| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,2,3,2,1,1,2,1,1,0,1,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1317'] |
| 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+41t^5+137t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1317'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 224*K1**4 - 32*K1**3*K3 - 1008*K1**2*K2**2 + 2240*K1**2*K2 - 1708*K1**2 + 880*K1*K2*K3 + 8*K1*K3*K4 - 72*K2**4 + 64*K2**2*K4 - 984*K2**2 - 188*K3**2 - 18*K4**2 + 1008 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1317'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71455', 'vk6.71508', 'vk6.71520', 'vk6.71977', 'vk6.71995', 'vk6.72034', 'vk6.72050', 'vk6.73223', 'vk6.73231', 'vk6.73256', 'vk6.73264', 'vk6.73657', 'vk6.73673', 'vk6.75150', 'vk6.75168', 'vk6.77084', 'vk6.77134', 'vk6.77144', 'vk6.77423', 'vk6.77439', 'vk6.78085', 'vk6.78091', 'vk6.78121', 'vk6.78127', 'vk6.81294', 'vk6.81538', 'vk6.81548', 'vk6.84653', 'vk6.84963', 'vk6.84976', 'vk6.85483', 'vk6.85486', 'vk6.86179', 'vk6.86195', 'vk6.86427', 'vk6.86884', 'vk6.87726', 'vk6.88126', 'vk6.89518', 'vk6.90036'] |
| 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 | O1O2O3U4O5O6U2U1O4U6U3U5 |
| R3 orbit | {'O1O2O3U4O5O6U2U1O4U6U3U5', 'O1O2O3U4O5U1O4U6U5U2O6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U1U5O6U3U2O5O4U6 |
| Gauss code of K* | O1O2O3U4U5U2O6U3U1O5O4U6 |
| Gauss code of -K* | O1O2O3U4O5O6U3U1O4U2U6U5 |
| 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 -2 1 0 2 1],[ 2 0 0 2 1 3 1],[ 2 0 0 1 2 2 0],[-1 -2 -1 0 0 0 -1],[ 0 -1 -2 0 0 1 1],[-2 -3 -2 0 -1 0 0],[-1 -1 0 1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 0 0 -1 -2 -3],[-1 0 0 1 -1 0 -1],[-1 0 -1 0 0 -1 -2],[ 0 1 1 0 0 -2 -1],[ 2 2 0 1 2 0 0],[ 2 3 1 2 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,0,0,1,2,3,-1,1,0,1,0,1,2,2,1,0] |
| Phi over symmetry | [-2,-2,0,1,1,2,0,0,2,3,2,1,1,2,1,1,0,1,1,1,1] |
| Phi of -K | [-2,-2,0,1,1,2,0,0,2,3,2,1,1,2,1,1,0,1,1,1,1] |
| Phi of K* | [-2,-1,-1,0,2,2,1,1,1,1,2,-1,1,1,2,0,2,3,1,0,0] |
| Phi of -K* | [-2,-2,0,1,1,2,0,1,1,2,3,2,0,1,2,1,0,1,1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+27t^4+80t^2+4 |
| Outer characteristic polynomial | t^7+41t^5+137t^3+7t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 224*K1**4 - 32*K1**3*K3 - 1008*K1**2*K2**2 + 2240*K1**2*K2 - 1708*K1**2 + 880*K1*K2*K3 + 8*K1*K3*K4 - 72*K2**4 + 64*K2**2*K4 - 984*K2**2 - 188*K3**2 - 18*K4**2 + 1008 |
| 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 |