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Flat knot 5.79

Min(phi) over symmetries of the knot is: [-2,-1,0,1,2,-1,1,2,3,0,2,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['5.79']
Arrow polynomial of the knot is: -4*K1**3 + 4*K1**2 + 4*K1*K2 + K1 - 2*K2 - K3 - 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.79', '7.3234', '7.3766', '7.4689', '7.6536', '7.7923', '7.7949', '7.8531', '7.8946', '7.10655', '7.11360', '7.11505', '7.11827', '7.11832', '7.12255', '7.12343', '7.12568', '7.12746', '7.12831', '7.12970', '7.13477', '7.13866', '7.13878', '7.13908', '7.13926', '7.14047', '7.14167', '7.14666', '7.14943', '7.15029', '7.15081', '7.15122', '7.15341', '7.15351', '7.15914', '7.15926', '7.15994', '7.16056', '7.16175', '7.16230', '7.16263', '7.16957', '7.17043', '7.17061', '7.17136', '7.17138', '7.17171', '7.17201', '7.17259', '7.17345', '7.17767', '7.17789', '7.18040', '7.18950', '7.18988', '7.19258', '7.19654', '7.19656', '7.19698', '7.19893', '7.20093', '7.20125', '7.20352', '7.20378', '7.20392', '7.21318', '7.21364', '7.21453', '7.21472', '7.21722', '7.22756', '7.23316', '7.23431', '7.24139', '7.24145', '7.24248', '7.24256', '7.24279', '7.24281', '7.24312', '7.24455', '7.24511', '7.24698', '7.24868', '7.25326', '7.25446', '7.25829', '7.25969', '7.26087', '7.26162', '7.26190', '7.26244', '7.26245', '7.26450', '7.26570', '7.26651', '7.26678', '7.26693', '7.26758', '7.26784', '7.26798', '7.26820', '7.27084', '7.27197', '7.27274', '7.27730', '7.27825', '7.27861', '7.27915', '7.27916', '7.28179', '7.28190', '7.28276', '7.28301', '7.28325', '7.28337', '7.28381', '7.28486', '7.28518', '7.28687', '7.29032', '7.29117', '7.29482', '7.29488', '7.29625', '7.29644', '7.30016', '7.30074', '7.30077', '7.30177', '7.30415', '7.30424', '7.30430', '7.30432', '7.30436', '7.30474', '7.30488', '7.30506', '7.30535', '7.30720', '7.30810', '7.30859', '7.30892', '7.30955', '7.30976', '7.31101', '7.31114', '7.31198', '7.31251', '7.31329', '7.31379', '7.31455', '7.31464', '7.31500', '7.31510', '7.31517', '7.31633', '7.31667', '7.31672', '7.31725', '7.31733', '7.31779', '7.31781', '7.31809', '7.31811', '7.31814', '7.31925', '7.31996', '7.32077', '7.32078', '7.32081', '7.32082', '7.32154', '7.32182', '7.32206', '7.32218', '7.32250', '7.32257', '7.32262', '7.32268', '7.32302', '7.32369', '7.32479', '7.32505', '7.32521', '7.32527', '7.32530', '7.32552', '7.32603', '7.32611', '7.32624', '7.32644', '7.32693', '7.32701', '7.32706', '7.32714', '7.32719', '7.32742', '7.32752', '7.32934', '7.33082', '7.33170', '7.33190', '7.33353', '7.33434', '7.33445', '7.33490', '7.33519', '7.33538', '7.33568', '7.33609', '7.33651', '7.33686', '7.33693', '7.33728', '7.33989', '7.34107', '7.34152', '7.34196', '7.34259', '7.34264', '7.34298', '7.34307', '7.34396', '7.34407', '7.34498', '7.34528', '7.34552', '7.34653', '7.34723', '7.34744', '7.34761', '7.34987', '7.35050', '7.35139', '7.35161', '7.35198', '7.35199', '7.35691', '7.35708', '7.35750', '7.35752', '7.35773', '7.35790', '7.35891', '7.35909', '7.35923', '7.35993', '7.36029', '7.36035', '7.36048', '7.36060', '7.36160', '7.36218', '7.36246', '7.36283', '7.36889', '7.36903', '7.36925', '7.36930', '7.36978', '7.36989', '7.37118', '7.37128', '7.37137', '7.37138', '7.37148', '7.37171', '7.37180', '7.37208', '7.37302', '7.37316', '7.37331', '7.37357', '7.37358', '7.37361', '7.37370', '7.37382', '7.37388', '7.37449', '7.37488', '7.37638', '7.37701', '7.37826', '7.37849', '7.37958', '7.37981', '7.37985', '7.38007', '7.38073', '7.38109', '7.38153', '7.38186', '7.38215', '7.38253', '7.38268', '7.38281', '7.38286', '7.38322', '7.38332', '7.38336', '7.38353', '7.38363', '7.38391', '7.38393', '7.38414', '7.38428', '7.38435', '7.38471', '7.38491', '7.38503', '7.38673', '7.38679', '7.38756', '7.38797', '7.38854', '7.38877', '7.38937', '7.38948', '7.39262', '7.39280', '7.39322', '7.39364', '7.39401', '7.39442', '7.39472', '7.39477', '7.39492', '7.39553', '7.39558', '7.39563', '7.39571', '7.39584', '7.39629', '7.39647', '7.39653', '7.39665', '7.39729', '7.39806', '7.39893', '7.39913', '7.39922', '7.39933', '7.39998', '7.40000', '7.40039', '7.40055', '7.40062', '7.40064', '7.40140', '7.40142', '7.40177', '7.40214', '7.40355', '7.40366', '7.40372', '7.40373', '7.40382', '7.40383', '7.40485', '7.40498', '7.40584', '7.40587', '7.40603', '7.40607', '7.40623', '7.40629', '7.40815', '7.40836', '7.40926', '7.40935', '7.40964', '7.41115', '7.41215', '7.41216', '7.41383', '7.41723', '7.41755', '7.41770', '7.42431', '7.42435', '7.42440', '7.42488', '7.42516', '7.42539', '7.42572', '7.42578', '7.42949', '7.43771', '7.43774', '7.43829', '7.43872', '7.43884', '7.44059', '7.44165', '7.44202', '7.44217', '7.44233']
Outer characteristic polynomial of the knot is: t^6+23t^4+27t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.79', '7.41148']
2-strand cable arrow polynomial of the knot is: -448*K1**2*K2**4 + 448*K1**2*K2**3 - 992*K1**2*K2**2 + 736*K1**2*K2 - 32*K1**2*K3**2 - 764*K1**2 + 576*K1*K2**3*K3 + 912*K1*K2*K3 + 96*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 416*K2**4 - 304*K2**2*K3**2 - 48*K2**2*K4**2 + 112*K2**2*K4 - 190*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 316*K3**2 - 84*K4**2 - 24*K5**2 - 2*K6**2 + 578
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.79']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.163', 'vk5.198', 'vk5.293', 'vk5.352', 'vk5.656', 'vk5.803', 'vk5.1179', 'vk5.1322', 'vk5.1376', 'vk5.1393', 'vk5.1410', 'vk5.1437', 'vk5.1669', 'vk5.1789', 'vk5.1887', 'vk5.1947']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3U1U2O4O5U4U3U5
R3 orbit {'O1O2O3U1U2O4O5U4U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1U5O4O5U2U3
Gauss code of K* O1O2O3U4U5U2O4O5U1U3
Gauss code of -K* O1O2O3U1U3O4O5U2U4U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 1 -1 2],[ 2 0 1 2 0 1],[ 0 -1 0 1 0 1],[-1 -2 -1 0 0 2],[ 1 0 0 0 0 1],[-2 -1 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -2],[-2 0 -2 -1 -1 -1],[-1 2 0 -1 0 -2],[ 0 1 1 0 0 -1],[ 1 1 0 0 0 0],[ 2 1 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,2,2,1,1,1,1,0,2,0,1,0]
Phi over symmetry [-2,-1,0,1,2,-1,1,2,3,0,2,1,1,1,1]
Phi of -K [-2,-1,0,1,2,1,1,1,3,1,2,2,0,1,-1]
Phi of K* [-2,-1,0,1,2,-1,1,2,3,0,2,1,1,1,1]
Phi of -K* [-2,-1,0,1,2,0,1,2,1,0,0,1,1,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial -5z-9
Enhanced Jones-Krushkal polynomial 6w^3z-11w^2z-9w
Inner characteristic polynomial t^5+13t^3+7t
Outer characteristic polynomial t^6+23t^4+27t^2
Flat arrow polynomial -4*K1**3 + 4*K1**2 + 4*K1*K2 + K1 - 2*K2 - K3 - 1
2-strand cable arrow polynomial -448*K1**2*K2**4 + 448*K1**2*K2**3 - 992*K1**2*K2**2 + 736*K1**2*K2 - 32*K1**2*K3**2 - 764*K1**2 + 576*K1*K2**3*K3 + 912*K1*K2*K3 + 96*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 416*K2**4 - 304*K2**2*K3**2 - 48*K2**2*K4**2 + 112*K2**2*K4 - 190*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 316*K3**2 - 84*K4**2 - 24*K5**2 - 2*K6**2 + 578
Genus of based matrix 1
Fillings of based matrix [[{1, 5}, {3, 4}, {2}], [{2, 5}, {3, 4}, {1}], [{3, 5}, {2, 4}, {1}], [{5}, {3, 4}, {1, 2}]]
If K is slice False
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