| Min(phi) over symmetries of the knot is: [-2,-1,1,2,-1,2,3,2,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1931', '6.1932', '7.40268', '7.40283'] |
| 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^5+21t^3+45t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1931', '6.1932'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 192*K1**2*K2**3 - 1216*K1**2*K2**2 + 1552*K1**2*K2 - 32*K1**2*K4**2 - 1496*K1**2 + 128*K1*K2**3*K3 + 1392*K1*K2*K3 + 128*K1*K3*K4 + 48*K1*K4*K5 - 1344*K2**6 + 1024*K2**4*K4 - 1328*K2**4 - 224*K2**2*K3**2 - 272*K2**2*K4**2 + 1056*K2**2*K4 - 136*K2**2 + 112*K2*K3*K5 + 48*K2*K4*K6 - 488*K3**2 - 204*K4**2 - 32*K5**2 + 1154 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1931'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72616', 'vk6.72759', 'vk6.73075', 'vk6.73162', 'vk6.73791', 'vk6.73926', 'vk6.75733', 'vk6.75929', 'vk6.77867', 'vk6.77911', 'vk6.78011', 'vk6.78744', 'vk6.78935', 'vk6.80348', 'vk6.81183', 'vk6.81777', 'vk6.82483', 'vk6.87303', 'vk6.87899', 'vk6.88419'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is -. |
| The reverse -K is |
| The 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 | O1O2O3U1U4U3O5O6O4U2U5U6 |
| R3 orbit | {'O1O2O3U1U4U3O5O6O4U2U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1U5O4O6O5U2U3U6 |
| Gauss code of K* | O1O2O3U4U1U5O4O6O5U2U3U6 |
| Gauss code of -K* | Same |
| Diagrammatic symmetry type | - |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -1 2 1 -1 1],[ 2 0 2 1 1 1 1],[ 1 -2 0 1 0 0 1],[-2 -1 -1 0 -2 -1 -1],[-1 -1 0 2 0 -1 0],[ 1 -1 0 1 1 0 1],[-1 -1 -1 1 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 -1 -2],[-2 0 -2 -1 -1],[-1 2 0 0 -1],[ 1 1 0 0 -2],[ 2 1 1 2 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-2,-1,1,2,2,1,1,0,1,2] |
| Phi over symmetry | [-2,-1,1,2,-1,2,3,2,2,-1] |
| Phi of -K | [-2,-1,1,2,-1,2,3,2,2,-1] |
| Phi of K* | [-2,-1,1,2,-1,2,3,2,2,-1] |
| Phi of -K* | [-2,-1,1,2,2,1,1,0,1,2] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z+9 |
| Enhanced Jones-Krushkal polynomial | 8w^4z-12w^3z+8w^3+8w^2z+w |
| Inner characteristic polynomial | t^4+11t^2+9 |
| Outer characteristic polynomial | t^5+21t^3+45t |
| Flat arrow polynomial | 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| 2-strand cable arrow polynomial | -64*K1**4 + 192*K1**2*K2**3 - 1216*K1**2*K2**2 + 1552*K1**2*K2 - 32*K1**2*K4**2 - 1496*K1**2 + 128*K1*K2**3*K3 + 1392*K1*K2*K3 + 128*K1*K3*K4 + 48*K1*K4*K5 - 1344*K2**6 + 1024*K2**4*K4 - 1328*K2**4 - 224*K2**2*K3**2 - 272*K2**2*K4**2 + 1056*K2**2*K4 - 136*K2**2 + 112*K2*K3*K5 + 48*K2*K4*K6 - 488*K3**2 - 204*K4**2 - 32*K5**2 + 1154 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | True |