| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,1,1,1,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1027'] |
| Arrow polynomial of the knot is: -10*K1**2 + 5*K2 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.947', '6.1027', '6.1399', '6.1430', '6.1433', '6.1442', '6.1465', '6.1469', '6.1476', '6.1505', '6.1529', '6.1606', '6.1612', '6.1613', '6.1616', '6.1649', '6.1694', '6.1736', '6.1768', '6.1771', '6.1774', '6.1884', '6.1886', '6.1887', '6.1889', '6.1960', '6.1962'] |
| Outer characteristic polynomial of the knot is: t^7+24t^5+25t^3+2t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1027'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**6 - 192*K1**4*K2**2 + 4256*K1**4*K2 - 8880*K1**4 + 128*K1**3*K2*K3 - 608*K1**3*K3 + 1856*K1**2*K2**3 - 11408*K1**2*K2**2 - 384*K1**2*K2*K4 + 12168*K1**2*K2 - 16*K1**2*K3**2 - 1328*K1**2 - 896*K1*K2**2*K3 + 6248*K1*K2*K3 + 96*K1*K3*K4 - 1656*K2**4 + 1128*K2**2*K4 - 2560*K2**2 - 720*K3**2 - 110*K4**2 + 3196 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1027'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13892', 'vk6.13987', 'vk6.14145', 'vk6.14368', 'vk6.14967', 'vk6.15088', 'vk6.15601', 'vk6.16071', 'vk6.16288', 'vk6.16311', 'vk6.17420', 'vk6.22603', 'vk6.22634', 'vk6.23928', 'vk6.33703', 'vk6.33778', 'vk6.34148', 'vk6.34261', 'vk6.34583', 'vk6.36205', 'vk6.36232', 'vk6.42279', 'vk6.53866', 'vk6.53907', 'vk6.54102', 'vk6.54411', 'vk6.54573', 'vk6.55565', 'vk6.59026', 'vk6.59045', 'vk6.60054', 'vk6.64558'] |
| 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 | O1O2O3O4U3U5O6U1O5U4U6U2 |
| R3 orbit | {'O1O2O3O4U3U5O6U1O5U4U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U5U1O6U4O5U6U2 |
| Gauss code of K* | O1O2O3U4U3U5U1O5O6U2O4U6 |
| Gauss code of -K* | O1O2O3U4O5U2O4O6U3U6U1U5 |
| 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 1 0 1],[ 2 0 2 0 1 2 1],[-1 -2 0 -1 0 0 0],[ 1 0 1 0 1 1 1],[-1 -1 0 -1 0 -1 0],[ 0 -2 0 -1 1 0 1],[-1 -1 0 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -1 -2],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -1],[ 0 0 1 1 0 -1 -2],[ 1 1 1 1 1 0 0],[ 2 2 1 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,0,1,2,0,1,1,1,1,1,1,1,2,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,1,1,1,0,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,0,1,2,2,0,1,1,1,1,0,0,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,1,2,0,0,1,2,1,1,1,0,0,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,1,1,1,0,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+16t^4+14t^2 |
| Outer characteristic polynomial | t^7+24t^5+25t^3+2t |
| Flat arrow polynomial | -10*K1**2 + 5*K2 + 6 |
| 2-strand cable arrow polynomial | -320*K1**6 - 192*K1**4*K2**2 + 4256*K1**4*K2 - 8880*K1**4 + 128*K1**3*K2*K3 - 608*K1**3*K3 + 1856*K1**2*K2**3 - 11408*K1**2*K2**2 - 384*K1**2*K2*K4 + 12168*K1**2*K2 - 16*K1**2*K3**2 - 1328*K1**2 - 896*K1*K2**2*K3 + 6248*K1*K2*K3 + 96*K1*K3*K4 - 1656*K2**4 + 1128*K2**2*K4 - 2560*K2**2 - 720*K3**2 - 110*K4**2 + 3196 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {3, 5}, {4}, {1, 2}]] |
| If K is slice | False |