| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,3,3,3,1,2,2,2,1,1,2,1,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.203'] |
| 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+82t^5+46t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.203'] |
| 2-strand cable arrow polynomial of the knot is: 320*K1**4*K2 - 2352*K1**4 - 544*K1**3*K3 - 2208*K1**2*K2**2 - 608*K1**2*K2*K4 + 7040*K1**2*K2 - 1008*K1**2*K3**2 - 128*K1**2*K3*K5 - 624*K1**2*K4**2 - 32*K1**2*K4*K6 - 6320*K1**2 - 544*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6240*K1*K2*K3 + 3152*K1*K3*K4 + 880*K1*K4*K5 + 32*K1*K5*K6 - 208*K2**4 - 224*K2**2*K3**2 - 112*K2**2*K4**2 + 1344*K2**2*K4 - 4916*K2**2 + 352*K2*K3*K5 + 96*K2*K4*K6 - 2932*K3**2 - 1664*K4**2 - 332*K5**2 - 28*K6**2 + 5486 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.203'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11013', 'vk6.11092', 'vk6.12179', 'vk6.12286', 'vk6.18210', 'vk6.18545', 'vk6.24670', 'vk6.25092', 'vk6.30578', 'vk6.30673', 'vk6.31848', 'vk6.31895', 'vk6.36798', 'vk6.37252', 'vk6.44039', 'vk6.44379', 'vk6.51812', 'vk6.51879', 'vk6.52676', 'vk6.52770', 'vk6.56004', 'vk6.56277', 'vk6.60541', 'vk6.60881', 'vk6.63492', 'vk6.63536', 'vk6.63970', 'vk6.64014', 'vk6.65663', 'vk6.65945', 'vk6.68709', 'vk6.68917'] |
| 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 | O1O2O3O4O5U2O6U3U5U6U1U4 |
| R3 orbit | {'O1O2O3O4O5U2O6U3U5U6U1U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U5U6U1U3O6U4 |
| Gauss code of K* | O1O2O3O4O5U4U6U1U5U2O6U3 |
| Gauss code of -K* | O1O2O3O4O5U3O6U4U1U5U6U2 |
| 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 -3 -2 3 1 2],[ 1 0 -2 -1 3 1 2],[ 3 2 0 1 3 2 2],[ 2 1 -1 0 3 1 2],[-3 -3 -3 -3 0 -1 1],[-1 -1 -2 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 1 -1 -3 -3 -3],[-2 -1 0 -1 -2 -2 -2],[-1 1 1 0 -1 -1 -2],[ 1 3 2 1 0 -1 -2],[ 2 3 2 1 1 0 -1],[ 3 3 2 2 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,-1,1,3,3,3,1,2,2,2,1,1,2,1,2,1] |
| Phi over symmetry | [-3,-2,-1,1,2,3,-1,1,3,3,3,1,2,2,2,1,1,2,1,2,1] |
| Phi of -K | [-3,-2,-1,1,2,3,0,0,2,3,3,0,2,2,2,1,1,1,0,1,2] |
| Phi of K* | [-3,-2,-1,1,2,3,2,1,1,2,3,0,1,2,3,1,2,2,0,0,0] |
| Phi of -K* | [-3,-2,-1,1,2,3,1,2,2,2,3,1,1,2,3,1,2,3,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+54t^4+16t^2+1 |
| Outer characteristic polynomial | t^7+82t^5+46t^3+6t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | 320*K1**4*K2 - 2352*K1**4 - 544*K1**3*K3 - 2208*K1**2*K2**2 - 608*K1**2*K2*K4 + 7040*K1**2*K2 - 1008*K1**2*K3**2 - 128*K1**2*K3*K5 - 624*K1**2*K4**2 - 32*K1**2*K4*K6 - 6320*K1**2 - 544*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6240*K1*K2*K3 + 3152*K1*K3*K4 + 880*K1*K4*K5 + 32*K1*K5*K6 - 208*K2**4 - 224*K2**2*K3**2 - 112*K2**2*K4**2 + 1344*K2**2*K4 - 4916*K2**2 + 352*K2*K3*K5 + 96*K2*K4*K6 - 2932*K3**2 - 1664*K4**2 - 332*K5**2 - 28*K6**2 + 5486 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}]] |
| If K is slice | False |