| Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,1,1,3,0,1,1,2,0,0,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1109'] |
| Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717'] |
| Outer characteristic polynomial of the knot is: t^7+32t^5+36t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1109'] |
| 2-strand cable arrow polynomial of the knot is: -448*K1**6 - 256*K1**4*K2**2 + 1408*K1**4*K2 - 4976*K1**4 + 704*K1**3*K2*K3 + 32*K1**3*K3*K4 - 928*K1**3*K3 - 4928*K1**2*K2**2 - 704*K1**2*K2*K4 + 10600*K1**2*K2 - 1136*K1**2*K3**2 - 304*K1**2*K4**2 - 5644*K1**2 - 608*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 7800*K1*K2*K3 + 1944*K1*K3*K4 + 272*K1*K4*K5 - 416*K2**4 - 448*K2**2*K3**2 - 112*K2**2*K4**2 + 1176*K2**2*K4 - 5404*K2**2 + 584*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2696*K3**2 - 908*K4**2 - 228*K5**2 - 52*K6**2 + 5578 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1109'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4131', 'vk6.4164', 'vk6.5373', 'vk6.5406', 'vk6.7491', 'vk6.7522', 'vk6.8996', 'vk6.9029', 'vk6.12436', 'vk6.12469', 'vk6.13355', 'vk6.13578', 'vk6.13611', 'vk6.14244', 'vk6.14693', 'vk6.14744', 'vk6.15182', 'vk6.15851', 'vk6.15904', 'vk6.30845', 'vk6.30878', 'vk6.32033', 'vk6.32066', 'vk6.33071', 'vk6.33104', 'vk6.33845', 'vk6.34305', 'vk6.48479', 'vk6.50258', 'vk6.53543', 'vk6.53932', 'vk6.54277'] |
| 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 | O1O2O3O4U2U4O5U1U5O6U3U6 |
| R3 orbit | {'O1O2O3O4U2U4O5U1U5O6U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U2O5U6U4O6U1U3 |
| Gauss code of K* | O1O2U3O4O3U5O6O5U4U1U6U2 |
| Gauss code of -K* | O1O2U1O3O4U3O5O6U5U2U6U4 |
| 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 -2 1 1 1 1],[ 2 0 -1 3 1 1 1],[ 2 1 0 2 1 0 1],[-1 -3 -2 0 0 0 1],[-1 -1 -1 0 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 1 -2 -2],[-1 0 1 0 0 -2 -3],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 2 2 1 0 1 0 1],[ 2 3 1 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,-1,2,2,-1,0,0,2,3,0,0,1,1,0,0,1,1,1,-1] |
| Phi over symmetry | [-2,-2,1,1,1,1,-1,1,1,1,3,0,1,1,2,0,0,0,0,-1,0] |
| Phi of -K | [-2,-2,1,1,1,1,-1,1,2,2,3,0,2,2,2,-1,0,0,0,0,0] |
| Phi of K* | [-1,-1,-1,-1,2,2,-1,0,0,2,2,0,0,0,1,0,2,2,2,3,-1] |
| Phi of -K* | [-2,-2,1,1,1,1,-1,1,1,1,3,0,1,1,2,0,0,0,0,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 2t^2-4t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+20t^4+14t^2+1 |
| Outer characteristic polynomial | t^7+32t^5+36t^3+8t |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -448*K1**6 - 256*K1**4*K2**2 + 1408*K1**4*K2 - 4976*K1**4 + 704*K1**3*K2*K3 + 32*K1**3*K3*K4 - 928*K1**3*K3 - 4928*K1**2*K2**2 - 704*K1**2*K2*K4 + 10600*K1**2*K2 - 1136*K1**2*K3**2 - 304*K1**2*K4**2 - 5644*K1**2 - 608*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 7800*K1*K2*K3 + 1944*K1*K3*K4 + 272*K1*K4*K5 - 416*K2**4 - 448*K2**2*K3**2 - 112*K2**2*K4**2 + 1176*K2**2*K4 - 5404*K2**2 + 584*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2696*K3**2 - 908*K4**2 - 228*K5**2 - 52*K6**2 + 5578 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |