| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,3,0,0,1,2,2,1,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1051'] |
| 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+34t^5+115t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1051'] |
| 2-strand cable arrow polynomial of the knot is: -1488*K1**4 - 64*K1**3*K3 - 416*K1**2*K2**2 + 2744*K1**2*K2 - 272*K1**2*K3**2 - 1576*K1**2 + 1304*K1*K2*K3 + 328*K1*K3*K4 - 8*K2**4 + 16*K2**2*K4 - 1336*K2**2 - 560*K3**2 - 102*K4**2 + 1428 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1051'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4407', 'vk6.4504', 'vk6.4848', 'vk6.5193', 'vk6.5793', 'vk6.5922', 'vk6.6409', 'vk6.6421', 'vk6.6842', 'vk6.7967', 'vk6.8374', 'vk6.8378', 'vk6.8805', 'vk6.9280', 'vk6.9401', 'vk6.9743', 'vk6.17888', 'vk6.17953', 'vk6.18283', 'vk6.18620', 'vk6.24395', 'vk6.25175', 'vk6.30035', 'vk6.30098', 'vk6.36901', 'vk6.37361', 'vk6.39835', 'vk6.39845', 'vk6.43822', 'vk6.44122', 'vk6.44447', 'vk6.46398', 'vk6.46403', 'vk6.47976', 'vk6.47982', 'vk6.48616', 'vk6.49075', 'vk6.49912', 'vk6.50604', 'vk6.51126'] |
| 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 | O1O2O3O4U5U6O5U2O6U4U1U3 |
| R3 orbit | {'O1O2O3O4U5U6O5U2O6U4U1U3', 'O1O2O3U4U5O4U1O5U6U2O6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U2U4U1O5U3O6U5U6 |
| Gauss code of K* | O1O2O3U2U4U3U1O5O6U5O4U6 |
| Gauss code of -K* | O1O2O3U4O5U6O4O6U3U1U5U2 |
| 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 -1 -1 2 1 -1 0],[ 1 0 0 2 1 0 1],[ 1 0 0 1 0 1 2],[-2 -2 -1 0 0 -3 -1],[-1 -1 0 0 0 -2 0],[ 1 0 -1 3 2 0 0],[ 0 -1 -2 1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -2 -3],[-1 0 0 0 0 -1 -2],[ 0 1 0 0 -2 -1 0],[ 1 1 0 2 0 0 1],[ 1 2 1 1 0 0 0],[ 1 3 2 0 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,1,1,2,3,0,0,1,2,2,1,0,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,2,3,0,0,1,2,2,1,0,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,-1,2,2,0,1,0,0,0,1,1,1,1,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,1,0,1,2,1,0,1,2,1,0,-1,0,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,2,3,0,2,0,1,1,1,2,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+26t^4+84t^2+1 |
| Outer characteristic polynomial | t^7+34t^5+115t^3+4t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | -1488*K1**4 - 64*K1**3*K3 - 416*K1**2*K2**2 + 2744*K1**2*K2 - 272*K1**2*K3**2 - 1576*K1**2 + 1304*K1*K2*K3 + 328*K1*K3*K4 - 8*K2**4 + 16*K2**2*K4 - 1336*K2**2 - 560*K3**2 - 102*K4**2 + 1428 |
| 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}], [{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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |