| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,4,1,0,0,1,0,0,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.583'] |
| 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+56t^5+29t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.583'] |
| 2-strand cable arrow polynomial of the knot is: -240*K1**4 + 64*K1**2*K2**3 - 368*K1**2*K2**2 + 632*K1**2*K2 - 80*K1**2*K3**2 - 576*K1**2 + 64*K1*K2**3*K3 + 736*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 136*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 88*K2**2*K4 - 400*K2**2 + 56*K2*K3*K5 - 324*K3**2 - 86*K4**2 - 12*K5**2 + 540 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.583'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11242', 'vk6.11320', 'vk6.12503', 'vk6.12614', 'vk6.18222', 'vk6.18558', 'vk6.24686', 'vk6.25104', 'vk6.30916', 'vk6.31039', 'vk6.32100', 'vk6.32219', 'vk6.36810', 'vk6.37268', 'vk6.44050', 'vk6.44391', 'vk6.51992', 'vk6.52087', 'vk6.52869', 'vk6.52916', 'vk6.56015', 'vk6.56290', 'vk6.60556', 'vk6.60896', 'vk6.63644', 'vk6.63689', 'vk6.64072', 'vk6.64117', 'vk6.65675', 'vk6.65961', 'vk6.68720', 'vk6.68929', 'vk6.71596', 'vk6.72141', 'vk6.73777', 'vk6.73915', 'vk6.77211', 'vk6.77523', 'vk6.78719', 'vk6.78752', 'vk6.78914', 'vk6.78924', 'vk6.79618', 'vk6.81359', 'vk6.85431', 'vk6.87984', 'vk6.88346', 'vk6.89320'] |
| 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 | O1O2O3O4U1O5U4O6U3U5U6U2 |
| R3 orbit | {'O1O2O3O4U1O5U4O6U3U5U6U2', 'O1O2O3O4U1U3O5U2O6U4U5U6', 'O1O2O3O4U1O5U4U2O6U5U3U6'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4U3U5U6U2O5U1O6U4 |
| Gauss code of K* | O1O2O3O4U5U4U1U6O5U2O6U3 |
| Gauss code of -K* | O1O2O3O4U2O5U3O6U5U4U1U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -3 1 -1 0 1 2],[ 3 0 3 2 1 2 1],[-1 -3 0 -2 -1 1 2],[ 1 -2 2 0 0 2 2],[ 0 -1 1 0 0 1 1],[-1 -2 -1 -2 -1 0 1],[-2 -1 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -2 -1 -2 -1],[-1 1 0 -1 -1 -2 -2],[-1 2 1 0 -1 -2 -3],[ 0 1 1 1 0 0 -1],[ 1 2 2 2 0 0 -2],[ 3 1 2 3 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,1,2,1,2,1,1,1,2,2,1,2,3,0,1,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,2,1,2,4,1,0,0,1,0,0,1,-1,-1,0] |
| Phi of -K | [-3,-1,0,1,1,2,0,2,1,2,4,1,0,0,1,0,0,1,-1,-1,0] |
| Phi of K* | [-2,-1,-1,0,1,3,-1,0,1,1,4,1,0,0,1,0,0,2,1,2,0] |
| Phi of -K* | [-3,-1,0,1,1,2,2,1,2,3,1,0,2,2,2,1,1,1,-1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+9w^2z+15w |
| Inner characteristic polynomial | t^6+40t^4 |
| Outer characteristic polynomial | t^7+56t^5+29t^3 |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -240*K1**4 + 64*K1**2*K2**3 - 368*K1**2*K2**2 + 632*K1**2*K2 - 80*K1**2*K3**2 - 576*K1**2 + 64*K1*K2**3*K3 + 736*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 136*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 88*K2**2*K4 - 400*K2**2 + 56*K2*K3*K5 - 324*K3**2 - 86*K4**2 - 12*K5**2 + 540 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}]] |
| If K is slice | False |