| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,2,0,0,1,2,1,0,-1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1758'] |
| 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+153t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1758'] |
| 2-strand cable arrow polynomial of the knot is: 1728*K1**2*K2**3 - 4864*K1**2*K2**2 - 96*K1**2*K2*K4 + 4608*K1**2*K2 - 3200*K1**2 - 1088*K1*K2**2*K3 + 3504*K1*K2*K3 + 112*K1*K3*K4 - 1592*K2**4 + 1128*K2**2*K4 - 1432*K2**2 - 704*K3**2 - 142*K4**2 + 2036 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1758'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16361', 'vk6.16404', 'vk6.18117', 'vk6.18453', 'vk6.22691', 'vk6.22792', 'vk6.24570', 'vk6.24987', 'vk6.34660', 'vk6.34747', 'vk6.36711', 'vk6.37131', 'vk6.42321', 'vk6.42368', 'vk6.43983', 'vk6.44296', 'vk6.54622', 'vk6.54649', 'vk6.55933', 'vk6.56227', 'vk6.59102', 'vk6.59164', 'vk6.60467', 'vk6.60828', 'vk6.64647', 'vk6.64695', 'vk6.65590', 'vk6.65896', 'vk6.68000', 'vk6.68026', 'vk6.68663', 'vk6.68873'] |
| 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 | O1O2O3U4U5O4O6U2U1O5U6U3 |
| R3 orbit | {'O1O2O3U4U5O4O6U2U1O5U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U3U2O4O6U5U6 |
| Gauss code of K* | O1O2U3O4O5U2U1U5O6O3U6U4 |
| Gauss code of -K* | O1O2U3O4O5U2U6O3O6U1U5U4 |
| 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 0 1],[ 1 0 0 2 1 1 1],[ 1 0 0 1 1 2 0],[-2 -2 -1 0 -3 0 -1],[ 1 -1 -1 3 0 0 2],[ 0 -1 -2 0 0 0 1],[-1 -1 0 1 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 0 -1 -2 -3],[-1 1 0 -1 0 -1 -2],[ 0 0 1 0 -2 -1 0],[ 1 1 0 2 0 0 1],[ 1 2 1 1 0 0 1],[ 1 3 2 0 -1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,1,0,1,2,3,1,0,1,2,2,1,0,0,-1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,0,1,2,0,0,1,2,1,0,-1,-1,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,-1,2,2,1,1,0,0,0,1,1,0,2,0] |
| Phi of K* | [-2,-1,0,1,1,1,0,2,0,1,2,0,0,1,2,1,0,-1,-1,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,-1,0,2,3,0,1,1,2,2,0,1,1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 8z^2+25z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+10w^3z^2-2w^3z+27w^2z+19w |
| Inner characteristic polynomial | t^6+28t^4+112t^2 |
| Outer characteristic polynomial | t^7+36t^5+153t^3+8t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | 1728*K1**2*K2**3 - 4864*K1**2*K2**2 - 96*K1**2*K2*K4 + 4608*K1**2*K2 - 3200*K1**2 - 1088*K1*K2**2*K3 + 3504*K1*K2*K3 + 112*K1*K3*K4 - 1592*K2**4 + 1128*K2**2*K4 - 1432*K2**2 - 704*K3**2 - 142*K4**2 + 2036 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |