Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 4.3

Min(phi) over symmetries of the knot is: [-3,1,1,1,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['4.3']
Arrow polynomial of the knot is: -2K1K2 + K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354']
Outer characteristic polynomial of the knot is: t^5+26t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.3']
2-strand cable arrow polynomial of the knot is: -72K12 + 96K1K2K3 + 48K1K3K4 - 8K2*2K4*2 + 40K2*2K4 - 86K22 + 8K2K4K6 - 72K32 - 44K4*2 - 2K6**2 + 90
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.3']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.26', 'vk4.28', 'vk4.45', 'vk4.47', 'vk4.80', 'vk4.81', 'vk6.10444', 'vk6.10447', 'vk6.10452', 'vk6.10455', 'vk6.10460', 'vk6.10463', 'vk6.10468', 'vk6.10471', 'vk6.10580', 'vk6.10583', 'vk6.10596', 'vk6.10599', 'vk6.10612', 'vk6.10615', 'vk6.10628', 'vk6.10631', 'vk6.10672', 'vk6.10675', 'vk6.10688', 'vk6.10691', 'vk6.10768', 'vk6.10771', 'vk6.10784', 'vk6.10787', 'vk6.10800', 'vk6.10803', 'vk6.10816', 'vk6.10819', 'vk6.10860', 'vk6.10863', 'vk6.10876', 'vk6.10879', 'vk6.11579', 'vk6.11581', 'vk6.11586', 'vk6.11588', 'vk6.11822', 'vk6.11825', 'vk6.11830', 'vk6.11833', 'vk6.11838', 'vk6.11841', 'vk6.11846', 'vk6.11849', 'vk6.11883', 'vk6.11893', 'vk6.11920', 'vk6.11923', 'vk6.11928', 'vk6.11931', 'vk6.11933', 'vk6.11936', 'vk6.11942', 'vk6.11945', 'vk6.12062', 'vk6.12065', 'vk6.12070', 'vk6.12073', 'vk6.12838', 'vk6.12841', 'vk6.12846', 'vk6.12849', 'vk6.12854', 'vk6.12857', 'vk6.12862', 'vk6.12865', 'vk6.12898', 'vk6.12908', 'vk6.12923', 'vk6.12925', 'vk6.12929', 'vk6.12935', 'vk6.13054', 'vk6.13057', 'vk6.13062', 'vk6.13065', 'vk6.13234', 'vk6.13236', 'vk6.13243', 'vk6.13245', 'vk6.14630', 'vk6.14634', 'vk6.14637', 'vk6.14641', 'vk6.16249', 'vk6.16251', 'vk6.16254', 'vk6.16256', 'vk6.16925', 'vk6.18118', 'vk6.19008', 'vk6.19010', 'vk6.19029', 'vk6.19031', 'vk6.19162', 'vk6.19164', 'vk6.19167', 'vk6.19169', 'vk6.20196', 'vk6.20198', 'vk6.20231', 'vk6.20504', 'vk6.20505', 'vk6.20506', 'vk6.20507', 'vk6.21069', 'vk6.21070', 'vk6.21077', 'vk6.21078', 'vk6.21089', 'vk6.21090', 'vk6.21091', 'vk6.21092', 'vk6.21219', 'vk6.21222', 'vk6.21225', 'vk6.21231', 'vk6.21234', 'vk6.21239', 'vk6.21242', 'vk6.21262', 'vk6.21268', 'vk6.21311', 'vk6.21314', 'vk6.21325', 'vk6.21328', 'vk6.21413', 'vk6.21416', 'vk6.21421', 'vk6.21424', 'vk6.21472', 'vk6.21474', 'vk6.21875', 'vk6.21876', 'vk6.21879', 'vk6.21880', 'vk6.22496', 'vk6.22497', 'vk6.22506', 'vk6.22507', 'vk6.22520', 'vk6.22521', 'vk6.22522', 'vk6.22523', 'vk6.22687', 'vk6.23315', 'vk6.24571', 'vk6.24768', 'vk6.24784', 'vk6.25225', 'vk6.25241', 'vk6.25528', 'vk6.25530', 'vk6.25547', 'vk6.25549', 'vk6.25627', 'vk6.25628', 'vk6.25644', 'vk6.25646', 'vk6.25728', 'vk6.25730', 'vk6.25731', 'vk6.25733', 'vk6.25884', 'vk6.25890', 'vk6.25900', 'vk6.25906', 'vk6.26960', 'vk6.26963', 'vk6.26968', 'vk6.26971', 'vk6.26976', 'vk6.26979', 'vk6.26984', 'vk6.26987', 'vk6.27026', 'vk6.27029', 'vk6.27032', 'vk6.27040', 'vk6.27043', 'vk6.27046', 'vk6.27089', 'vk6.27092', 'vk6.27103', 'vk6.27106', 'vk6.27180', 'vk6.27186', 'vk6.27195', 'vk6.27201', 'vk6.27222', 'vk6.27225', 'vk6.27230', 'vk6.27233', 'vk6.27241', 'vk6.27249', 'vk6.27259', 'vk6.27265', 'vk6.27275', 'vk6.27281', 'vk6.27307', 'vk6.27313', 'vk6.27323', 'vk6.27329', 'vk6.27348', 'vk6.27350', 'vk6.27353', 'vk6.27356', 'vk6.27358', 'vk6.27361', 'vk6.27439', 'vk6.27920', 'vk6.27923', 'vk6.27926', 'vk6.27929', 'vk6.28192', 'vk6.28204', 'vk6.28509', 'vk6.28510', 'vk6.28515', 'vk6.28523', 'vk6.28529', 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'vk6.88494', 'vk6.88496', 'vk6.88499', 'vk6.88503', 'vk6.88511', 'vk6.88513', 'vk6.88515', 'vk6.88516', 'vk6.88535', 'vk6.88539', 'vk6.88555', 'vk6.88629', 'vk6.88637', 'vk6.88649', 'vk6.88681', 'vk6.88684', 'vk6.88687', 'vk6.88690', 'vk6.89725', 'vk6.89726', 'vk6.89867', 'vk6.89869']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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

Min(phi) over symmetries of the knot is: [-3,1,1,1,1,2,3,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['4.3']

Arrow polynomial of the knot is: -2*K1*K2 + K1 + K3 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354']

Outer characteristic polynomial of the knot is: t^5+26t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.3']

2-strand cable arrow polynomial of the knot is: -72*K1**2 + 96*K1*K2*K3 + 48*K1*K3*K4 - 8*K2**2*K4**2 + 40*K2**2*K4 - 86*K2**2 + 8*K2*K4*K6 - 72*K3**2 - 44*K4**2 - 2*K6**2 + 90

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.3']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.26', 'vk4.28', 'vk4.45', 'vk4.47', 'vk4.80', 'vk4.81', 'vk6.10444', 'vk6.10447', 'vk6.10452', 'vk6.10455', 'vk6.10460', 'vk6.10463', 'vk6.10468', 'vk6.10471', 'vk6.10580', 'vk6.10583', 'vk6.10596', 'vk6.10599', 'vk6.10612', 'vk6.10615', 'vk6.10628', 'vk6.10631', 'vk6.10672', 'vk6.10675', 'vk6.10688', 'vk6.10691', 'vk6.10768', 'vk6.10771', 'vk6.10784', 'vk6.10787', 'vk6.10800', 'vk6.10803', 'vk6.10816', 'vk6.10819', 'vk6.10860', 'vk6.10863', 'vk6.10876', 'vk6.10879', 'vk6.11579', 'vk6.11581', 'vk6.11586', 'vk6.11588', 'vk6.11822', 'vk6.11825', 'vk6.11830', 'vk6.11833', 'vk6.11838', 'vk6.11841', 'vk6.11846', 'vk6.11849', 'vk6.11883', 'vk6.11893', 'vk6.11920', 'vk6.11923', 'vk6.11928', 'vk6.11931', 'vk6.11933', 'vk6.11936', 'vk6.11942', 'vk6.11945', 'vk6.12062', 'vk6.12065', 'vk6.12070', 'vk6.12073', 'vk6.12838', 'vk6.12841', 'vk6.12846', 'vk6.12849', 'vk6.12854', 'vk6.12857', 'vk6.12862', 'vk6.12865', 'vk6.12898', 'vk6.12908', 'vk6.12923', 'vk6.12925', 'vk6.12929', 'vk6.12935', 'vk6.13054', 'vk6.13057', 'vk6.13062', 'vk6.13065', 'vk6.13234', 'vk6.13236', 'vk6.13243', 'vk6.13245', 'vk6.14630', 'vk6.14634', 'vk6.14637', 'vk6.14641', 'vk6.16249', 'vk6.16251', 'vk6.16254', 'vk6.16256', 'vk6.16925', 'vk6.18118', 'vk6.19008', 'vk6.19010', 'vk6.19029', 'vk6.19031', 'vk6.19162', 'vk6.19164', 'vk6.19167', 'vk6.19169', 'vk6.20196', 'vk6.20198', 'vk6.20231', 'vk6.20504', 'vk6.20505', 'vk6.20506', 'vk6.20507', 'vk6.21069', 'vk6.21070', 'vk6.21077', 'vk6.21078', 'vk6.21089', 'vk6.21090', 'vk6.21091', 'vk6.21092', 'vk6.21219', 'vk6.21222', 'vk6.21225', 'vk6.21231', 'vk6.21234', 'vk6.21239', 'vk6.21242', 'vk6.21262', 'vk6.21268', 'vk6.21311', 'vk6.21314', 'vk6.21325', 'vk6.21328', 'vk6.21413', 'vk6.21416', 'vk6.21421', 'vk6.21424', 'vk6.21472', 'vk6.21474', 'vk6.21875', 'vk6.21876', 'vk6.21879', 'vk6.21880', 'vk6.22496', 'vk6.22497', 'vk6.22506', 'vk6.22507', 'vk6.22520', 'vk6.22521', 'vk6.22522', 'vk6.22523', 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'vk6.56770', 'vk6.56776', 'vk6.56778', 'vk6.56784', 'vk6.56960', 'vk6.56968', 'vk6.56987', 'vk6.56998', 'vk6.57001', 'vk6.57012', 'vk6.57022', 'vk6.57028', 'vk6.57030', 'vk6.57036', 'vk6.57106', 'vk6.57112', 'vk6.57114', 'vk6.57120', 'vk6.57369', 'vk6.57370', 'vk6.57371', 'vk6.57372', 'vk6.57494', 'vk6.57500', 'vk6.57504', 'vk6.57761', 'vk6.57771', 'vk6.57775', 'vk6.57787', 'vk6.57867', 'vk6.57870', 'vk6.57880', 'vk6.57888', 'vk6.57890', 'vk6.57893', 'vk6.57900', 'vk6.57904', 'vk6.57906', 'vk6.57909', 'vk6.57916', 'vk6.57933', 'vk6.57937', 'vk6.57974', 'vk6.57977', 'vk6.57982', 'vk6.57985', 'vk6.58098', 'vk6.58101', 'vk6.58106', 'vk6.58109', 'vk6.58116', 'vk6.58121', 'vk6.58282', 'vk6.58294', 'vk6.58298', 'vk6.58310', 'vk6.58672', 'vk6.58682', 'vk6.58683', 'vk6.58693', 'vk6.59468', 'vk6.60344', 'vk6.60355', 'vk6.60358', 'vk6.60369', 'vk6.60468', 'vk6.60958', 'vk6.60974', 'vk6.61077', 'vk6.61089', 'vk6.61091', 'vk6.61103', 'vk6.61220', 'vk6.61223', 'vk6.61236', 'vk6.61239', 'vk6.61248', 'vk6.61250', 'vk6.61253', 'vk6.61258', 'vk6.61262', 'vk6.61264', 'vk6.61267', 'vk6.61272', 'vk6.61307', 'vk6.61310', 'vk6.61313', 'vk6.61314', 'vk6.61317', 'vk6.61320', 'vk6.61357', 'vk6.61360', 'vk6.61365', 'vk6.61368', 'vk6.61462', 'vk6.61468', 'vk6.61470', 'vk6.61476', 'vk6.61513', 'vk6.61516', 'vk6.61521', 'vk6.61524', 'vk6.61687', 'vk6.61699', 'vk6.61702', 'vk6.61714', 'vk6.62177', 'vk6.62185', 'vk6.62189', 'vk6.62196', 'vk6.62280', 'vk6.62286', 'vk6.62288', 'vk6.62294', 'vk6.62342', 'vk6.62348', 'vk6.62349', 'vk6.62355', 'vk6.62451', 'vk6.62457', 'vk6.62459', 'vk6.62465', 'vk6.62488', 'vk6.62492', 'vk6.62609', 'vk6.62615', 'vk6.62616', 'vk6.62622', 'vk6.62647', 'vk6.62652', 'vk6.62658', 'vk6.62660', 'vk6.62663', 'vk6.62666', 'vk6.62667', 'vk6.62673', 'vk6.62679', 'vk6.62685', 'vk6.62691', 'vk6.62697', 'vk6.62699', 'vk6.62705', 'vk6.62707', 'vk6.62711', 'vk6.62842', 'vk6.62848', 'vk6.62850', 'vk6.62856', 'vk6.63131', 'vk6.63137', 'vk6.63139', 'vk6.63144', 'vk6.63217', 'vk6.63222', 'vk6.63407', 'vk6.63410', 'vk6.63415', 'vk6.63418', 'vk6.63903', 'vk6.63906', 'vk6.63911', 'vk6.63914', 'vk6.63919', 'vk6.63922', 'vk6.63927', 'vk6.63930', 'vk6.63960', 'vk6.63965', 'vk6.64218', 'vk6.64221', 'vk6.64226', 'vk6.64229', 'vk6.64235', 'vk6.64238', 'vk6.64241', 'vk6.64270', 'vk6.64275', 'vk6.64915', 'vk6.65588', 'vk6.65750', 'vk6.65764', 'vk6.66516', 'vk6.66520', 'vk6.66626', 'vk6.66632', 'vk6.66634', 'vk6.66640', 'vk6.67080', 'vk6.67086', 'vk6.67247', 'vk6.67250', 'vk6.67254', 'vk6.67258', 'vk6.67261', 'vk6.67266', 'vk6.67269', 'vk6.67295', 'vk6.67298', 'vk6.67301', 'vk6.67302', 'vk6.67305', 'vk6.67308', 'vk6.67341', 'vk6.67344', 'vk6.67348', 'vk6.67351', 'vk6.67411', 'vk6.67416', 'vk6.67418', 'vk6.67423', 'vk6.67441', 'vk6.67444', 'vk6.67445', 'vk6.67448', 'vk6.67449', 'vk6.67457', 'vk6.67944', 'vk6.67949', 'vk6.68216', 'vk6.68661', 'vk6.69111', 'vk6.69114', 'vk6.69118', 'vk6.69121', 'vk6.69125', 'vk6.69128', 'vk6.69133', 'vk6.69136', 'vk6.69157', 'vk6.69161', 'vk6.69197', 'vk6.69200', 'vk6.69204', 'vk6.69207', 'vk6.69291', 'vk6.69294', 'vk6.69295', 'vk6.69298', 'vk6.81991', 'vk6.81993', 'vk6.82139', 'vk6.82140', 'vk6.82142', 'vk6.82143', 'vk6.82145', 'vk6.82146', 'vk6.82718', 'vk6.82720', 'vk6.82723', 'vk6.82849', 'vk6.82852', 'vk6.82971', 'vk6.82973', 'vk6.83290', 'vk6.83295', 'vk6.83391', 'vk6.83393', 'vk6.83395', 'vk6.83446', 'vk6.83452', 'vk6.83461', 'vk6.83467', 'vk6.83469', 'vk6.83471', 'vk6.83901', 'vk6.83903', 'vk6.83904', 'vk6.83905', 'vk6.83907', 'vk6.83908', 'vk6.84517', 'vk6.84523', 'vk6.84547', 'vk6.84551', 'vk6.84673', 'vk6.84679', 'vk6.84759', 'vk6.84765', 'vk6.84768', 'vk6.84777', 'vk6.84795', 'vk6.84806', 'vk6.84836', 'vk6.84838', 'vk6.84839', 'vk6.84841', 'vk6.84842', 'vk6.84843', 'vk6.84845', 'vk6.84848', 'vk6.86225', 'vk6.86227', 'vk6.86228', 'vk6.86236', 'vk6.86238', 'vk6.86242', 'vk6.86244', 'vk6.86248', 'vk6.86250', 'vk6.86268', 'vk6.86274', 'vk6.86278', 'vk6.86842', 'vk6.86845', 'vk6.88462', 'vk6.88468', 'vk6.88472', 'vk6.88486', 'vk6.88494', 'vk6.88496', 'vk6.88499', 'vk6.88503', 'vk6.88511', 'vk6.88513', 'vk6.88515', 'vk6.88516', 'vk6.88535', 'vk6.88539', 'vk6.88555', 'vk6.88629', 'vk6.88637', 'vk6.88649', 'vk6.88681', 'vk6.88684', 'vk6.88687', 'vk6.88690', 'vk6.89725', 'vk6.89726', 'vk6.89867', 'vk6.89869']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3O4U1U4U3U2
R3 orbit	{'O1O2O3O4U1U4U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U2U1U4
Gauss code of K*	Same
Gauss code of -K*	O1O2O3O4U3U2U1U4
Diagrammatic symmetry type	+
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -3 1 1 1],[ 3 0 3 2 1],[-1 -3 0 0 0],[-1 -2 0 0 0],[-1 -1 0 0 0]]
Primitive based matrix	[[ 0 1 1 1 -3],[-1 0 0 0 -1],[-1 0 0 0 -2],[-1 0 0 0 -3],[ 3 1 2 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,3,0,0,1,0,2,3]
Phi over symmetry	[-3,1,1,1,1,2,3,0,0,0]
Phi of -K	[-3,1,1,1,1,2,3,0,0,0]
Phi of K*	[-1,-1,-1,3,0,0,1,0,2,3]
Phi of -K*	[-3,1,1,1,1,2,3,0,0,0]
Symmetry type of based matrix	+
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	z+3
Enhanced Jones-Krushkal polynomial	-4w^3z+5w^2z+3w
Inner characteristic polynomial	t^4+14t^2
Outer characteristic polynomial	t^5+26t^3+6t
Flat arrow polynomial	-2K1K2 + K1 + K3 + 1
2-strand cable arrow polynomial	-72K12 + 96K1K2K3 + 48K1K3K4 - 8K2*2K4*2 + 40K2*2K4 - 86K22 + 8K2K4K6 - 72K32 - 44K4*2 - 2K6**2 + 90
Genus of based matrix	1
Fillings of based matrix	[[{1, 4}, {2, 3}], [{2, 4}, {1, 3}], [{2, 4}, {3}, {1}], [{3, 4}, {1, 2}]]
If K is slice	False

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