| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,3,3,-1,0,1,2,0,0,0,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.637'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+52t^5+63t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.637'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 128*K1**4 + 160*K1**2*K2**3 - 976*K1**2*K2**2 - 32*K1**2*K2*K4 + 1896*K1**2*K2 - 1596*K1**2 + 64*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1208*K1*K2*K3 + 48*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 312*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 464*K2**2*K4 - 1120*K2**2 + 16*K2*K3*K5 - 340*K3**2 - 118*K4**2 + 1092 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.637'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11062', 'vk6.11142', 'vk6.12224', 'vk6.12333', 'vk6.18337', 'vk6.18676', 'vk6.24773', 'vk6.25232', 'vk6.30633', 'vk6.30730', 'vk6.31865', 'vk6.31937', 'vk6.36955', 'vk6.37417', 'vk6.44144', 'vk6.44467', 'vk6.51863', 'vk6.51910', 'vk6.52726', 'vk6.52835', 'vk6.56123', 'vk6.56348', 'vk6.60640', 'vk6.60981', 'vk6.63520', 'vk6.63566', 'vk6.63998', 'vk6.64044', 'vk6.65773', 'vk6.66034', 'vk6.68776', 'vk6.68986', 'vk6.83621', 'vk6.83669', 'vk6.84299', 'vk6.85233', 'vk6.85598', 'vk6.86753', 'vk6.88707', 'vk6.88974'] |
| 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 | O1O2O3O4U1O5U3U5O6U4U2U6 |
| R3 orbit | {'O1O2O3O4U5U2O6U4U3U1O5U6', 'O1O2O3O4U1O5U3U5O6U4U2U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3U1O5U6U2O6U4 |
| Gauss code of K* | O1O2O3U4U2U5U1O4U6O5O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U4O6U3U5U2U6 |
| 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 -3 0 -1 1 1 2],[ 3 0 3 1 2 1 2],[ 0 -3 0 -2 1 1 2],[ 1 -1 2 0 2 1 1],[-1 -2 -1 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-2 -2 -2 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -2 -1 -2],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 -1 -2 -2],[ 0 2 1 1 0 -2 -3],[ 1 1 1 2 2 0 -1],[ 3 2 1 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,1,2,1,2,0,1,1,1,1,2,2,2,3,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,0,2,3,3,-1,0,1,2,0,0,0,0,0,1] |
| Phi of -K | [-3,-1,0,1,1,2,1,0,2,3,3,-1,0,1,2,0,0,0,0,0,1] |
| Phi of K* | [-2,-1,-1,0,1,3,0,1,0,2,3,0,0,0,2,0,1,3,-1,0,1] |
| Phi of -K* | [-3,-1,0,1,1,2,1,3,1,2,2,2,1,2,1,1,1,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+36t^4+28t^2+1 |
| Outer characteristic polynomial | t^7+52t^5+63t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 128*K1**4 + 160*K1**2*K2**3 - 976*K1**2*K2**2 - 32*K1**2*K2*K4 + 1896*K1**2*K2 - 1596*K1**2 + 64*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1208*K1*K2*K3 + 48*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 312*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 464*K2**2*K4 - 1120*K2**2 + 16*K2*K3*K5 - 340*K3**2 - 118*K4**2 + 1092 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |