| Min(phi) over symmetries of the knot is: [-3,-1,1,1,2,1,1,3,2,1,2,2,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['5.27'] |
| Arrow polynomial of the knot is: 6*K1**2 + 2*K1*K2 - K1 - 3*K2 - K3 - 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['5.25', '5.27', '5.31', '5.36', '5.41', '7.3240', '7.3361', '7.5140', '7.5153', '7.5264', '7.6535', '7.6651', '7.6900', '7.7155', '7.7183', '7.7436', '7.7444', '7.7505', '7.7565', '7.7630', '7.7690', '7.7786', '7.7859', '7.7885', '7.8418', '7.8806', '7.8907', '7.9068', '7.9940', '7.10163', '7.10634', '7.10651', '7.10950', '7.11008', '7.11019', '7.11114', '7.11306', '7.11487', '7.11702', '7.11831', '7.11845', '7.11848', '7.11875', '7.11954', '7.11963', '7.11994', '7.12036', '7.12054', '7.12094', '7.12138', '7.12148', '7.12164', '7.12212', '7.12237', '7.12368', '7.12393', '7.12469', '7.12860', '7.12924', '7.12930', '7.12938', '7.12939', '7.12942', '7.13009', '7.13010', '7.13615', '7.15148', '7.15857', '7.15970', '7.15975', '7.16033', '7.16042', '7.16251', '7.16253', '7.16256', '7.16291', '7.16297', '7.16303', '7.16685', '7.16707', '7.16733', '7.16785', '7.16826', '7.16914', '7.16934', '7.16946', '7.16947', '7.16970', '7.16986', '7.16992', '7.17129', '7.17164', '7.17220', '7.17340', '7.17362', '7.17363', '7.17756', '7.17763', '7.17771', '7.17892', '7.18066', '7.18069', '7.18074', '7.18096', '7.18104', '7.18170', '7.18259', '7.18267', '7.18281', '7.18347', '7.18356', '7.18453', '7.18505', '7.18527', '7.18545', '7.18550', '7.18612', '7.18616', '7.18635', '7.18637', '7.18638', '7.18672', '7.18673', '7.18725', '7.18758', '7.18801', '7.18809', '7.18820', '7.18821', '7.18834', '7.18855', '7.18900', '7.18901', '7.19063', '7.19067', '7.19129', '7.19130', '7.19251', '7.19390', '7.19560', '7.19837', '7.19946', '7.20089', '7.20090', '7.20229', '7.20523', '7.20529', '7.20536', '7.20544', '7.20551', '7.20553', '7.20598', '7.20606', '7.20615', '7.20701', '7.20752', '7.20803', '7.20865', '7.20898', '7.20971', '7.20973', '7.21019', '7.21023', '7.21031', '7.21037', '7.21047', '7.21060', '7.21062', '7.21079', '7.21082', '7.21083', '7.21099', '7.21113', '7.21114', '7.21148', '7.21204', '7.21285', '7.21315', '7.21320', '7.21331', '7.21342', '7.21349', '7.21378', '7.21382', '7.21392', '7.21399', '7.21401', '7.21402', '7.21413', '7.21432', '7.21450', '7.21464', '7.21481', '7.21493', '7.21586', '7.21635', '7.21643', '7.21664', '7.21667', '7.21682', '7.21691', '7.21710', '7.21731', '7.21758', '7.21799', '7.21800', '7.21829', '7.21873', '7.21896', '7.22006', '7.22424', '7.22549', '7.22581', '7.22613', '7.22685', '7.22722', '7.22941', '7.22946', '7.22956', '7.22959', '7.22986', '7.23024', '7.23027', '7.23051', '7.23094', '7.23141', '7.23143', '7.23209', '7.23235', '7.23279', '7.23301', '7.23355', '7.23565', '7.23753', '7.23853', '7.24333', '7.24349', '7.24404', '7.25199', '7.25201', '7.25531', '7.25536', '7.25581', '7.25648', '7.25812', '7.26842', '7.26873', '7.27234', '7.27315', '7.27468', '7.27479', '7.27670', '7.28098', '7.29083', '7.31009', '7.31211', '7.31505', '7.31514', '7.31521', '7.31539', '7.31544', '7.31611', '7.31616', '7.31636', '7.31639', '7.31642', '7.31643', '7.31665', '7.31683', '7.31748', '7.31784', '7.31820', '7.31838', '7.31841', '7.31848', '7.31853', '7.31906', '7.31914', '7.31922', '7.31932', '7.31933', '7.31940', '7.31945', '7.31951', '7.31982', '7.31992', '7.31993', '7.32000', '7.32008', '7.32015', '7.32017', '7.32031', '7.32033', '7.32042', '7.32044', '7.32174', '7.32313', '7.32330', '7.32381', '7.32405', '7.32829', '7.32888', '7.32937', '7.32993', '7.33208', '7.33225', '7.33369', '7.33484', '7.33506', '7.33585', '7.33778', '7.33985', '7.34029', '7.34795', '7.34796', '7.34853', '7.34882', '7.34895', '7.35021', '7.35099', '7.35107', '7.35188'] |
| Outer characteristic polynomial of the knot is: t^6+42t^4+16t^2+1 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.27', '7.20797'] |
| 2-strand cable arrow polynomial of the knot is: 256*K1**4*K2 - 864*K1**4 + 64*K1**3*K2*K3 - 256*K1**3*K3 + 160*K1**2*K2**3 - 880*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 2168*K1**2*K2 - 160*K1**2*K3**2 - 1328*K1**2 - 256*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 1352*K1*K2*K3 + 168*K1*K3*K4 - 152*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 192*K2**2*K4 - 1030*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 428*K3**2 - 74*K4**2 - 12*K5**2 - 2*K6**2 + 1064 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.27'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.435', 'vk5.444', 'vk5.461', 'vk5.471', 'vk5.689', 'vk5.694', 'vk5.841', 'vk5.848', 'vk5.964', 'vk5.972', 'vk5.1201', 'vk5.1497', 'vk5.1500', 'vk5.1537', 'vk5.1817', 'vk5.1889'] |
| 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 | O1O2O3O4U1O5U3U5U4U2 |
| R3 orbit | {'O1O2O3O4U1O5U3U5U4U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U1U5U2O5U4 |
| Gauss code of K* | O1O2O3O4U5U4U1U3O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5U2U4U1U5 |
| 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 -3 1 -1 2 1],[ 3 0 3 1 2 1],[-1 -3 0 -2 1 1],[ 1 -1 2 0 2 1],[-2 -2 -1 -2 0 0],[-1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -3],[-2 0 0 -1 -2 -2],[-1 0 0 -1 -1 -1],[-1 1 1 0 -2 -3],[ 1 2 1 2 0 -1],[ 3 2 1 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,3,0,1,2,2,1,1,1,2,3,1] |
| Phi over symmetry | [-3,-1,1,1,2,1,1,3,2,1,2,2,-1,0,1] |
| Phi of -K | [-3,-1,1,1,2,1,1,3,3,0,1,1,-1,0,1] |
| Phi of K* | [-2,-1,-1,1,3,0,1,1,3,1,0,1,1,3,1] |
| Phi of -K* | [-3,-1,1,1,2,1,1,3,2,1,2,2,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | -z^2-12z-19 |
| Enhanced Jones-Krushkal polynomial | -w^3z^2-12w^2z-19w |
| Inner characteristic polynomial | t^5+26t^3+3t |
| Outer characteristic polynomial | t^6+42t^4+16t^2+1 |
| Flat arrow polynomial | 6*K1**2 + 2*K1*K2 - K1 - 3*K2 - K3 - 2 |
| 2-strand cable arrow polynomial | 256*K1**4*K2 - 864*K1**4 + 64*K1**3*K2*K3 - 256*K1**3*K3 + 160*K1**2*K2**3 - 880*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 2168*K1**2*K2 - 160*K1**2*K3**2 - 1328*K1**2 - 256*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 1352*K1*K2*K3 + 168*K1*K3*K4 - 152*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 192*K2**2*K4 - 1030*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 428*K3**2 - 74*K4**2 - 12*K5**2 - 2*K6**2 + 1064 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 5}, {2, 4}, {3}], [{1, 5}, {3, 4}, {2}], [{1, 5}, {4}, {2, 3}], [{1, 5}, {4}, {3}, {2}], [{2, 5}, {1, 4}, {3}], [{2, 5}, {3, 4}, {1}], [{2, 5}, {4}, {1, 3}], [{2, 5}, {4}, {3}, {1}], [{3, 5}, {1, 4}, {2}], [{3, 5}, {2, 4}, {1}], [{3, 5}, {4}, {1, 2}], [{3, 5}, {4}, {2}, {1}], [{4, 5}, {1, 3}, {2}], [{4, 5}, {2, 3}, {1}], [{4, 5}, {3}, {1, 2}], [{4, 5}, {3}, {2}, {1}], [{5}, {1, 4}, {2, 3}], [{5}, {1, 4}, {3}, {2}], [{5}, {2, 4}, {1, 3}], [{5}, {2, 4}, {3}, {1}], [{5}, {3, 4}, {1, 2}], [{5}, {3, 4}, {2}, {1}], [{5}, {4}, {1, 3}, {2}], [{5}, {4}, {2, 3}, {1}], [{5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |