| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,3,1,0,1,1,0,1,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.547'] |
| 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+40t^5+28t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.547', '6.1382'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 192*K1**4*K2 - 816*K1**4 + 128*K1**3*K2*K3 - 880*K1**2*K2**2 + 1552*K1**2*K2 - 208*K1**2*K3**2 - 800*K1**2 + 1008*K1*K2*K3 + 272*K1*K3*K4 + 16*K1*K4*K5 - 160*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 144*K2**2*K4 - 724*K2**2 + 128*K2*K3*K5 + 32*K2*K4*K6 - 396*K3**2 - 144*K4**2 - 52*K5**2 - 12*K6**2 + 878 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.547'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4438', 'vk6.4533', 'vk6.4850', 'vk6.5194', 'vk6.5820', 'vk6.5947', 'vk6.6419', 'vk6.6436', 'vk6.6851', 'vk6.7988', 'vk6.8380', 'vk6.8397', 'vk6.8806', 'vk6.9299', 'vk6.9418', 'vk6.9742', 'vk6.17890', 'vk6.17955', 'vk6.18282', 'vk6.18617', 'vk6.24393', 'vk6.24693', 'vk6.25168', 'vk6.30033', 'vk6.30096', 'vk6.30903', 'vk6.31028', 'vk6.32087', 'vk6.32208', 'vk6.36892', 'vk6.37277', 'vk6.37350', 'vk6.39826', 'vk6.39846', 'vk6.43820', 'vk6.44109', 'vk6.44432', 'vk6.46404', 'vk6.47961', 'vk6.47981', 'vk6.49076', 'vk6.49910', 'vk6.50609', 'vk6.51129', 'vk6.51971', 'vk6.52068', 'vk6.60579', 'vk6.65986'] |
| 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 | O1O2O3O4O5U3U5U2O6U4U1U6 |
| R3 orbit | {'O1O2O3O4U2O5U3U5O6U4U1U6', 'O1O2O3O4O5U3U5U2O6U4U1U6', 'O1O2O3U1O4O5U3U5O6U2U4U6'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4O5U6U5U2O6U4U1U3 |
| Gauss code of K* | O1O2O3U4O5O6O4U6U3U1U5U2 |
| Gauss code of -K* | O1O2O3U1O4O5O6U5U3U6U4U2 |
| 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 -1 -1 -2 1 1 2],[ 1 0 -1 -2 2 1 2],[ 1 1 0 -1 2 1 1],[ 2 2 1 0 2 1 1],[-1 -2 -2 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-2 -2 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -2 -1],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 -2 -2 -2],[ 1 1 1 2 0 1 -1],[ 1 2 1 2 -1 0 -2],[ 2 1 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,1,1,2,1,0,1,1,1,2,2,2,-1,1,2] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,2,3,1,0,1,1,0,1,2,0,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,0,1,2,3,1,0,1,1,0,1,2,0,0,1] |
| Phi of K* | [-2,-1,-1,1,1,2,0,1,1,2,3,0,0,0,1,1,1,2,-1,-1,0] |
| Phi of -K* | [-2,-1,-1,1,1,2,1,2,1,2,1,1,1,2,1,1,2,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 9z+19 |
| Enhanced Jones-Krushkal polynomial | 9w^2z+19w |
| Inner characteristic polynomial | t^6+28t^4+8t^2 |
| Outer characteristic polynomial | t^7+40t^5+28t^3 |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 192*K1**4*K2 - 816*K1**4 + 128*K1**3*K2*K3 - 880*K1**2*K2**2 + 1552*K1**2*K2 - 208*K1**2*K3**2 - 800*K1**2 + 1008*K1*K2*K3 + 272*K1*K3*K4 + 16*K1*K4*K5 - 160*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 144*K2**2*K4 - 724*K2**2 + 128*K2*K3*K5 + 32*K2*K4*K6 - 396*K3**2 - 144*K4**2 - 52*K5**2 - 12*K6**2 + 878 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |