| Min(phi) over symmetries of the knot is: [-3,0,1,2,1,3,3,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.116', '6.458', '6.1180', '7.28052'] |
| Arrow polynomial of the knot is: -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.458', '6.601', '6.611', '6.995', '6.1026', '6.1037'] |
| Outer characteristic polynomial of the knot is: t^5+35t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.116', '6.458'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 + 352*K1**4*K2 - 1376*K1**4 + 64*K1**3*K2*K3 - 816*K1**2*K2**2 + 1912*K1**2*K2 - 704*K1**2*K3**2 - 288*K1**2*K4**2 - 928*K1**2 + 1376*K1*K2*K3 + 912*K1*K3*K4 + 312*K1*K4*K5 - 56*K2**4 - 112*K2**2*K3**2 - 56*K2**2*K4**2 + 152*K2**2*K4 - 1042*K2**2 + 232*K2*K3*K5 + 104*K2*K4*K6 - 636*K3**2 - 386*K4**2 - 172*K5**2 - 38*K6**2 + 1320 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.458'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4450', 'vk6.4545', 'vk6.5832', 'vk6.5959', 'vk6.6390', 'vk6.6821', 'vk6.8008', 'vk6.8347', 'vk6.9319', 'vk6.9438', 'vk6.11626', 'vk6.11979', 'vk6.12972', 'vk6.13422', 'vk6.13519', 'vk6.13710', 'vk6.14068', 'vk6.15041', 'vk6.15161', 'vk6.17773', 'vk6.17804', 'vk6.18849', 'vk6.19442', 'vk6.19737', 'vk6.24320', 'vk6.25444', 'vk6.25475', 'vk6.26616', 'vk6.33276', 'vk6.33337', 'vk6.37568', 'vk6.39290', 'vk6.39755', 'vk6.41470', 'vk6.44891', 'vk6.46319', 'vk6.47896', 'vk6.48637', 'vk6.49879', 'vk6.53237'] |
| 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 | O1O2O3O4O5U6U3O6U4U1U5U2 |
| R3 orbit | {'O1O2O3O4O5U4U6U3O6U1U5U2', 'O1O2O3O4O5U6U3O6U4U1U5U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U1U5U2O6U3U6 |
| Gauss code of K* | O1O2O3O4U2U4U5U1U3O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U2U4U6U1U3 |
| Diagrammatic symmetry type | c |
| 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 -1 0 3 -1],[ 2 0 2 0 1 3 1],[-1 -2 0 -1 0 2 -2],[ 1 0 1 0 0 1 1],[ 0 -1 0 0 0 1 0],[-3 -3 -2 -1 -1 0 -3],[ 1 -1 2 -1 0 3 0]] |
| Primitive based matrix | [[ 0 3 0 -1 -2],[-3 0 -1 -3 -3],[ 0 1 0 0 -1],[ 1 3 0 0 -1],[ 2 3 1 1 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-3,0,1,2,1,3,3,0,1,1] |
| Phi over symmetry | [-3,0,1,2,1,3,3,0,1,1] |
| Phi of -K | [-2,-1,0,3,0,1,2,1,1,2] |
| Phi of K* | [-3,0,1,2,2,1,2,1,1,0] |
| Phi of -K* | [-2,-1,0,3,1,1,3,0,3,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 11z+23 |
| Enhanced Jones-Krushkal polynomial | 11w^2z+23w |
| Inner characteristic polynomial | t^4+21t^2+4 |
| Outer characteristic polynomial | t^5+35t^3+7t |
| Flat arrow polynomial | -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4 |
| 2-strand cable arrow polynomial | -128*K1**6 + 352*K1**4*K2 - 1376*K1**4 + 64*K1**3*K2*K3 - 816*K1**2*K2**2 + 1912*K1**2*K2 - 704*K1**2*K3**2 - 288*K1**2*K4**2 - 928*K1**2 + 1376*K1*K2*K3 + 912*K1*K3*K4 + 312*K1*K4*K5 - 56*K2**4 - 112*K2**2*K3**2 - 56*K2**2*K4**2 + 152*K2**2*K4 - 1042*K2**2 + 232*K2*K3*K5 + 104*K2*K4*K6 - 636*K3**2 - 386*K4**2 - 172*K5**2 - 38*K6**2 + 1320 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}]] |
| If K is slice | False |