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Flat knot 5.104

Min(phi) over symmetries of the knot is: [-1,-1,0,1,1,-1,0,1,1,0,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['5.89', '5.104', '7.44274']
Arrow polynomial of the knot is: 8*K1**2 - 4*K2 - 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.65', '5.104', '7.6963', '7.7928', '7.7938', '7.10369', '7.11302', '7.11515', '7.18015', '7.18064', '7.18234', '7.18357', '7.18419', '7.18454', '7.18456', '7.18524', '7.18556', '7.18790', '7.18887', '7.18903', '7.18968', '7.19002', '7.19038', '7.19075', '7.19097', '7.19126', '7.19780', '7.20264', '7.20695', '7.20700', '7.20722', '7.20874', '7.20903', '7.20990', '7.21240', '7.21246', '7.21355', '7.21440', '7.21604', '7.22161', '7.22246', '7.22694', '7.22838', '7.22873', '7.22877', '7.23079', '7.23124', '7.23205', '7.23229', '7.23788', '7.23798', '7.24311', '7.24334', '7.24419', '7.24444', '7.24453', '7.24790', '7.25000', '7.25335', '7.25470', '7.25855', '7.25888', '7.25936', '7.26053', '7.26123', '7.26548', '7.26736', '7.27008', '7.27061', '7.27085', '7.27158', '7.27192', '7.27298', '7.27378', '7.27411', '7.27504', '7.27595', '7.27604', '7.27609', '7.27620', '7.27628', '7.27728', '7.27731', '7.27737', '7.27740', '7.27746', '7.27756', '7.27774', '7.27803', '7.27824', '7.27852', '7.27860', '7.27908', '7.27935', '7.27936', '7.28220', '7.28356', '7.28459', '7.28541', '7.28630', '7.28666', '7.28705', '7.28774', '7.28950', '7.28989', '7.29020', '7.29050', '7.29819', '7.30132', '7.30595', '7.30646', '7.30900', '7.30970', '7.31105', '7.31368', '7.31474', '7.31513', '7.31525', '7.31652', '7.31700', '7.31704', '7.31871', '7.31994', '7.32038', '7.32039', '7.32332', '7.32335', '7.32340', '7.32474', '7.32515', '7.32580', '7.32587', '7.32600', '7.32676', '7.32697', '7.32760', '7.32768', '7.32781', '7.32831', '7.32836', '7.32859', '7.32863', '7.32963', '7.33004', '7.33005', '7.33025', '7.33052', '7.33054', '7.33061', '7.33080', '7.33106', '7.33121', '7.33128', '7.33137', '7.33152', '7.33226', '7.33235', '7.33257', '7.33262', '7.33325', '7.33719', '7.33953', '7.34184', '7.34186', '7.34340', '7.34548', '7.34558', '7.34707', '7.34718', '7.34885', '7.34932', '7.35001', '7.35059', '7.35104', '7.35134', '7.35148', '7.35150', '7.35200', '7.35201', '7.35205', '7.35207', '7.35216', '7.35221', '7.35227', '7.35229', '7.35233', '7.35236', '7.35241', '7.35242', '7.35243', '7.35245', '7.35263', '7.35267', '7.35268', '7.35273', '7.35276', '7.35277', '7.35281', '7.35284', '7.35296', '7.35297', '7.35306', '7.35310', '7.35312', '7.35315', '7.35320', '7.35321', '7.35334', '7.35343', '7.35344', '7.35348', '7.35351', '7.35364', '7.35365', '7.35369', '7.35379', '7.35416', '7.35417', '7.35496', '7.35509', '7.35510', '7.35512', '7.35513', '7.35548', '7.35549', '7.35550', '7.35564', '7.35565', '7.35567', '7.35631', '7.35632', '7.35635', '7.35654', '7.35677', '7.35699', '7.35740', '7.35764', '7.35940', '7.35975', '7.35979', '7.35982', '7.36008', '7.36010', '7.36017', '7.36039', '7.36044', '7.36054', '7.36075', '7.36145', '7.36173', '7.36209', '7.36215', '7.36243', '7.36261', '7.36268', '7.36281', '7.36288', '7.36295', '7.36327', '7.36333', '7.36335', '7.36342', '7.36412', '7.36413', '7.36427', '7.36430', '7.36439', '7.36444', '7.36450', '7.36457', '7.36460', '7.36471', '7.36475', '7.36485', '7.36496', '7.36505', '7.36507', '7.36534', '7.36545', '7.36554', '7.36584', '7.36585', '7.36588', '7.36592', '7.36594', '7.36598', '7.36602', '7.36603', '7.36614', '7.36615', '7.36616', '7.36627', '7.36631', '7.36643', '7.36647', '7.36659', '7.36660', '7.36663', '7.36678', '7.36703', '7.36706', '7.36714', '7.36719', '7.36752', '7.36762', '7.36823', '7.36910', '7.36938', '7.36958', '7.36983', '7.37023', '7.37049', '7.37050', '7.37057', '7.37058', '7.37061', '7.37078', '7.37079', '7.37086', '7.37095', '7.37104', '7.37105', '7.37111', '7.37281', '7.37530', '7.37648', '7.37665', '7.37667', '7.37678', '7.37686', '7.37784', '7.37790', '7.37802', '7.37901', '7.37909', '7.37960', '7.38010', '7.38119', '7.38128', '7.38240', '7.38579', '7.38581', '7.38594', '7.38598', '7.38605', '7.38618', '7.38647', '7.38654', '7.38660', '7.38696', '7.38726', '7.38817', '7.38903', '7.38923', '7.38926', '7.38933', '7.38953', '7.39040', '7.39047', '7.39065', '7.39077', '7.39082', '7.39092', '7.39119', '7.39122', '7.39138', '7.39147', '7.39177', '7.39184', '7.39187', '7.39189', '7.39193', '7.39194', '7.39199', '7.39200', '7.39202', '7.39204', '7.39209', '7.39211', '7.39213', '7.39218', '7.39222', '7.39223', '7.39226', '7.39229', '7.39234', '7.39246', '7.39255', '7.39269', '7.39291', '7.39372', '7.39379', '7.39380', '7.39387', '7.39391', '7.39395', '7.39460', '7.39464', '7.39479', '7.39520', '7.39536', '7.39585', '7.39607', '7.39675', '7.39702', '7.39712', '7.39716', '7.39751', '7.39934', '7.39988', '7.40097', '7.40098', '7.40287', '7.40386', '7.40420', '7.40464', '7.40467', '7.40531', '7.40547', '7.40549', '7.40550', '7.40636', '7.40671', '7.40690', '7.40737', '7.40747', '7.40776', '7.40781', '7.40784', '7.40786', '7.40828', '7.40830', '7.40859', '7.40870', '7.40890', '7.40907', '7.40945', '7.40974', '7.40989', '7.41009', '7.41066', '7.41308', '7.41359', '7.41386', '7.41452', '7.41487', '7.41543', '7.41554', '7.41586', '7.41603', '7.41615', '7.41695', '7.41699', '7.41746', '7.41791', '7.41888', '7.41892', '7.41893', '7.41922', '7.41953', '7.41959', '7.41973', '7.41979', '7.42000', '7.42007', '7.42028', '7.42054', '7.42062', '7.42070', '7.42079', '7.42081', '7.42083', '7.42084', '7.42099', '7.42102', '7.42116', '7.42117', '7.42140', '7.42144', '7.42146', '7.42156', '7.42161', '7.42166', '7.42173', '7.42174', '7.42180', '7.42184', '7.42185', '7.42190', '7.42193', '7.42198', '7.42202', '7.42263', '7.42267', '7.42268', '7.42269', '7.42275', '7.42294', '7.42302', '7.42310', '7.42322', '7.42335', '7.42347', '7.42349', '7.42394', '7.42397', '7.42598', '7.42658', '7.42694', '7.42712', '7.42719', '7.42779', '7.42841', '7.42875', '7.42884', '7.42897', '7.42909', '7.42917', '7.42926', '7.42959', '7.42965', '7.42974', '7.42982', '7.43014', '7.43022', '7.43039', '7.43093', '7.43101', '7.43104', '7.43106', '7.43107', '7.43110', '7.43113', '7.43115', '7.43116', '7.43117', '7.43122', '7.43127', '7.43135', '7.43143', '7.43148', '7.43155', '7.43162', '7.43182', '7.43188', '7.43192', '7.43199', '7.43201', '7.43204', '7.43205', '7.43223', '7.43260', '7.43265', '7.43288', '7.43410', '7.43424', '7.43431', '7.43452', '7.43468', '7.43490', '7.43491', '7.43522', '7.43526', '7.43533', '7.43571', '7.43572', '7.43581', '7.43585', '7.43657', '7.43658', '7.43662', '7.43663', '7.43667', '7.43670', '7.43671', '7.43674', '7.43675', '7.43676', '7.43677', '7.43690', '7.43692', '7.43694', '7.43695', '7.43696', '7.43697', '7.43698', '7.43700', '7.43706', '7.43707', '7.43722', '7.43733', '7.43738', '7.43740', '7.43742', '7.43747', '7.43982', '7.44026', '7.44082', '7.44084', '7.44142', '7.44263', '7.44266', '7.44277', '7.44279', '7.44291', '7.44324', '7.44333', '7.44334', '7.44343', '7.44344', '7.44349', '7.44352', '7.44354', '7.44356', '7.44363', '7.44364', '7.44366', '7.44374', '7.44376', '7.44378', '7.44380', '7.44384', '7.44394', '7.44416', '7.44419', '7.44428', '7.44429', '7.44430', '7.44431', '7.44432', '7.44434', '7.44436', '7.44455', '7.44484', '7.44509', '7.44521', '7.44522', '7.44526', '7.44527', '7.44565', '7.44580', '7.44588', '7.44594', '7.44599', '7.44606', '7.44609', '7.44633', '7.44636', '7.44641', '7.44646', '7.44649', '7.44650', '7.44654', '7.44696', '7.44725', '7.44747', '7.44769', '7.44808', '7.44817', '7.44825', '7.44829', '7.44846', '7.44879', '7.44896', '7.44910', '7.44937', '7.44938', '7.44959', '7.44968', '7.44983', '7.44984', '7.44996', '7.44999', '7.45001', '7.45006', '7.45010', '7.45012', '7.45014', '7.45017', '7.45018', '7.45020', '7.45027', '7.45033', '7.45044', '7.45045', '7.45046', '7.45052', '7.45053', '7.45054', '7.45055', '7.45064', '7.45066', '7.45068', '7.45069', '7.45071', '7.45074', '7.45081', '7.45082', '7.45091', '7.45092', '7.45094', '7.45095', '7.45096', '7.45097', '7.45098', '7.45099', '7.45102', '7.45105', '7.45114', '7.45118', '7.45119', '7.45123', '7.45124', '7.45130', '7.45131', '7.45132', '7.45145', '7.45153', '7.45167', '7.45172', '7.45176', '7.45178', '7.45186', '7.45188', '7.45290', '7.45293', '7.45296', '7.45298', '7.45303', '7.45304', '7.45305', '7.45306', '7.45308', '7.45310', '7.45311', '7.45312', '7.45317', '7.45318', '7.45319', '7.45324', '7.45328', '7.45330', '7.45331', '7.45332', '7.45333', '7.45336', '7.45337', '7.45338', '7.45341', '7.45342', '7.45346', '7.45348', '7.45349', '7.45352', '7.45353', '7.45360', '7.45361', '7.45363', '7.45364', '7.45365', '7.45366', '7.45367', '7.45370', '7.45371', '7.45372', '7.45376', '7.45383', '7.45389', '7.45391', '7.45396', '7.45397', '7.45398', '7.45402', '7.45408', '7.45410', '7.45412', '7.45413', '7.45416', '7.45420', '7.45468', '7.45469', '7.45474', '7.45475', '7.45476', '7.45478', '7.45479', '7.45484', '7.45489', '7.45498', '7.45499', '7.45500', '7.45504', '7.45509', '7.45516', '7.45518', '7.45522', '7.45523', '7.45527', '7.45536', '7.45537', '7.45539', '7.45546', '7.45547', '7.45551', '7.45560', '7.45562', '7.45568', '7.45575', '7.45579', '7.45582', '7.45583', '7.45585', '7.45588', '7.45592', '7.45607', '7.45608', '7.45613', '7.45614', '7.45615', '7.45621', '7.45622', '7.45628', '7.45629', '7.45632', '7.45643', '7.45645', '7.45647', '7.45651', '7.45654', '7.45657', '7.45661', '7.45662', '7.45664', '7.45665', '7.45669', '7.45671', '7.45673', '7.45677', '7.45678', '7.45679', '7.45680', '7.45688', '7.45689', '7.45690', '7.45692', '7.45695', '7.45696', '7.45700', '7.45701', '7.45702', '7.45703', '7.45708', '7.45712', '7.45714', '7.45715', '7.45716', '7.45718', '7.45719', '7.45721', '7.45726', '7.45727', '7.45729', '7.45730', '7.45733', '7.45734', '7.45736', '7.45742', '7.45743', '7.45757', '7.45759', '7.45763', '7.45765', '7.45769', '7.45770', '7.45771', '7.45774', '7.45776', '7.45777', '7.45781', '7.45782', '7.45787', '7.45788', '7.45802', '7.45804', '7.45807', '7.45808', '7.45809', '7.45811', '7.45813', '7.45814', '7.45815', '7.45816', '7.45819', '7.45820', '7.45821', '7.45822', '7.45828', '7.45839', '7.45843', '7.45847', '7.45848', '7.45851', '7.45852', '7.45853', '7.45861', '7.45864', '7.45868', '7.45875', '7.45876', '7.45882', '7.45885', '7.45888', '7.45894', '7.45942', '7.45973', '7.45979', '7.45983', '7.45984', '7.46003', '7.46008', '7.46013', '7.46014', '7.46027', '7.46044', '7.46079', '7.46089', '7.46091', '7.46109', '7.46131', '7.46141', '7.46153', '7.46158', '7.46161', '7.46174', '7.46177', '7.46208', '7.46216', '7.46220']
Outer characteristic polynomial of the knot is: t^6+12t^4+11t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.24', '5.89', '5.104', '6.971', '7.9322', '7.22878', '7.39897', '7.42720', '7.44274']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 448*K1**4*K2 - 896*K1**4 + 64*K1**3*K2*K3 + 320*K1**2*K2**3 - 1312*K1**2*K2**2 + 1744*K1**2*K2 - 128*K1**2*K3**2 - 936*K1**2 + 1024*K1*K2*K3 + 96*K1*K3*K4 - 192*K2**4 + 64*K2**2*K4 - 696*K2**2 - 296*K3**2 - 40*K4**2 + 862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.104']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.441', 'vk5.467', 'vk5.688', 'vk5.842', 'vk5.970', 'vk5.1200', 'vk5.1503', 'vk5.1540', 'vk5.1819', 'vk5.1888']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3U4U3O5U2O4U1U5
R3 orbit {'O1O2O3U4U3O5U2O4U1U5'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2U3O4U1O5O3U5U4U2
Gauss code of -K* O1O2U3O4U1O5O3U5U4U2
Diagrammatic symmetry type r
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 0 1 -1 1],[ 1 0 1 1 -1 1],[ 0 -1 0 0 -1 0],[-1 -1 0 0 -1 -1],[ 1 1 1 1 0 1],[-1 -1 0 1 -1 0]]
Primitive based matrix [[ 0 1 1 0 -1 -1],[-1 0 1 0 -1 -1],[-1 -1 0 0 -1 -1],[ 0 0 0 0 -1 -1],[ 1 1 1 1 0 1],[ 1 1 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,1,1,-1,0,1,1,0,1,1,1,1,-1]
Phi over symmetry [-1,-1,0,1,1,-1,0,1,1,0,1,1,1,1,-1]
Phi of -K [-1,-1,0,1,1,-1,0,1,1,0,1,1,1,1,-1]
Phi of K* [-1,-1,0,1,1,-1,1,1,1,1,1,1,0,0,-1]
Phi of -K* [-1,-1,0,1,1,-1,1,1,1,1,1,1,0,0,-1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial -9z-17
Enhanced Jones-Krushkal polynomial 2w^3z-11w^2z-17w
Inner characteristic polynomial t^5+8t^3+3t
Outer characteristic polynomial t^6+12t^4+11t^2
Flat arrow polynomial 8*K1**2 - 4*K2 - 3
2-strand cable arrow polynomial -128*K1**4*K2**2 + 448*K1**4*K2 - 896*K1**4 + 64*K1**3*K2*K3 + 320*K1**2*K2**3 - 1312*K1**2*K2**2 + 1744*K1**2*K2 - 128*K1**2*K3**2 - 936*K1**2 + 1024*K1*K2*K3 + 96*K1*K3*K4 - 192*K2**4 + 64*K2**2*K4 - 696*K2**2 - 296*K3**2 - 40*K4**2 + 862
Genus of based matrix 1
Fillings of based matrix [[{1, 5}, {3, 4}, {2}], [{2, 5}, {1, 4}, {3}], [{3, 5}, {2, 4}, {1}], [{4, 5}, {1, 3}, {2}]]
If K is slice False
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