| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,1,2,2,-1,-1,0,0,1,-1,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1600'] |
| 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+29t^5+61t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1600'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 704*K1**4 - 64*K1**3*K3 - 320*K1**2*K2**2 - 64*K1**2*K2*K4 + 2840*K1**2*K2 - 2320*K1**2 - 64*K1*K2**2*K3 + 984*K1*K2*K3 + 64*K1*K3*K4 - 120*K2**4 + 200*K2**2*K4 - 1560*K2**2 - 384*K3**2 - 70*K4**2 + 1548 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1600'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71492', 'vk6.71495', 'vk6.71550', 'vk6.71554', 'vk6.72022', 'vk6.72023', 'vk6.72073', 'vk6.72075', 'vk6.72530', 'vk6.72535', 'vk6.72638', 'vk6.72660', 'vk6.72922', 'vk6.72961', 'vk6.73110', 'vk6.73134', 'vk6.73647', 'vk6.73683', 'vk6.73692', 'vk6.77115', 'vk6.77118', 'vk6.77170', 'vk6.77172', 'vk6.77461', 'vk6.77465', 'vk6.77946', 'vk6.77964', 'vk6.78587', 'vk6.81429', 'vk6.86911', 'vk6.87246', 'vk6.89345'] |
| 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 | O1O2O3U1O4U3U5O6U4O5U2U6 |
| R3 orbit | {'O1O2O3U1O4U3U5O6U4O5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2O5U6O4U5U1O6U3 |
| Gauss code of K* | O1O2U3O4U2O5O3U6U5U1O6U4 |
| Gauss code of -K* | O1O2U3O4U1O3O5U4O6U5U2U6 |
| 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 0 0 1 0 1],[ 2 0 2 1 1 2 1],[ 0 -2 0 -1 2 -1 1],[ 0 -1 1 0 1 -1 1],[-1 -1 -2 -1 0 -1 0],[ 0 -2 1 1 1 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 0 -1 -1 -1 -1],[-1 0 0 -1 -1 -2 -1],[ 0 1 1 0 1 1 -2],[ 0 1 1 -1 0 1 -1],[ 0 1 2 -1 -1 0 -2],[ 2 1 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,0,1,1,1,1,1,1,2,1,-1,-1,2,-1,1,2] |
| Phi over symmetry | [-2,0,0,0,1,1,0,0,1,2,2,-1,-1,0,0,1,-1,0,0,0,0] |
| Phi of -K | [-2,0,0,0,1,1,0,0,1,2,2,-1,-1,0,0,1,-1,0,0,0,0] |
| Phi of K* | [-1,-1,0,0,0,2,0,-1,0,0,2,0,0,0,2,-1,-1,0,-1,1,0] |
| Phi of -K* | [-2,0,0,0,1,1,1,2,2,1,1,-1,1,1,1,1,1,1,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+23t^4+36t^2+4 |
| Outer characteristic polynomial | t^7+29t^5+61t^3+8t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 704*K1**4 - 64*K1**3*K3 - 320*K1**2*K2**2 - 64*K1**2*K2*K4 + 2840*K1**2*K2 - 2320*K1**2 - 64*K1*K2**2*K3 + 984*K1*K2*K3 + 64*K1*K3*K4 - 120*K2**4 + 200*K2**2*K4 - 1560*K2**2 - 384*K3**2 - 70*K4**2 + 1548 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |