| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,1,2,-1,0,1,1,0,0,0,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1797'] |
| Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.313', '6.623', '6.1031', '6.1201', '6.1327', '6.1378', '6.1640', '6.1697', '6.1797', '6.1833'] |
| Outer characteristic polynomial of the knot is: t^7+18t^5+36t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1797'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 128*K1**4 - 192*K1**2*K2**4 + 640*K1**2*K2**3 - 5216*K1**2*K2**2 - 128*K1**2*K2*K4 + 6096*K1**2*K2 - 4460*K1**2 + 352*K1*K2**3*K3 - 608*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4680*K1*K2*K3 + 112*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 1432*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 1312*K2**2*K4 - 2624*K2**2 + 8*K2*K3*K5 - 1044*K3**2 - 278*K4**2 + 3004 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1797'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11066', 'vk6.11146', 'vk6.12228', 'vk6.12337', 'vk6.18335', 'vk6.18673', 'vk6.24770', 'vk6.25227', 'vk6.30645', 'vk6.30740', 'vk6.31873', 'vk6.31944', 'vk6.36960', 'vk6.37419', 'vk6.44147', 'vk6.44469', 'vk6.51857', 'vk6.51902', 'vk6.52718', 'vk6.52823', 'vk6.56121', 'vk6.56345', 'vk6.60638', 'vk6.60976', 'vk6.63516', 'vk6.63562', 'vk6.63994', 'vk6.64040', 'vk6.65776', 'vk6.66036', 'vk6.68779', 'vk6.68988'] |
| 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 | O1O2O3U4U2O5O6U5U1O4U6U3 |
| R3 orbit | {'O1O2O3U4U2O5O6U5U1O4U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U3U6O4O6U2U5 |
| Gauss code of K* | O1O2U3O4O5U2U6U5O3O6U1U4 |
| Gauss code of -K* | O1O2U3O4O5U2U5O6O3U1U6U4 |
| 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 0 2 -1 -1 1],[ 1 0 0 2 0 0 1],[ 0 0 0 0 0 0 -1],[-2 -2 0 0 -1 -1 0],[ 1 0 0 1 0 -1 0],[ 1 0 0 1 1 0 1],[-1 -1 1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 0 -1 -1 -2],[-1 0 0 1 0 -1 -1],[ 0 0 -1 0 0 0 0],[ 1 1 0 0 0 -1 0],[ 1 1 1 0 1 0 0],[ 1 2 1 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,0,1,1,2,-1,0,1,1,0,0,0,1,0,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,1,2,-1,0,1,1,0,0,0,1,0,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,2,2,1,1,1,2,2,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,2,1,2,2,2,1,1,2,1,1,1,0,0,1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,0,1,0,0,1,1,0,1,2,-1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2-4w^3z+27w^2z+23w |
| Inner characteristic polynomial | t^6+10t^4+15t^2+4 |
| Outer characteristic polynomial | t^7+18t^5+36t^3+12t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 128*K1**4 - 192*K1**2*K2**4 + 640*K1**2*K2**3 - 5216*K1**2*K2**2 - 128*K1**2*K2*K4 + 6096*K1**2*K2 - 4460*K1**2 + 352*K1*K2**3*K3 - 608*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4680*K1*K2*K3 + 112*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 1432*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 1312*K2**2*K4 - 2624*K2**2 + 8*K2*K3*K5 - 1044*K3**2 - 278*K4**2 + 3004 |
| 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}], [{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}], [{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}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |