| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,3,0,0,1,1,-1,0,0,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1793'] |
| Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070'] |
| Outer characteristic polynomial of the knot is: t^7+25t^5+38t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1793'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 960*K1**4*K2 - 2544*K1**4 + 288*K1**3*K2*K3 - 704*K1**3*K3 + 384*K1**2*K2**3 - 3872*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 7528*K1**2*K2 - 368*K1**2*K3**2 - 5152*K1**2 + 128*K1*K2**3*K3 - 960*K1*K2**2*K3 - 192*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 6128*K1*K2*K3 + 1024*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 592*K2**4 - 64*K2**3*K6 - 352*K2**2*K3**2 - 128*K2**2*K4**2 + 1360*K2**2*K4 - 4386*K2**2 + 520*K2*K3*K5 + 104*K2*K4*K6 - 2096*K3**2 - 688*K4**2 - 176*K5**2 - 14*K6**2 + 4342 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1793'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17109', 'vk6.17351', 'vk6.20576', 'vk6.21983', 'vk6.23498', 'vk6.23837', 'vk6.28042', 'vk6.29499', 'vk6.35643', 'vk6.36083', 'vk6.39452', 'vk6.41651', 'vk6.43006', 'vk6.43317', 'vk6.46040', 'vk6.47706', 'vk6.55260', 'vk6.55511', 'vk6.57442', 'vk6.58611', 'vk6.59664', 'vk6.60012', 'vk6.62117', 'vk6.63085', 'vk6.65060', 'vk6.65254', 'vk6.66978', 'vk6.67841', 'vk6.68320', 'vk6.68469', 'vk6.69597', 'vk6.70288'] |
| 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 | O1O2O3U4U2O5O6U1U6O4U5U3 |
| R3 orbit | {'O1O2O3U4U2O5O6U1U6O4U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U6U3O6O4U2U5 |
| Gauss code of K* | O1O2U3O4O5U1U6U5O3O6U4U2 |
| Gauss code of -K* | O1O2U3O4O5U4U2O6O3U1U6U5 |
| 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 0 2 -1 0 1],[ 2 0 0 3 0 1 1],[ 0 0 0 0 0 -1 0],[-2 -3 0 0 -1 -1 0],[ 1 0 0 1 0 0 1],[ 0 -1 1 1 0 0 0],[-1 -1 0 0 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -1 -3],[-1 0 0 0 0 -1 -1],[ 0 0 0 0 -1 0 0],[ 0 1 0 1 0 0 -1],[ 1 1 1 0 0 0 0],[ 2 3 1 0 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,1,1,3,0,0,1,1,1,0,0,0,1,0] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,1,3,0,0,1,1,-1,0,0,0,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,1,1,2,2,1,1,1,1,2,-1,1,1,1,2,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,2,2,1,1,1,1,2,1,1,1,1,2,1] |
| Phi of -K* | [-2,-1,0,0,1,2,0,0,1,1,3,0,0,1,1,-1,0,0,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | z^2+18z+33 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+18w^2z+33w |
| Inner characteristic polynomial | t^6+15t^4+20t^2+4 |
| Outer characteristic polynomial | t^7+25t^5+38t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 960*K1**4*K2 - 2544*K1**4 + 288*K1**3*K2*K3 - 704*K1**3*K3 + 384*K1**2*K2**3 - 3872*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 7528*K1**2*K2 - 368*K1**2*K3**2 - 5152*K1**2 + 128*K1*K2**3*K3 - 960*K1*K2**2*K3 - 192*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 6128*K1*K2*K3 + 1024*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 592*K2**4 - 64*K2**3*K6 - 352*K2**2*K3**2 - 128*K2**2*K4**2 + 1360*K2**2*K4 - 4386*K2**2 + 520*K2*K3*K5 + 104*K2*K4*K6 - 2096*K3**2 - 688*K4**2 - 176*K5**2 - 14*K6**2 + 4342 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | True |