| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,1,1,1,0,1,1,1,0,0,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1713'] |
| 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+30t^5+54t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1713'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 224*K1**4*K2 - 384*K1**4 + 416*K1**3*K2*K3 - 256*K1**3*K3 + 2368*K1**2*K2**3 - 5120*K1**2*K2**2 - 896*K1**2*K2*K4 + 4152*K1**2*K2 - 256*K1**2*K3**2 - 2904*K1**2 - 672*K1*K2**2*K3 + 3896*K1*K2*K3 + 520*K1*K3*K4 - 1592*K2**4 + 1080*K2**2*K4 - 1192*K2**2 - 832*K3**2 - 278*K4**2 + 1980 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1713'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73977', 'vk6.73988', 'vk6.74494', 'vk6.74509', 'vk6.75952', 'vk6.75973', 'vk6.76710', 'vk6.76724', 'vk6.78944', 'vk6.78965', 'vk6.79488', 'vk6.79511', 'vk6.80472', 'vk6.80491', 'vk6.80950', 'vk6.80963', 'vk6.83015', 'vk6.83099', 'vk6.83657', 'vk6.83790', 'vk6.83943', 'vk6.84117', 'vk6.84266', 'vk6.85187', 'vk6.85540', 'vk6.85871', 'vk6.86259', 'vk6.86591', 'vk6.86748', 'vk6.87445', 'vk6.88311', 'vk6.89744'] |
| 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 | O1O2O3U1U4O5O6U2U3O4U6U5 |
| R3 orbit | {'O1O2O3U1U4O5O6U2U3O4U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O6U1U2O5O4U6U3 |
| Gauss code of K* | O1O2U3O4O5U6U1U2O6O3U5U4 |
| Gauss code of -K* | O1O2U3O4O5U2U1O3O6U4U5U6 |
| 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 -1 1 0 1 1],[ 2 0 1 2 0 2 2],[ 1 -1 0 1 0 2 1],[-1 -2 -1 0 -1 1 0],[ 0 0 0 1 0 0 1],[-1 -2 -2 -1 0 0 0],[-1 -2 -1 0 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -2],[-1 -1 0 0 0 -2 -2],[-1 0 0 0 -1 -1 -2],[ 0 1 0 1 0 0 0],[ 1 1 2 1 0 0 -1],[ 2 2 2 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,2,2,1,1,2,0,0,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,1,1,1,0,1,1,1,0,0,0,1,0] |
| Phi of -K | [-2,-1,0,1,1,1,0,2,1,1,1,1,0,1,1,1,0,0,0,1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,0,1,0,0,1,1,0,1,1,1,2,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,0,2,2,2,0,1,1,2,1,1,0,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 7z^2+24z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+9w^3z^2-2w^3z+26w^2z+21w |
| Inner characteristic polynomial | t^6+22t^4+29t^2+1 |
| Outer characteristic polynomial | t^7+30t^5+54t^3+9t |
| Flat arrow polynomial | -2*K1**2 + K2 + 2 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 224*K1**4*K2 - 384*K1**4 + 416*K1**3*K2*K3 - 256*K1**3*K3 + 2368*K1**2*K2**3 - 5120*K1**2*K2**2 - 896*K1**2*K2*K4 + 4152*K1**2*K2 - 256*K1**2*K3**2 - 2904*K1**2 - 672*K1*K2**2*K3 + 3896*K1*K2*K3 + 520*K1*K3*K4 - 1592*K2**4 + 1080*K2**2*K4 - 1192*K2**2 - 832*K3**2 - 278*K4**2 + 1980 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |