| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,2,2,2,1,0,1,2,2,2,2,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1220'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+30t^5+58t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1220'] |
| 2-strand cable arrow polynomial of the knot is: 736*K1**4*K2 - 2384*K1**4 + 160*K1**3*K2*K3 - 128*K1**3*K3 + 512*K1**2*K2**3 - 4592*K1**2*K2**2 - 480*K1**2*K2*K4 + 7336*K1**2*K2 - 304*K1**2*K3**2 - 4968*K1**2 + 128*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 160*K1*K2**2*K5 + 32*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6944*K1*K2*K3 + 1704*K1*K3*K4 + 88*K1*K4*K5 + 8*K1*K5*K6 - 592*K2**4 - 784*K2**2*K3**2 - 112*K2**2*K4**2 + 1832*K2**2*K4 - 4892*K2**2 - 160*K2*K3**2*K4 + 888*K2*K3*K5 + 152*K2*K4*K6 - 64*K3**4 + 120*K3**2*K6 - 2588*K3**2 - 1064*K4**2 - 252*K5**2 - 52*K6**2 + 4726 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1220'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4868', 'vk6.5211', 'vk6.6450', 'vk6.6869', 'vk6.8415', 'vk6.8834', 'vk6.9767', 'vk6.10058', 'vk6.11682', 'vk6.12035', 'vk6.13028', 'vk6.20502', 'vk6.20755', 'vk6.21871', 'vk6.27916', 'vk6.29410', 'vk6.29724', 'vk6.32679', 'vk6.33022', 'vk6.39339', 'vk6.39795', 'vk6.46359', 'vk6.47609', 'vk6.47936', 'vk6.48826', 'vk6.49095', 'vk6.51349', 'vk6.51560', 'vk6.53281', 'vk6.57363', 'vk6.64350', 'vk6.66916'] |
| 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 | O1O2O3O4U3U1U4O5O6U2U6U5 |
| R3 orbit | {'O1O2O3O4U3U1U4O5O6U2U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6U3O6O5U1U4U2 |
| Gauss code of K* | O1O2O3U4U5O6O5O4U2U6U1U3 |
| Gauss code of -K* | O1O2O3U2U1O4O5O6U4U6U3U5 |
| 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 2 1 1],[ 2 0 2 0 2 1 1],[ 1 -2 0 -1 1 2 1],[ 1 0 1 0 1 0 0],[-2 -2 -1 -1 0 0 0],[-1 -1 -2 0 0 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 -1 -1 -2],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -2 -1],[ 1 1 0 0 0 1 0],[ 1 1 1 2 -1 0 -2],[ 2 2 1 1 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,0,1,1,2,0,0,1,1,0,2,1,-1,0,2] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,1,2,2,2,1,0,1,2,2,2,2,0,1,1] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,1,2,2,2,1,0,1,2,2,2,2,0,1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,1,1,2,2,2,0,0,2,2,1,2,2,-1,-1,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,2,1,1,2,1,0,0,1,1,2,1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+18t^4+20t^2+1 |
| Outer characteristic polynomial | t^7+30t^5+58t^3+11t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | 736*K1**4*K2 - 2384*K1**4 + 160*K1**3*K2*K3 - 128*K1**3*K3 + 512*K1**2*K2**3 - 4592*K1**2*K2**2 - 480*K1**2*K2*K4 + 7336*K1**2*K2 - 304*K1**2*K3**2 - 4968*K1**2 + 128*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 160*K1*K2**2*K5 + 32*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6944*K1*K2*K3 + 1704*K1*K3*K4 + 88*K1*K4*K5 + 8*K1*K5*K6 - 592*K2**4 - 784*K2**2*K3**2 - 112*K2**2*K4**2 + 1832*K2**2*K4 - 4892*K2**2 - 160*K2*K3**2*K4 + 888*K2*K3*K5 + 152*K2*K4*K6 - 64*K3**4 + 120*K3**2*K6 - 2588*K3**2 - 1064*K4**2 - 252*K5**2 - 52*K6**2 + 4726 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | False |