods = False
self.print_results = False
self.print_results = True
# saving HOMFLYPT polynomials into self.f_homfly_lm_out
self.save_homfly = True
self.save_homfly = False
# reuse HOMFLYPT polynomials previously saved
self.input_file_with_homflypt = True
# self.input_file_with_homflypt = False
self.check_old_results = True
# self.check_old_results = False
if self.input_file_with_homflypt:
if not os.path.isfile(self.f_homfly_lm_in):
warnings.warn("No input file with HOMFLYPT polynomials")
self.input_file_with_homflypt = False
def get_set(self):
set_to_check = set()
periodic_burde = set(["3_1", "5_1", "7_1", "8_19", "9_1",
"9_35", "9_40", "9_41", "9_47", "9_49",
"10_3", "10_123", "10_124"])
set_to_check |= periodic_burde
# knots that fail Borodzik criterion
self.fails_dict = {
"12a100": 3,
"12a348": 3,
"13a4648": 3,
"13n3659": 3,
"14a7583": 3,
"14a7948": 3,
"14a8670": 3,
"14a9356": 3,
"14a14971": 3,
"14a16311": 3,
"14a17173": 3,
"14a17260": 3,
"14a18647": 3,
"14n908": 3,
"14n913": 3,
"14n2451": 3,
"14n2458": 3,
"14n6565": 3,
"14n9035": 3,
"14n11989": 3,
"14n14577": 3,
"14n23051": 3,
"14n24618": 3,
"15a6030": 3,
"15a6066": 3,
"15a10622": 3,
"15a15077": 3,
"15a33910": 3,
"15a36983": 3,
"15a46768": 3,
"15a72333": 3,
"15a82771": 3,
"15n3147": 3,
"15n3369": 3,
"15n3372": 3,
"15n4496": 3,
"15n4514": 3,
"15n4517": 3,
"15n11293": 3,
"15n11533": 3,
"15n14173": 3,
"15n15251": 3,
"15n19351": 3,
"15n19989": 3,
"15n20691": 3,
"15n33684": 3,
"15n34725": 3,
"15n36715": 3,
"15n45612": 3,
"15n49287": 3,
"15n55026": 3,
"15n58771": 3,
"15n59908": 3,
"15n61622": 3,
"15n61790": 3,
"15n61833": 3,
"15n63397": 3,
"15n67585": 3,
"15n69848": 3,
"15n90233": 3,
"15n90525": 3,
"15n112198": 3,
"15n115648": 3,
"15n116414": 3,
"15n120198": 3,
"15n120375": 3,
"15n133302": 3,
"15n135864": 3,
"15n135918": 3,
"15n148509": 3,
"15n155150": 3,
"15n158831": 3,
"15n162066": 3,
"15n162237": 3,
"15n163140": 3,
"15n165092": 3,
"15n165622": 3,
"15n167650": 3,
"14n26993": 5,
"15a80526": 5,
"15n83514": 5,
"15n95792": 5,
}
set_to_check |= set(self.fails_dict.keys())
# knots that pass Borodzik criterion
self.success_dict = {
"9_40": 3,
"9_41": 3,
"9_49": 3,
"11a297": 3,
"11a321": 3,
"11n133": 3,
"12a561": 3,
"12a780": 3,
"12a1019": 3,
"12a1202": 3,
"12a1206": 3,
"12n706": 3,
"12n837": 3,
"12n839": 3,
"12n843": 3,
"12n844": 3,
"12n847": 3,
"12n881": 3,
"13n2694": 3,
"14a2160": 3,
"14a7206": 3,
"14a10416": 3,
"14a12671": 3,
"14a15296": 3,
"14a16592": 3,
"14a18362": 3,
"14n945": 3,
"14n3276": 3,
"14n3888": 3,
"14n4912": 3,
"14n9288": 3,
"14n10327": 3,
"14n11309": 3,
"14n11898": 3,
"14n13447": 3,
"14n13863": 3,
"14n14083": 3,
"14n14183": 3,
"14n14497": 3,
"14n16414": 3,
"14n16415": 3,
"14n16428": 3,
"14n16682": 3,
"14n17032": 3,
"14n17183": 3,
"14n17871": 3,
"14n17959": 3,
"14n21996": 3,
"14n23568": 3,
"14n24905": 3,
"15a8033": 3,
"15a15545": 3,
"15a20833": 3,
"15a22423": 3,
"15a23751": 3,
"15a24566": 3,
"15a24687": 3,
"15a33565": 3,
"15a35274": 3,
"15a39992": 3,
"15a40971": 3,
"15a54610": 3,
"15a74206": 3,
"15a74381": 3,
"15a77993": 3,
"15a81135": 3,
"15a81151": 3,
"15a81179": 3,
"15a81370": 3,
"15a81477": 3,
"15a81796": 3,
"15a82451": 3,
"15a82698": 3,
"15a83361": 3,
"15a83814": 3,
"15a85128": 3,
"15a85145": 3,
"15a85169": 3,
"15a85223": 3,
"15a85254": 3,
"15a85257": 3,
"15n15450": 3,
"15n15810": 3,
"15n17487": 3,
"15n17658": 3,
"15n18682": 3,
"15n20353": 3,
"15n28777": 3,
"15n29526": 3,
"15n31070": 3,
"15n33167": 3,
"15n39609": 3,
"15n39756": 3,
"15n39792": 3,
"15n39829": 3,
"15n39838": 3,
"15n39866": 3,
"15n40188": 3,
"15n40203": 3,
"15n45334": 3,
"15n47776": 3,
"15n48100": 3,
"15n50732": 3,
"15n52424": 3,
"15n52723": 3,
"15n55025": 3,
"15n59277": 3,
"15n59777": 3,
"15n59987": 3,
"15n60899": 3,
"15n61859": 3,
"15n62066": 3,
"15n68367": 3,
"15n68469": 3,
"15n72570": 3,
"15n75241": 3,
"15n77155": 3,
"15n78784": 3,
"15n78786": 3,
"15n81011": 3,
"15n84209": 3,
"15n85291": 3,
"15n93105": 3,
"15n95263": 3,
"15n95294": 3,
"15n98814": 3,
"15n99593": 3,
"15n100351": 3,
"15n105142": 3,
"15n122147": 3,
"15n126255": 3,
"15n127744": 3,
"15n132539": 3,
"15n134183": 3,
"15n134435": 3,
"15n135170": 3,
"15n137023": 3,
"15n142082": 3,
"15n145384": 3,
"15n146140": 3,
"15n147033": 3,
"15n151780": 3,
"15n152852": 3,
"15n153976": 3,
"15n154660": 3,
"15n154766": 3,
"15n159959": 3,
"15n160415": 3,
"15n160533": 3,
"15n165706": 3,
"15n165708": 3,
"15n165735": 3,
"15n165748": 3,
"15n166307": 3,
"15n167633": 3,
"15n167645": 3,
"15n168004": 3,
"15n168014": 3,
"10_123": 5,
"14n7478": 5,
"15a40549": 5,
"15a53966": 5,
"15a64035": 5,
"15a69121": 5,
"15a76651": 5,
"15a84903": 5,
"15a85262": 5,
"15n35157": 5,
"15n113162": 5,
"15n142117": 5,
"14a19470": 7,
"15n162490": 7,
"15a74206": 9,
"15n154766": 9,
"15n160415": 9,
"15n165706": 9,
}
set_to_check |= set(self.success_dict.keys())
# knots that are known to be (not) periodic
self.periods_dict = {
"3_1": [3],
"5_1": [5],
"7_1": [7],
"8_19": [3],
"9_1": [3, 9],
"9_35": [3],
"9_40": [3],
"9_41": [3],
"9_47": [3],
"9_49": [3],
"10_3": [3],
"10_123": [5],
"10_124": [3, 5],
"11a367": [11],
"12a503": [3],
"12a561": [3],
"12a615": [3],
"12a1019": [3],
"12a1022": [3],
"12a1202": [3],
"14a19470": [7],
"15a64035": [5],
"15a84903": [5],
"15a85262": [5],
"15a85263": [5],
"12a100": [-3],
"12a348": [-3],
"12a376": [-3],
"12a1206": [-3],
"13a2142": [-5],
"13a2907": [-5],
"13a3010": [-5],
"15a23599": [-5],
"15a23902": [-5],
"15a40549": [-5],
"15a53966": [-5]
}
set_to_check |= set(self.periods_dict.keys())
# knots that have Alexander polynomial = 1
self.alexander_1 = set(["11n34",
"11n42",
"12n313",
"12n430",
"13n65",
"13n71",
"13n866",
"13n1019",
"13n1496",
"13n1756",
"13n1757",
"13n3871",
"13n3872",
"13n3897",
"13n3934",
"13n3936",
"13n3938",
"13n4582",
"13n4591",
"14n3798",
"14n4425",
"14n5152",
"14n5486",
"14n6082",
"14n7469",
"14n7708",
"14n9023",
"14n9290",
"14n9684",
"14n9773",
"14n9882",
"14n10011",
"14n10119",
"14n10990",
"14n11063",
"14n11129",
"14n11515",
"14n11680",
"14n12763",
"14n14735",
"14n14833",
"14n15285",
"14n15581",
"14n18909",
"14n18911",
"14n21673",
"14n22185",
"14n22589",
"14n23325",
"14n23411",
"14n23417",
"14n23940",
"14n24036",
"14n24396",
"14n25281",
"15n2810",
"15n3240",
"15n4018",
"15n4646",
"15n11287",
"15n11296",
"15n11568",
"15n11570",
"15n15829",
"15n16056",
"15n17501",
"15n21288",
"15n21905",
"15n21939",
"15n21944",
"15n24436",
"15n25044",
"15n27582",
"15n27824",
"15n28998",
"15n29401",
"15n29559",
"15n29563",
"15n30723",
"15n31075",
"15n34773",
"15n36113",
"15n37062",
"15n38863",
"15n40132",
"15n40402",
"15n40938",
"15n42200",
"15n42279",
"15n42516",
"15n44873",
"15n45781",
"15n45782",
"15n46093",
"15n46536",
"15n48362",
"15n49081",
"15n49735",
"15n49992",
"15n50050",
"15n50051",
"15n50147",
"15n50819",
"15n51748",
"15n51847",
"15n52282",
"15n52651",
"15n54221",
"15n58433",
"15n58501",
"15n59917",
"15n59918",
"15n61482",
"15n62093",
"15n62150",
"15n63468",
"15n64468",
"15n65084",
"15n65980",
"15n71170",
"15n73226",
"15n74185",
"15n77245",
"15n77247",
"15n80534",
"15n82843",
"15n83995",
"15n85041",
"15n85314",
"15n87941",
"15n88033",
"15n89822",
"15n91092",
"15n91760",
"15n95983",
"15n95989",
"15n95995",
"15n96014",
"15n103703",
"15n108850",
"15n108966",
"15n110439",
"15n113775",
"15n115135",
"15n115375",
"15n117232",
"15n120055",
"15n120219",
"15n121343",
"15n121598",
"15n121834",
"15n121916",
"15n122603",
"15n123337",
"15n123414",
"15n123479",
"15n124496",
"15n124511",
"15n124640",
"15n124838",
"15n124849",
"15n125351",
"15n126042",
"15n126050",
"15n127500",
"15n128163",
"15n130096",
"15n130504",
"15n130528",
"15n131977",
"15n132396",
"15n132965",
"15n134216",
"15n135221",
"15n135706",
"15n138033",
"15n138051",
"15n139247",
"15n139256",
"15n139840",
"15n140327",
"15n140449",
"15n142843",
"15n143482",
"15n143825",
"15n143856",
"15n143985",
"15n144034",
"15n144436",
"15n144439",
"15n145339",
"15n145981",
"15n146982",
"15n151010",
"15n154389",
"15n155056",
"15n155464",
"15n156539",
"15n163337",
"15n165398",
])
set_to_check |= self.alexander_1
set_to_check |= set(["10_123"])
set_to_check |= set(["15n166130"])
set_to_check = set(self.fails_dict.keys())
return set_to_check
class PeriodicityTester(object):
def __init__(self, name, pd_code, A=None, f_homfly_in=None):
self.results = []
'''
To results for each period q a list in following form will be appended:
[q, murasugi, naik_1, naik_2, borodzik, przytycki]
Crierion is set to be:
-1 if it is not applicable (details in check_naik_2, check_przytycki,
1 if criterion doesn't exclude periodic,
0 if criterion excludes periodicity.
murasugi, naik_1, naik_2 or borodzik is also set to be:
2 if alexander_polynomial == 1.
0 if previous criterion in the list is 0.
'''
self.name = name
self.pd_code = pd_code
self.smith = None
self.reset_results()
if pd_code is not None:
self.K = Link(pd_code)
self.seifert = self.K.seifert_matrix()
else:
self.seifert = A
# delta := Alexander polynomial
delta = (self.seifert.transpose() - t * self.seifert).determinant()
self.delta = delta.shift(-delta.exponents()[0])
self.delta_factors = self.set_delta_factors()
self.przytycki_tester = self.get_przytycki_tester(f_homfly_in)
def reset_results(self):
self.murasugi = 0
self.naik_1 = 0
self.naik_2 = 0
self.borodzik = 0
self.przytycki = 0
self.murasugi_fulfilling = set()
self.naik_1_fulfilling = []
self.naik_2_fulfilling = []
def set_smith(self):
symetric_from_seifert = self.seifert + self.seifert.transpose()
assert symetric_from_seifert.determinant() != 0, \
"The determinant of A + A^T is zero."
self.smith = symetric_from_seifert.smith_form()
D, U, V = self.smith
self.diagonal = D.diagonal()
self.maximum_in_diagonal = max(self.diagonal)
C = U.inverse()
E_inverse = V
self.C_tran_E_inv_D_inv = C.transpose() * E_inverse * D.inverse()
self.matrix_C = C
self.matrix_E_inverse = E_inverse
def get_przytycki_tester(self, f_homfly_in):
if self.pd_code is not None:
try:
return PrzytyckiTester(self.K, self.name, f_homfly_in)
except ImportError as e:
if settings.debugging:
print "Error by checking Przytycki criterion.\n" + str(e)
return None
def get_C_tran_E_inv_D_inv(self):
if self.smith is None:
self.set_smith()
return self.C_tran_E_inv_D_inv
def get_maximum_in_diagonal(self):
if self.smith is None:
self.set_smith()
return self.maximum_in_diagonal
def set_delta_factors(self):
# find all delta (alexander polynomial) factors
lst_of_factors = [[f[0]] * f[1] for f in self.delta.factor()]
# flattening a list
lst_of_factors = [el for sublist in lst_of_factors for el in sublist]
delta_candidates = set()
for s in get_subsets(lst_of_factors):
d = t^0
for el in s:
d *= el
delta_candidates.add(d)
return delta_candidates
def check_criteria_for_period(self, q):
self.reset_results()
self.przytycki = self.check_przytycki(q)
if self.delta == 1:
self.murasugi = 2
self.naik_1 = 2
self.naik_2 = 2
self.borodzik = 2
return 2
self.murasugi = self.check_murasugi(q)
self.naik_1 = self.check_naik_1(q)
self.naik_2 = self.check_naik_2(q)
self.borodzik = self.check_borodzik(q)
if settings.debugging:
print ("\n" + "#" * 30 + " Calculations for knot " + self.name +
" and q = " + str(q) + " " + "#" * 30 + "\n")
self.print_data_for_murasugi(q)
self.print_data_for_naik_1(q)
self.print_data_for_naik_2(q)
self.print_data_for_borodzik(q)
return self.borodzik * self.przytycki
def check_murasugi(self, q):
'''
Select these delta factors and natural number r such that:
delta = delta_prime^q * (1 + t^1 + ... + t^(r-1))^(q-1) mod q_factor
where "delta_prime" is a delta factor.
'''
q_factor = factor(q)[0][0]
quotient_delta = self.delta.change_ring(GF(q_factor))
# Underlying polynomial of quotient_delta:
quotient_delta = quotient_delta.polynomial_construction()[0]
delta_degree = quotient_delta.degree()
for candidate in self.delta_factors:
quotient_candidate = candidate.change_ring(GF(q_factor))
power_candidate = quotient_candidate^q
power_candidate = power_candidate.polynomial_construction()[0]
# (r - 1) - possible t-polynomial degree
r = (delta_degree - power_candidate.degree()) / (q - 1) + 1
if r < 1 or not r.is_integer():
continue
t_polynomial = get_t_polynomial(q, r)
right_side = t_polynomial * power_candidate
if quotient_delta != right_side and -quotient_delta != right_side:
continue
self.murasugi_fulfilling.add((candidate, r))
return int(bool(self.murasugi_fulfilling))
def check_naik_1_candidate(self, delta_prime, delta_evaluated, q):
t_delta = delta_evaluated / delta_prime(-1)
t_delta_dict = {f[0]: f[1] for f in factor(t_delta)}
t_delta_factors = [f for f in t_delta_dict.keys()
if f != 2 and gcd(q, f) == 1]
for f in t_delta_factors:
q_f = naik_number_dict.setdefault((q, f), get_naik_number(q, f))
if not (t_delta_dict[f] / (2 * q_f)).is_integer():
return None
return t_delta_factors
def check_naik_1(self, q):
'''
For each delta' find a set P of prime numbers p such that:
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*[q|p].
If it holds for at least one delta' candidate, set naik_1 = True.
'''
# Proposition 2.7.
delta_evaluated = self.delta(-1)
for delta_prime, _ in self.murasugi_fulfilling:
t_delta_factors = self.check_naik_1_candidate(delta_prime,
delta_evaluated, q)
if t_delta_factors is not None:
self.naik_1_fulfilling.append((delta_prime, t_delta_factors))
return int(bool(self.naik_1_fulfilling))
def check_naik_2_candidate(self, q, p_list):
delta_prime_bases = []
maximum_in_diagonal = self.get_maximum_in_diagonal()
for p in p_list:
q_p = naik_number_dict[(q, p)]
bases_for_p_torsion = []
factor_power = p
# find all p^k torsion parts
while (maximum_in_diagonal / factor_power).is_integer():
basis_for_p_k_part = []
for el in self.diagonal:
to_be_append = el / factor_power
is_int = (to_be_append / p).is_integer()
if to_be_append.is_integer() and not is_int:
basis_for_p_k_part.append(to_be_append)
else:
basis_for_p_k_part.append(0)
len_non_zero = sum(x != 0 for x in basis_for_p_k_part)
# check if dimension is multiple of 2 * naik_number
if not (len_non_zero / (2 * q_p)).is_integer():
return None
factor_power *= p
bases_for_p_torsion.append(basis_for_p_k_part)
delta_prime_bases.append((p, bases_for_p_torsion))
return delta_prime_bases
def check_naik_2(self, q):
'''
For each delta' consider a set P of primes p such that: gcd(p, q) == 1,
p != 2, p| delta(-1)/delta'(-1) (self.naik_1_fulfilling) and p is not
a factor of delta'(-1). Check if dimension of p^k torsion part
is divisible by 2*[p|q] for all k and all p from P.
If it holds for at least one delta' candidate, we set naik_2 to be True.
In particular naik_2 is set to be -1 if the criterion passes,
but only in cases where P is an empty set.
'''
# Proposition 2.8.
for delta_prime, p_list in self.naik_1_fulfilling:
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]
if not p_list:
self.naik_2 = -1
self.borodzik = -1
continue
delta_prime_bases = self.check_naik_2_candidate(q, p_list)
if delta_prime_bases is not None:
self.naik_2_fulfilling.append((delta_prime,
delta_prime_bases))
if self.naik_2_fulfilling:
return 1
return self.naik_2
def check_borodzik(self, q):
'''
Consider all delta' that meet criterion Naik 2.
For all p from a set P (defined as in check_naik_2)
and all k consider p^k torsion part.
For each p^k torsion check if eta == epsilon_1 * epsilon_2
(see check_borodzik_candidate()).
If it holds for at least one delta' candidate, set borodzik to be True.
In particular borodzik is set to be -1 if the criterion passes,
but only in cases where P is an empty set.
'''
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
borodzik_pass = True
for p, bases_for_p in delta_prime_bases:
# if len(bases_for_p) > 1:
# print "HURA" # more than one p^k part - not found yet
if not self.check_borodzik_candidate(q, p, bases_for_p):
borodzik_pass = False
break
if borodzik_pass:
return 1
return self.borodzik
def check_borodzik_candidate(self, q, p, bases):
'''
For each p^k torsion check if eta == epsilon_1 * epsilon_2.
If determinant of corsesponding matrix P is square modulo p, then:
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:
epsilon_2 = -1, else: epsilon_2 = 1.
eta = naik_sign^d, where d = n / (2 * [p, q]).
If p^([p, q]) % q == 1, then: naik_sign = 1, else: naik_sign = -1.
'''
for k, p_k_basis in enumerate(bases):
X = np.diagflat(p_k_basis)
# columns that up to zero (element in diagonal is zero):
zero_columns = np.nonzero(X.sum(axis=0) == 0)
X = np.delete(X, zero_columns, axis=1)
n = X.shape[1]
X = matrix(X)
P = p^(k + 1) * X.transpose() * self.get_C_tran_E_inv_D_inv() * X
P_det = P.determinant()
if P_det % p == 0:
raise ValueError("P determinant is 0 modulo p.")
if p % 4 == 3 and n % 4 == 2: # epsilon_1
epsilon = -1
else:
epsilon = 1
if not mod(P_det, p).is_square():
epsilon *= -1 # epsilon = epsilon_1 * epsilon_2
q_p = naik_number_dict[(q, p)]
d = n / (2 * q_p)
# sign(q_p) - whether rest is -1 or 1
if sign(q_p)^d != epsilon:
return False
return True
def check_przytycki(self, q):
if self.przytycki_tester is not None and q in prime_numbers:
try:
return self.przytycki_tester.check_congruence(q)
except (AttributeError, OverflowError) as e:
pass
return -1
def save_results(self, f_out, f_homfly_out=None):
for result in self.results:
line_to_write = self.name + "," + ",".join(map(str, result))
if settings.check_old_results and (result[0] in [3, 5, 7, 9, 11]):
line = f_old_results.readline()
# name, q, murasugi, naik_1, naik_2, borodzik, przytycki
while line:
line = line.split(',')
if line[0] == self.name and line[1] == str(result[0]):
old_results = [int(x) for x in line[2:]]
# if old_results[:-1] != result[1:-1]:
if old_results[:] != result[1:]:
print ("#" * 30 + " ERROR " + line[0] + " " +
"#" * 30)
print "q = " + line[1]
print "result " + str(result[1:])
print "old_results " + str(old_results)
break
line = f_old_results.readline()
if not line:
print "No data to compare."
f_out.writelines(line_to_write + "\n")
if self.przytycki_tester is not None and f_homfly_out is not None:
lm_polynomial = self.przytycki_tester.homflypt_polynomial
line_to_write = self.name + "," + str(lm_polynomial) + "\n"
f_homfly_out.writelines(line_to_write)
def print_results(self):
print "\n" + "#" * 15 + " " + str(self.name) + " " + "#" * 15
if self.name in settings.periods_dict:
print "periods: " + str(settings.periods_dict[self.name])
for result in self.results:
q = result[0]
print
self.print_przytycki_result(q, result[5])
if result[1] == 2:
print "Alexander polynomial is 1"
continue
if not result[1]:
print "\t\tMurasugi: fail, q = " + str(q)
continue
print "Murasugi: pass, q = " + str(q)
if not result[2]:
print "\t\tNaik 1: fail, q = " + str(q)
continue
print "Naik 1: pass, q = " + str(q)
if not result[3]:
print "\t\tNaik 2: fail, q = " + str(q)
continue
if result[3] == -1:
print "Naik 2: not applicable, q = " + str(q)
continue
print "Naik 2: pass, q = " + str(q)
if not result[4]:
print ("\t\tBorodzik: fail, q = " + str(q))
continue
if result[4] == -1:
print ("Borodzik: not applicable, q = " + str(q))
continue
print ("Borodzik: pass, q = " + str(q))
def print_przytycki_result(self, q, result):
if not result:
print "\t\tPrzytycki: fail, q = " + str(q)
elif result == -1:
print "Przytycki: not applicable, q = " + str(q)
else:
print "Przytycki: pass, q = " + str(q)
def print_data_for_murasugi(self, q):
if self.murasugi:
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
" passes Murasugi condition for q = " +
str(q) + " " + "#" * 30)
else:
print ("\nKnot " + str(self.name) +
" fails Murasugi condition for q = " + str(q))
quotient_delta = self.delta.change_ring(GF(q))
quotient_delta = quotient_delta.polynomial_construction()[0]
print "delta: " + str(self.delta)
print "delta factors: " + str(self.delta.factor())
print "delta mod q = " + str(quotient_delta)
delta_degree = quotient_delta.degree()
self.print_murasugi_fulfilling(q)
# self.print_candidates_that_fail_murasugi(q)
def print_murasugi_fulfilling(self, q):
quotient_delta = self.delta.change_ring(GF(q))
quotient_delta = quotient_delta.polynomial_construction()[0]
delta_degree = quotient_delta.degree()
print ("\nNumber of candidates that pass Murasugi = " +
str(len(self.murasugi_fulfilling)))
for i, (delta_prime, r) in enumerate(self.murasugi_fulfilling):
print "\n" + str(i + 1) + ". delta_prime:\t" + str(delta_prime)
t_polynomial = get_t_polynomial(q, r)
print "polynomial^(q-1) = " + str(t_polynomial)
right_side = t_polynomial * delta_prime^q
print "*" * 50
print "delta == delta_prime^q * polynomial^(q-1) mod q"
print "right side:\t" + str(right_side.factor())
print "left side:\t" + str(quotient_delta.factor())
def print_candidates_that_fail_murasugi(self, q):
quotient_delta = self.delta.change_ring(GF(q))
quotient_delta = quotient_delta.polynomial_construction()[0]
delta_degree = quotient_delta.degree()
for candidate in self.delta_factors:
quotient_candidate = candidate.change_ring(GF(q))
power_candidate = quotient_candidate^q
shifted_candidate = power_candidate.polynomial_construction()[0]
r = (delta_degree - shifted_candidate.degree()) / (q - 1) + 1
if r > 0 and r.is_integer():
t_polynomial = get_t_polynomial(q, r)
right_side = t_polynomial * shifted_candidate
if (quotient_delta == right_side or
(-quotient_delta) == right_side):
continue
print "\nFor candidate = " + str(candidate)
print "quotient_candidate = " + str(quotient_candidate)
print "candidate^q = " + str(power_candidate)
print "shifted = " + str(shifted_candidate)
print "delta degree = " + str(delta_degree)
print "candidate^q degree " + str(shifted_candidate.degree())
print "r = " + str(r)
if r > 0 and r.is_integer():
print "right_side = " + str(right_side)
print "delta mod q = " + str(quotient_delta)
def print_data_for_naik_1(self, q):
if not self.murasugi:
return None
if not self.naik_1:
print ("\nKnot " + str(self.name) +
" fails Naik 1 condition for q = " + str(q))
else:
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
" passes Naik 1 condition for q = " + str(q) +
" " + "#" * 30)
print "delta: " + str(self.delta)
print "delta at -1: " + str(self.delta(-1))
print "factors for evaluated: " + str(self.delta(-1).factor())
self.print_naik_1_fulfilling(q)
def print_naik_1_fulfilling(self, q):
print ("\nNumber of candidates that pass Naik 1 = " +
str(len(self.naik_1_fulfilling)))
for delta_prime, p_list in self.naik_1_fulfilling:
print "delta prime: " + str(delta_prime)
print "delta prime at -1: " + str(delta_prime(-1))
t_delta = self.delta(-1)/delta_prime(-1)
print "delta/delta_prime(-1):\t\t" + str(t_delta)
print "delta/delta_prime(-1) factors:\t" + str(t_delta.factor())
if not p_list:
print "List of factors was empty."
for p in p_list:
g = abs(naik_number_dict[(q, p)])
print "factor of del/del'(-1): " + str(p)
print "Naik number: " + str(g)
print "2 * Naik number:\t" + str(2 * g)
test_naik_number = p^g % q
print (str(p) + "^" + str(g) + " % " + str(q) + " = " +
str(test_naik_number) + " = " +
str(test_naik_number - q))
t_delta_dict = {i[0]: i[1] for i in factor(t_delta)}
print "The power of factor:\t" + str(t_delta_dict[p])
def print_data_for_naik_2(self, q):
if not self.naik_1:
return None
if not self.naik_2:
return None
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
" passes Naik 2 condition for q = " + str(q) + " " + "#" * 30)
print "delta:\t\t\t" + str(self.delta)
print "delta at -1:\t\t" + str(self.delta(-1))
print "factors for evaluated:\t" + str(self.delta(-1).factor())
if self.naik_2 == -1:
self.print_naik_2_not_applicable(q)
return None
self.print_naik_2_fulfilling(q)
def print_naik_2_not_applicable(self, q):
for delta_prime, p_list in self.naik_1_fulfilling:
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]
if not p_list:
print ("\nChecking Naik 2 condition for candidate " +
str(delta_prime) + " and q = " + str(q)) + "."
print ("The list of factors was empty or all factors " +
"were dela'(-1) factors.")
print "Naik 2 and Borodzik can not exclude periodicity.\n"
def print_naik_2_fulfilling(self, q):
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
print "\ndelta prime:\t\t\t" + str(delta_prime)
print "delta prime at -1:\t\t" + str(delta_prime(-1))
t_delta = self.delta(-1)/delta_prime(-1)
print "delta/delta_prime(-1):\ " + str(t_delta)
print "delta/delta_prime(-1) factors: " + str(t_delta.factor())
for p, bases_for_p in delta_prime_bases:
print "\nfactor p for delta prime:\t\t\t" + str(p)
g = abs(naik_number_dict[(q, p)])
print "Naik number:\t\t" + str(g)
print "2 * Naik number:\t" + str(2 * g)
test_naik_number = p^g % q
print (str(p) + "^" + str(g) + " % " + str(q) + " = " +
str(test_naik_number) + " = " +
str(test_naik_number - q))
t_delta_dict = {i[0]: i[1] for i in factor(t_delta)}
print "The power of factor:\t" + str(t_delta_dict[p])
print "diagonal: " + str(self.diagonal)
print "p^k basis"
for k, b in enumerate(bases_for_p):
print "k = " + str(k + 1)
print "basis:\t" + str(b)
def print_data_for_borodzik(self, q):
if self.naik_2 != 1:
return None
if self.borodzik:
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
" passes Borodzik condition for q = " +
str(q) + " " + "#" * 30)
else:
print "%" * 200
print ("\nKnot " + str(self.name) +
" fails Borodzik condition for q = " + str(q))
if settings.print_matrices:
self.print_matrices_for_borodzik(q)
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
print "\nResults for candidate delta_prime = " + str(delta_prime)
for p, bases_for_p in delta_prime_bases:
print "Results for p = " + str(p)
for k, p_k_basis in enumerate(bases_for_p):
self.print_borodzik_for_p_k_basis(p, k, p_k_basis, q)
print "%" * 200 + "\n" * 3
def print_matrices_for_borodzik(self, q):
print "\n\nSeifert matrix A:"
print str(self.seifert)
print "\n\nA + A^T:"
print str(self.seifert + self.seifert.transpose())
print "\n\nC"
print str(self.matrix_C)
# print "\nE^(-1)"
# print str(self.E_inverse)
print "\n\nD - diagonal"
print str(self.diagonal)
print "\n\nE"
print str(self.matrix_E_inverse.inverse())
print "\n\nC^T * E^{-1} * D^{-1}"
print self.get_C_tran_E_inv_D_inv()
def print_borodzik_for_p_k_basis(self, p, k, p_k_basis, q):
# X matrix
X = np.diagflat(p_k_basis)
zero_columns = np.nonzero(X.sum(axis=0) == 0)
X = np.delete(X, zero_columns, axis=1)
n = X.shape[1]
X = matrix(X)
# P deterinant and epsilon_1
P = p^(k + 1) * X.transpose() * self.get_C_tran_E_inv_D_inv() * X
P_det = P.determinant()
if settings.print_matrices:
print "\nsubmatrix:"
print self.C_tran_E_inv_D_inv[-n:, -n:]
print "\nP\n" + str(P)
print "\ndet(P) = " + str(P_det)
if mod(P_det, p).is_square():
print ("det(P) % p = " + str(P_det % p) +
" is a square => epsilon_1 := 1")
epsilon_1 = 1
else:
print ("det(P) % p = " + str(P_det % p) +
" isn't a square => episilon_1 := -1")
epsilon_1 = -1
# p % 4 and n % 4, and epsilon_2
print "\np % 4 = " + str(p) + " % 4 = " + str(p % 4)
print "n % 4 = " + str(n) + " % 4 = " + str(n % 4)
if p % 4 == 3 and n % 4 == 2:
print "(p % 4 == 3 and n % 4 == 2) => episilon_2 := -1"
epsilon_2 = -1
else:
print "(p % 4 != 3 or n % 4 != 2) => episilon_2 := 1"
epsilon_2 = 1
# epsilon and eta
print "epsilon = epsilon_1 * epsilon_2 = " + str(epsilon_1 * epsilon_2)
q_p = naik_number_dict[(q, p)]
d = n / (2 * abs(q_p))
print "\nnaik_sign = " + str(sign(q_p))
print "eta = naik_sign^d = " + str(sign(q_p)^d)
if sign(q_p)^d == epsilon_1 * epsilon_2:
print "eta == epsilon\n"
else:
print "eta != epsilon\n"
class PrzytyckiTester(object):
def __init__(self, K, name, f_homfly_in=None):
self.verbose = True
self.verbose = False
self.verbose = settings.debugging
homflypt = self.get_homflypt_polynomial(K, name, f_homfly_in)
homfly_difference = homflypt(a, -z) - homflypt(a^-1, -z)
self.homfly_difference = z * homfly_difference
self.homflypt_polynomial = homflypt
if self.verbose:
print "\n" + "Knot " + name
print "HOMFLYPT = " + str(homflypt)
print ("HOMFLYPT(a, -z) - HOMFLYPT(a^-1, -z) = " +
str(homfly_difference))
print
def get_homflypt_polynomial(self, K, name, f_homfly_in=None):
if f_homfly_in is not None:
try:
current_name, homflypt = f_homfly_in.readline().split(',')
while current_name != name:
current_name, homflypt = f_homfly_in.readline().split(',')
homflypt = sage_eval(homflypt, locals={'a': a, 'z': z})
return homflypt
except (AttributeError, ValueError) as e:
if self.verbose:
print "The file with HOMFLYPT is incorect!\n" + str(e)
return K.homfly_polynomial('a', 'z', 'lm')
def check_congruence(self, q):
for i in range(q + 1):
z_coefficient = self.homfly_difference.coefficient(z^(i+1))
ideal = (a + a^-1)^(q - i) # for i == q will be 1
coefficient_modulo_ideal = z_coefficient.quo_rem(ideal)[1]
coefficient_modulo_q = coefficient_modulo_ideal.change_ring(GF(q))
if self.verbose:
print "\nv_" + str(i) + " = " + str(z_coefficient)
print ("v_" + str(i) + " mod (a + a^-1)^(q - i) = " +
str(coefficient_modulo_ideal))
print ("(v_" + str(i) + " mod (a + a^-1)^(q - i)) mod q = " +
str(coefficient_modulo_q))
if coefficient_modulo_q != 0:
return 0
return 1
def check_criteria(name, pd_code, f_homfly_in=None):
if settings.only_chosen and name not in settings.set_to_check:
return None
tester = PeriodicityTester(name, pd_code, None, f_homfly_in)
for i, q in enumerate(settings.periods):
if settings.only_periods:
if tester.name not in settings.periods_dict:
continue
if (q not in settings.periods_dict[tester.name] and
(-q) not in settings.periods_dict[tester.name]):
continue
if settings.only_periods_where_borodzik:
if tester.name not in settings.fails_dict:
if tester.name not in settings.success_dict:
continue
if q != settings.success_dict[tester.name]:
continue
else:
if q != settings.fails_dict[tester.name]:
continue
tester.check_criteria_for_period(q)
tester.results.append([q, tester.murasugi, tester.naik_1,
tester.naik_2, tester.borodzik,
tester.przytycki])
if settings.print_results:
tester.print_results()
return tester
def get_naik_number(q, p):
'''
Calculate the smallest integer i = [q, p] such that p^i == +/-1 mod q.
Signum of i shows whether rest is -1 or 1
'''
if gcd(q, p) > 1:
return 0
p_power = p
for i in xrange(1, sys.maxint):
pq = p_power % q
if pq == 1:
return i
if pq == q - 1:
return -i
p_power *= p
def get_t_polynomial(q, r): # for check_murasugi(), r coresponds to l in paper
t_polynomial = sum([t^i for i in range(r)])
t_polynomial = t_polynomial.change_ring(GF(q))
t_polynomial ^= (q - 1)
return t_polynomial
def get_subsets(myset):
return reduce(lambda z, x: z + [y + [x] for y in z], myset, [[]])
def parse_pd_code(pd_code_from_file):
set = '0987654321[],'
pd_code = ''.join([c for c in pd_code_from_file if c in set])
return eval(pd_code)
def parse_knot_name(name):
data = name[5: -2].split(',')
name = data[0].strip() + data[1].strip().lower()[:1] + data[2].strip()
return name
def check_11_to_15(f_out, f_homfly_out=None, f_homfly_in=None):
with open(settings.f_pd_knot_11_15, 'r') as f:
line = f.readline()
while line:
name = parse_knot_name(line)
pd_code = parse_pd_code(f.readline())
line = f.readline()
tester = check_criteria(name, pd_code, f_homfly_in)
if tester is None:
continue
tester.save_results(f_out, f_homfly_out)
def check_up_to_10(f_out, f_homfly_out=None, f_homfly_in=None):
with open(settings.f_knot_up_to_10, 'r') as f:
line = f.readline()
while line:
line = line.split(" = ")
name = str(line[0])[5:]
pd_code = parse_pd_code(str(line[1]))
line = f.readline()
tester = check_criteria(name, pd_code, f_homfly_in)
if tester is None:
continue
tester.save_results(f_out, f_homfly_out)
def test_all(f_out, f_homfly_out=None, f_homfly_in=None):
check_up_to_10(f_out, f_homfly_out, f_homfly_in)
if f_homfly_out is not None:
f_homfly_out.flush()
if f_out is not None:
f_out.flush()
check_11_to_15(f_out, f_homfly_out, f_homfly_in)
if __name__ == '__main__':
settings = MySettings()
S. = LaurentPolynomialRing(ZZ)
R. = LaurentPolynomialRing(ZZ)
prime_numbers = Primes()
naik_number_dict = {}
if not os.path.isfile(settings.f_old_results) \
or not settings.check_old_results:
settings.check_old_results = False
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:
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)