| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,1,0,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1436'] |
| Arrow polynomial of the knot is: -2*K1**2 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.6', '4.8', '6.780', '6.804', '6.914', '6.931', '6.946', '6.960', '6.1002', '6.1016', '6.1019', '6.1051', '6.1058', '6.1078', '6.1102', '6.1115', '6.1217', '6.1294', '6.1306', '6.1317', '6.1321', '6.1324', '6.1336', '6.1377', '6.1416', '6.1420', '6.1427', '6.1429', '6.1434', '6.1436', '6.1437', '6.1439', '6.1441', '6.1444', '6.1450', '6.1451', '6.1458', '6.1459', '6.1477', '6.1482', '6.1490', '6.1503', '6.1504', '6.1511', '6.1521', '6.1547', '6.1560', '6.1561', '6.1562', '6.1597', '6.1598', '6.1600', '6.1601', '6.1608', '6.1620', '6.1622', '6.1624', '6.1634', '6.1635', '6.1637', '6.1638', '6.1713', '6.1725', '6.1758', '6.1846', '6.1933', '6.1944', '6.1949', '6.1950', '6.1951'] |
| Outer characteristic polynomial of the knot is: t^7+26t^5+49t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1436'] |
| 2-strand cable arrow polynomial of the knot is: -1216*K1**4*K2**2 + 2400*K1**4*K2 - 5264*K1**4 + 800*K1**3*K2*K3 - 288*K1**3*K3 + 1312*K1**2*K2**3 - 7504*K1**2*K2**2 - 256*K1**2*K2*K4 + 8216*K1**2*K2 - 80*K1**2*K3**2 - 1332*K1**2 - 768*K1*K2**2*K3 + 4456*K1*K2*K3 + 136*K1*K3*K4 - 408*K2**4 + 352*K2**2*K4 - 2112*K2**2 - 612*K3**2 - 54*K4**2 + 2220 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1436'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19946', 'vk6.20046', 'vk6.21191', 'vk6.21332', 'vk6.26915', 'vk6.27107', 'vk6.28669', 'vk6.28800', 'vk6.38339', 'vk6.38496', 'vk6.40479', 'vk6.40701', 'vk6.45208', 'vk6.45392', 'vk6.47031', 'vk6.47144', 'vk6.56742', 'vk6.56854', 'vk6.57843', 'vk6.57997', 'vk6.61179', 'vk6.61377', 'vk6.62417', 'vk6.62544', 'vk6.66446', 'vk6.66567', 'vk6.67216', 'vk6.67362', 'vk6.69094', 'vk6.69215', 'vk6.69875', 'vk6.69960'] |
| 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 | O1O2O3U4O5U1O4U2U3O6U5U6 |
| R3 orbit | {'O1O2O3U4O5U1O4U2U3O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U1U2O6U3O5U6 |
| Gauss code of K* | O1O2U3O4O3U5U1U2O6U4O5U6 |
| Gauss code of -K* | O1O2U1O3O4U5O6U2O5U3U4U6 |
| 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 0 1 1],[ 2 0 0 1 2 2 1],[ 1 0 0 1 1 1 1],[-1 -1 -1 0 -1 0 1],[ 0 -2 -1 1 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 0 -1 -1],[-1 0 1 0 -1 -1 -2],[ 0 1 0 1 0 -1 -2],[ 1 1 1 1 1 0 0],[ 2 1 1 2 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,1,1,0,1,1,1,1,2,1,2,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,1,0,1,1,-1,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,0,1,2,2,0,1,1,1,0,0,1,0,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,1,1,2,0,0,1,1,0,1,2,0,0,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,1,0,1,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+18t^4+32t^2+4 |
| Outer characteristic polynomial | t^7+26t^5+49t^3+7t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | -1216*K1**4*K2**2 + 2400*K1**4*K2 - 5264*K1**4 + 800*K1**3*K2*K3 - 288*K1**3*K3 + 1312*K1**2*K2**3 - 7504*K1**2*K2**2 - 256*K1**2*K2*K4 + 8216*K1**2*K2 - 80*K1**2*K3**2 - 1332*K1**2 - 768*K1*K2**2*K3 + 4456*K1*K2*K3 + 136*K1*K3*K4 - 408*K2**4 + 352*K2**2*K4 - 2112*K2**2 - 612*K3**2 - 54*K4**2 + 2220 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{6}, {4, 5}, {1, 3}, {2}]] |
| If K is slice | False |