| Min(phi) over symmetries of the knot is: [-3,-2,0,0,2,3,0,1,1,2,3,0,2,3,3,0,0,0,1,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.613'] |
| Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.206', '6.236', '6.575', '6.580', '6.613', '6.619', '6.810', '6.819', '6.831', '6.838', '6.957', '6.1018', '6.1028', '6.1046', '6.1073', '6.1279', '6.1507', '6.1532', '6.1556', '6.1639', '6.1688', '6.1924', '6.1931'] |
| Outer characteristic polynomial of the knot is: t^7+77t^5+232t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.613'] |
| 2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 1024*K1**4*K2 - 2208*K1**4 + 128*K1**3*K2**3*K3 - 256*K1**3*K2**2*K3 + 416*K1**3*K2*K3 - 352*K1**3*K3 - 256*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2144*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 8272*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 8952*K1**2*K2 - 320*K1**2*K3**2 - 5244*K1**2 - 128*K1*K2**4*K3 + 1856*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 - 352*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 7160*K1*K2*K3 + 752*K1*K3*K4 - 64*K2**6 + 320*K2**4*K4 - 2224*K2**4 - 1120*K2**2*K3**2 - 288*K2**2*K4**2 + 1552*K2**2*K4 - 2920*K2**2 + 336*K2*K3*K5 - 1668*K3**2 - 344*K4**2 + 3966 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.613'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73275', 'vk6.73418', 'vk6.74014', 'vk6.74554', 'vk6.75187', 'vk6.75419', 'vk6.76032', 'vk6.76764', 'vk6.78148', 'vk6.78385', 'vk6.78995', 'vk6.79554', 'vk6.79977', 'vk6.80132', 'vk6.80521', 'vk6.80987', 'vk6.81878', 'vk6.82148', 'vk6.82177', 'vk6.82596', 'vk6.83586', 'vk6.83761', 'vk6.84039', 'vk6.84616', 'vk6.84947', 'vk6.85588', 'vk6.85710', 'vk6.85923', 'vk6.86739', 'vk6.87673', 'vk6.88941', 'vk6.89964'] |
| 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 | O1O2O3O4U5O6U1O5U2U3U6U4 |
| R3 orbit | {'O1O2O3O4U5O6U1O5U2U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U2U3O6U4O5U6 |
| Gauss code of K* | O1O2O3O4U5U1U2U4O6U3O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6U2O5U1U3U4U6 |
| 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 -2 0 3 0 2],[ 3 0 0 1 3 3 2],[ 2 0 0 1 3 2 1],[ 0 -1 -1 0 2 0 0],[-3 -3 -3 -2 0 -2 -1],[ 0 -3 -2 0 2 0 2],[-2 -2 -1 0 1 -2 0]] |
| Primitive based matrix | [[ 0 3 2 0 0 -2 -3],[-3 0 -1 -2 -2 -3 -3],[-2 1 0 0 -2 -1 -2],[ 0 2 0 0 0 -1 -1],[ 0 2 2 0 0 -2 -3],[ 2 3 1 1 2 0 0],[ 3 3 2 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,0,2,3,1,2,2,3,3,0,2,1,2,0,1,1,2,3,0] |
| Phi over symmetry | [-3,-2,0,0,2,3,0,1,1,2,3,0,2,3,3,0,0,0,1,2,1] |
| Phi of -K | [-3,-2,0,0,2,3,1,0,2,3,3,0,1,3,2,0,0,1,2,1,0] |
| Phi of K* | [-3,-2,0,0,2,3,0,1,1,2,3,0,2,3,3,0,0,0,1,2,1] |
| Phi of -K* | [-3,-2,0,0,2,3,0,1,3,2,3,1,2,1,3,0,0,2,2,2,1] |
| 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+51t^4+136t^2+1 |
| Outer characteristic polynomial | t^7+77t^5+232t^3+11t |
| Flat arrow polynomial | 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| 2-strand cable arrow polynomial | -512*K1**4*K2**2 + 1024*K1**4*K2 - 2208*K1**4 + 128*K1**3*K2**3*K3 - 256*K1**3*K2**2*K3 + 416*K1**3*K2*K3 - 352*K1**3*K3 - 256*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2144*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 8272*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 8952*K1**2*K2 - 320*K1**2*K3**2 - 5244*K1**2 - 128*K1*K2**4*K3 + 1856*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 - 352*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 7160*K1*K2*K3 + 752*K1*K3*K4 - 64*K2**6 + 320*K2**4*K4 - 2224*K2**4 - 1120*K2**2*K3**2 - 288*K2**2*K4**2 + 1552*K2**2*K4 - 2920*K2**2 + 336*K2*K3*K5 - 1668*K3**2 - 344*K4**2 + 3966 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |