| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1201'] |
| 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+32t^5+36t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.806', '6.1201'] |
| 2-strand cable arrow polynomial of the knot is: -32*K1**4 - 192*K1**2*K2**4 + 416*K1**2*K2**3 - 2688*K1**2*K2**2 - 96*K1**2*K2*K4 + 2760*K1**2*K2 - 1908*K1**2 + 448*K1*K2**3*K3 - 448*K1*K2**2*K3 - 96*K1*K2**2*K5 + 2360*K1*K2*K3 + 104*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 792*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 696*K2**2*K4 - 1080*K2**2 + 48*K2*K3*K5 - 500*K3**2 - 142*K4**2 + 1300 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1201'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10513', 'vk6.10517', 'vk6.10586', 'vk6.10594', 'vk6.10773', 'vk6.10781', 'vk6.10888', 'vk6.10892', 'vk6.17680', 'vk6.17682', 'vk6.17729', 'vk6.17731', 'vk6.24290', 'vk6.24292', 'vk6.24761', 'vk6.25220', 'vk6.30198', 'vk6.30202', 'vk6.30273', 'vk6.30281', 'vk6.30400', 'vk6.30408', 'vk6.30654', 'vk6.30746', 'vk6.36516', 'vk6.36951', 'vk6.43620', 'vk6.43622', 'vk6.43724', 'vk6.43726', 'vk6.52736', 'vk6.52846', 'vk6.60348', 'vk6.60350', 'vk6.60620', 'vk6.60953', 'vk6.63458', 'vk6.63462', 'vk6.65417', 'vk6.65755'] |
| 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 | O1O2O3O4U2U3U5O6O5U4U1U6 |
| R3 orbit | {'O1O2O3U1O4U3U5O6O5U2U4U6', 'O1O2O3O4U2U3U5O6O5U4U1U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U4U1O6O5U6U2U3 |
| Gauss code of K* | O1O2O3U4U3O5O6O4U6U1U2U5 |
| Gauss code of -K* | O1O2O3U4U1O4O5O6U3U5U6U2 |
| 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 -2 0 1 1 1],[ 1 0 -2 0 2 1 1],[ 2 2 0 1 2 2 1],[ 0 0 -1 0 1 0 1],[-1 -2 -2 -1 0 -1 0],[-1 -1 -2 0 1 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[-1 -1 0 0 -1 -2 -2],[ 0 0 1 1 0 0 -1],[ 1 1 1 2 0 0 -2],[ 2 2 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,1,1,1,1,2,2,0,1,2] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,1,0,-1] |
| Phi of -K | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,1,0,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,0,1,1,1,1,1,0,1,2,1,1,-1] |
| Phi of -K* | [-2,-1,0,1,1,1,2,1,1,2,2,0,1,1,2,1,0,1,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+24t^4+11t^2 |
| Outer characteristic polynomial | t^7+32t^5+36t^3+3t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -32*K1**4 - 192*K1**2*K2**4 + 416*K1**2*K2**3 - 2688*K1**2*K2**2 - 96*K1**2*K2*K4 + 2760*K1**2*K2 - 1908*K1**2 + 448*K1*K2**3*K3 - 448*K1*K2**2*K3 - 96*K1*K2**2*K5 + 2360*K1*K2*K3 + 104*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 792*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 696*K2**2*K4 - 1080*K2**2 + 48*K2*K3*K5 - 500*K3**2 - 142*K4**2 + 1300 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |