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Flat knot 5.57

Min(phi) over symmetries of the knot is: [-3,-1,1,1,2,1,1,3,2,1,2,1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['5.57']
Arrow polynomial of the knot is: -4*K1**3 + 2*K1**2 + 2*K1*K2 + 2*K1 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.14', '5.26', '5.28', '5.29', '5.33', '5.38', '5.40', '5.57', '7.375', '7.379', '7.875', '7.1597', '7.1811', '7.2286', '7.2287', '7.2292', '7.2293', '7.2663', '7.3321', '7.3333', '7.3357', '7.4202', '7.4861', '7.4867', '7.4873', '7.4962', '7.4968', '7.4971', '7.4975', '7.4987', '7.5376', '7.5647', '7.6531', '7.6738', '7.6742', '7.6759', '7.6858', '7.7003', '7.7179', '7.7642', '7.7773', '7.8497', '7.8593', '7.8600', '7.8613', '7.8682', '7.8717', '7.8741', '7.8807', '7.8898', '7.8922', '7.9058', '7.9687', '7.9690', '7.9691', '7.9906', '7.10051', '7.10633', '7.11007', '7.11014', '7.11018', '7.11699', '7.11742', '7.11815', '7.11953', '7.11973', '7.11991', '7.12027', '7.12305', '7.12306', '7.12533', '7.12734', '7.12750', '7.12752', '7.12835', '7.12838', '7.12916', '7.13505', '7.13613', '7.13753', '7.13769', '7.13816', '7.13933', '7.13937', '7.14089', '7.14433', '7.15094', '7.15095', '7.15100', '7.15144', '7.15157', '7.15642', '7.15700', '7.15707', '7.15708', '7.15860', '7.15862', '7.15879', '7.15894', '7.15897', '7.15960', '7.16116', '7.16191', '7.16271', '7.16290', '7.16292', '7.16400', '7.16495', '7.16668', '7.16698', '7.16781', '7.16830', '7.16980', '7.16998', '7.17022', '7.17039', '7.17158', '7.17182', '7.17185', '7.17214', '7.17310', '7.17337', '7.17377', '7.17694', '7.18094', '7.18130', '7.18132', '7.18172', '7.18173', '7.18184', '7.18253', '7.18254', '7.18255', '7.18272', '7.18369', '7.18377', '7.18381', '7.18392', '7.18398', '7.18481', '7.18493', '7.18498', '7.18544', '7.18584', '7.18589', '7.18641', '7.18661', '7.18662', '7.18711', '7.18712', '7.18717', '7.18736', '7.18816', '7.18894', '7.18906', '7.18907', '7.18918', '7.18929', '7.18991', '7.19023', '7.19024', '7.19035', '7.19113', '7.19114', '7.19294', '7.19348', '7.19362', '7.19435', '7.19472', '7.19566', '7.19571', '7.19585', '7.19607', '7.19613', '7.19681', '7.19683', '7.19687', '7.19691', '7.19717', '7.19720', '7.19846', '7.19870', '7.19952', '7.20130', '7.20219', '7.20227', '7.20516', '7.20530', '7.20537', '7.20539', '7.20542', '7.20548', '7.20550', '7.20566', '7.20572', '7.20605', '7.20635', '7.20685', '7.20738', '7.20788', '7.20869', '7.20871', '7.20886', '7.20906', '7.20953', '7.20959', '7.20968', '7.20977', '7.21003', '7.21020', '7.21077', '7.21097', '7.21111', '7.21116', '7.21170', '7.21174', '7.21266', '7.21295', '7.21305', '7.21307', '7.21322', '7.21348', '7.21381', '7.21403', '7.21494', '7.21569', '7.21595', '7.21601', '7.21620', '7.21644', '7.21723', '7.21724', '7.21756', '7.21760', '7.21765', '7.21794', '7.21795', '7.21803', '7.21810', '7.21812', '7.21828', '7.21835', '7.21916', '7.22029', '7.22034', '7.22042', '7.22050', '7.22079', '7.22210', '7.22260', '7.22270', '7.22275', '7.22314', '7.22415', '7.22446', '7.22477', '7.22490', '7.22499', '7.22854', '7.22921', '7.22936', '7.22937', '7.22938', '7.22948', '7.22955', '7.22971', '7.22989', '7.23263', '7.23264', '7.23275', '7.23306', '7.23337', '7.23569', '7.23570', '7.23595', '7.23692', '7.23698', '7.23727', '7.23730', '7.23944', '7.23976', '7.24171', '7.25160', '7.25330', '7.25350', '7.25351', '7.25356', '7.25415', '7.25422', '7.25451', '7.25479', '7.26037', '7.26260', '7.26859', '7.27334', '7.27966', '7.27997', '7.28077', '7.29013', '7.29095', '7.29249', '7.29538', '7.29673', '7.29984', '7.30209', '7.30257', '7.31037', '7.31226', '7.31503', '7.31512', '7.31524', '7.31533', '7.31546', '7.31562', '7.31563', '7.31566', '7.31571', '7.31613', '7.31646', '7.31651', '7.31690', '7.31787', '7.31916', '7.32021', '7.32024', '7.32026', '7.32066', '7.32125', '7.32142', '7.32144', '7.32161', '7.32312', '7.32376', '7.32399', '7.32407', '7.32834', '7.32876', '7.32962', '7.33223', '7.33314', '7.33323', '7.33403', '7.33461', '7.33494', '7.33737', '7.33741', '7.33742', '7.33757', '7.33910', '7.34799', '7.34857']
Outer characteristic polynomial of the knot is: t^6+44t^4+80t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.57']
2-strand cable arrow polynomial of the knot is: 288*K1**2*K2**3 - 1248*K1**2*K2**2 - 32*K1**2*K2*K4 + 1168*K1**2*K2 - 880*K1**2 + 192*K1*K2**3*K3 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1072*K1*K2*K3 + 16*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 552*K2**4 - 160*K2**2*K3**2 - 72*K2**2*K4**2 + 472*K2**2*K4 - 456*K2**2 + 48*K2*K3*K5 + 24*K2*K4*K6 - 240*K3**2 - 98*K4**2 + 648
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.57']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.429', 'vk5.457', 'vk5.635', 'vk5.777', 'vk5.922', 'vk5.960', 'vk5.1145', 'vk5.1298', 'vk5.1489', 'vk5.1528', 'vk5.1646', 'vk5.1764', 'vk5.1798', 'vk5.1813', 'vk5.1866', 'vk5.1932']
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 O1O2O3O4U5U2O5U1U3U4
R3 orbit {'O1O2O3O4U5U2O5U1U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U4O5U3U5
Gauss code of K* O1O2O3U1U4U2U3O5O4U5
Gauss code of -K* O1O2O3U4O5O4U1U2U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -1 1 3 -1],[ 2 0 1 2 3 1],[ 1 -1 0 0 1 1],[-1 -2 0 0 1 -1],[-3 -3 -1 -1 0 -3],[ 1 -1 -1 1 3 0]]
Primitive based matrix [[ 0 3 1 -1 -1 -2],[-3 0 -1 -1 -3 -3],[-1 1 0 0 -1 -2],[ 1 1 0 0 1 -1],[ 1 3 1 -1 0 -1],[ 2 3 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,1,1,2,1,1,3,3,0,1,2,-1,1,1]
Phi over symmetry [-3,-1,1,1,2,1,1,3,2,1,2,1,-1,0,0]
Phi of -K [-2,-1,-1,1,3,0,0,1,2,-1,2,3,1,1,1]
Phi of K* [-3,-1,1,1,2,1,1,3,2,1,2,1,-1,0,0]
Phi of -K* [-2,-1,-1,1,3,1,1,2,3,-1,1,3,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial -4z^2-15z-13
Enhanced Jones-Krushkal polynomial -4w^3z^2-15w^2z-13w
Inner characteristic polynomial t^5+28t^3+43t
Outer characteristic polynomial t^6+44t^4+80t^2+1
Flat arrow polynomial -4*K1**3 + 2*K1**2 + 2*K1*K2 + 2*K1 - K2
2-strand cable arrow polynomial 288*K1**2*K2**3 - 1248*K1**2*K2**2 - 32*K1**2*K2*K4 + 1168*K1**2*K2 - 880*K1**2 + 192*K1*K2**3*K3 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1072*K1*K2*K3 + 16*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 552*K2**4 - 160*K2**2*K3**2 - 72*K2**2*K4**2 + 472*K2**2*K4 - 456*K2**2 + 48*K2*K3*K5 + 24*K2*K4*K6 - 240*K3**2 - 98*K4**2 + 648
Genus of based matrix 1
Fillings of based matrix [[{3, 5}, {1, 4}, {2}], [{3, 5}, {2, 4}, {1}], [{3, 5}, {4}, {1, 2}]]
If K is slice False
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