| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,2,0,2,-1,-1,1,-1,0,0,1,0,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1562'] |
| 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+31t^5+150t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1562'] |
| 2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 128*K1**4 - 64*K1**3*K3 + 128*K1**2*K2**3 - 1024*K1**2*K2**2 + 1880*K1**2*K2 - 1560*K1**2 - 192*K1*K2**2*K3 + 1160*K1*K2*K3 + 16*K1*K3*K4 - 120*K2**4 + 136*K2**2*K4 - 960*K2**2 - 312*K3**2 - 14*K4**2 + 956 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1562'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4223', 'vk6.4304', 'vk6.5488', 'vk6.5602', 'vk6.7591', 'vk6.7686', 'vk6.9092', 'vk6.9173', 'vk6.18376', 'vk6.18716', 'vk6.24829', 'vk6.25288', 'vk6.37021', 'vk6.37471', 'vk6.44190', 'vk6.44511', 'vk6.48543', 'vk6.48600', 'vk6.49244', 'vk6.49360', 'vk6.50334', 'vk6.50393', 'vk6.51073', 'vk6.51106', 'vk6.56157', 'vk6.56386', 'vk6.60686', 'vk6.61039', 'vk6.65829', 'vk6.66083', 'vk6.68818', 'vk6.69028', 'vk6.83740', 'vk6.83866', 'vk6.85055', 'vk6.85351', 'vk6.86670', 'vk6.86961', 'vk6.87407', 'vk6.89533'] |
| 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 | O1O2O3U4O5U6U1O4O6U3U2U5 |
| R3 orbit | {'O1O2O3U4O5U6U1O4O6U3U2U5', 'O1O2O3U4U5O4U6O5U2U1O6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U2U1O5O6U3U5O4U6 |
| Gauss code of K* | O1O2O3U4U2U1O5U3O6O4U5U6 |
| Gauss code of -K* | O1O2O3U4U5O6O4U1O5U3U2U6 |
| 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 0 0 -1 2 0],[ 1 0 0 -1 1 1 2],[ 0 0 0 0 -1 1 1],[ 0 1 0 0 -1 0 1],[ 1 -1 1 1 0 3 0],[-2 -1 -1 0 -3 0 -2],[ 0 -2 -1 -1 0 2 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 0 -1 -2 -1 -3],[ 0 0 0 0 1 1 -1],[ 0 1 0 0 1 0 -1],[ 0 2 -1 -1 0 -2 0],[ 1 1 -1 0 2 0 1],[ 1 3 1 1 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,0,1,2,1,3,0,-1,-1,1,-1,0,1,2,0,-1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,2,0,2,-1,-1,1,-1,0,0,1,0,2,-1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,-1,1,2,2,1,0,0,0,1,1,0,0,1,2] |
| Phi of K* | [-2,0,0,0,1,1,0,1,2,0,2,-1,-1,1,-1,0,0,1,0,2,-1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,0,1,1,3,2,-1,0,1,-1,-1,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+25t^4+111t^2+4 |
| Outer characteristic polynomial | t^7+31t^5+150t^3+7t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | 96*K1**4*K2 - 128*K1**4 - 64*K1**3*K3 + 128*K1**2*K2**3 - 1024*K1**2*K2**2 + 1880*K1**2*K2 - 1560*K1**2 - 192*K1*K2**2*K3 + 1160*K1*K2*K3 + 16*K1*K3*K4 - 120*K2**4 + 136*K2**2*K4 - 960*K2**2 - 312*K3**2 - 14*K4**2 + 956 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{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}, {4, 5}, {3}, {2}, {1}], [{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 |