| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,2,3,0,0,0,2,0,-1,0,-1,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1416'] |
| 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+36t^5+31t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1329', '6.1416'] |
| 2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 1472*K1**4 - 128*K1**3*K3 - 240*K1**2*K2**2 + 2576*K1**2*K2 - 1124*K1**2 + 368*K1*K2*K3 - 8*K2**4 + 8*K2**2*K4 - 952*K2**2 - 108*K3**2 - 2*K4**2 + 952 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1416'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11002', 'vk6.11083', 'vk6.12172', 'vk6.12281', 'vk6.18199', 'vk6.18536', 'vk6.24659', 'vk6.25085', 'vk6.30571', 'vk6.30668', 'vk6.31845', 'vk6.31894', 'vk6.36793', 'vk6.37247', 'vk6.44036', 'vk6.44378', 'vk6.51813', 'vk6.51882', 'vk6.52681', 'vk6.52777', 'vk6.56005', 'vk6.56280', 'vk6.60546', 'vk6.60888', 'vk6.63497', 'vk6.63543', 'vk6.63979', 'vk6.64025', 'vk6.65670', 'vk6.65956', 'vk6.68718', 'vk6.68928', 'vk6.83179', 'vk6.83603', 'vk6.84127', 'vk6.84346', 'vk6.86493', 'vk6.86497', 'vk6.88746', 'vk6.88904'] |
| 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 | O1O2O3U1O4U2O5U3U4O6U5U6 |
| R3 orbit | {'O1O2O3U1O4U2O5U3U4O6U5U6', 'O1O2O3U4U1O5U2O6U3U5O4U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U5O4U6U1O5U2O6U3 |
| Gauss code of K* | O1O2U3O4O3U5U6U1O5U2O6U4 |
| Gauss code of -K* | O1O2U1O3O4U2O5U3O6U4U5U6 |
| 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 0 1 1 1],[ 2 0 1 2 2 1 0],[ 1 -1 0 1 2 2 0],[ 0 -2 -1 0 1 2 1],[-1 -2 -2 -1 0 1 1],[-1 -1 -2 -2 -1 0 1],[-1 0 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 -1 -2 -2],[-1 -1 0 1 -2 -2 -1],[-1 -1 -1 0 -1 0 0],[ 0 1 2 1 0 -1 -2],[ 1 2 2 0 1 0 -1],[ 2 2 1 0 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,1,2,2,-1,2,2,1,1,0,0,1,2,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,2,3,0,0,0,2,0,-1,0,-1,-1,-1] |
| Phi of -K | [-2,-1,0,1,1,1,0,0,1,2,3,0,0,0,2,0,-1,0,-1,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,0,2,3,-1,-1,0,2,0,0,1,0,0,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,2,0,1,2,1,0,2,2,1,2,1,-1,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+28t^4+12t^2+1 |
| Outer characteristic polynomial | t^7+36t^5+31t^3+4t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | 128*K1**4*K2 - 1472*K1**4 - 128*K1**3*K3 - 240*K1**2*K2**2 + 2576*K1**2*K2 - 1124*K1**2 + 368*K1*K2*K3 - 8*K2**4 + 8*K2**2*K4 - 952*K2**2 - 108*K3**2 - 2*K4**2 + 952 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |