| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,2,4,-1,0,1,2,-1,1,0,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.634'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+56t^5+85t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.634'] |
| 2-strand cable arrow polynomial of the knot is: -96*K1**4 + 96*K1**2*K2**3 - 592*K1**2*K2**2 + 744*K1**2*K2 - 700*K1**2 + 96*K1*K2**3*K3 + 832*K1*K2*K3 + 32*K1*K3*K4 - 288*K2**6 + 160*K2**4*K4 - 568*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 360*K2**2*K4 - 176*K2**2 + 136*K2*K3*K5 - 300*K3**2 - 46*K4**2 - 24*K5**2 + 564 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.634'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71615', 'vk6.71771', 'vk6.72192', 'vk6.72344', 'vk6.72621', 'vk6.72624', 'vk6.72767', 'vk6.72773', 'vk6.73082', 'vk6.73091', 'vk6.73165', 'vk6.73170', 'vk6.73785', 'vk6.73921', 'vk6.75721', 'vk6.77235', 'vk6.77313', 'vk6.77562', 'vk6.77870', 'vk6.77914', 'vk6.77924', 'vk6.78012', 'vk6.78728', 'vk6.80341', 'vk6.81119', 'vk6.81169', 'vk6.81186', 'vk6.81244', 'vk6.81334', 'vk6.81523', 'vk6.82484', 'vk6.85459', 'vk6.86348', 'vk6.86918', 'vk6.87130', 'vk6.87294', 'vk6.87893', 'vk6.88105', 'vk6.88423', 'vk6.88670'] |
| 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 | O1O2O3O4U1O5U3U4O6U5U2U6 |
| R3 orbit | {'O1O2O3O4U1U2O5U4O6U3U5U6', 'O1O2O3O4U1O5U3U4O6U5U2U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3U6O5U1U2O6U4 |
| Gauss code of K* | O1O2O3U4U2U5U6O4U1O5O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U4U5U2U6 |
| 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 -3 0 -1 1 1 2],[ 3 0 3 1 2 2 1],[ 0 -3 0 -2 0 2 2],[ 1 -1 2 0 1 2 1],[-1 -2 0 -1 0 1 1],[-1 -2 -2 -2 -1 0 1],[-2 -1 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 -2 -1 -1],[-1 1 0 1 0 -1 -2],[-1 1 -1 0 -2 -2 -2],[ 0 2 0 2 0 -2 -3],[ 1 1 1 2 2 0 -1],[ 3 1 2 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,1,1,2,1,1,-1,0,1,2,2,2,2,2,3,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,0,2,2,4,-1,0,1,2,-1,1,0,1,0,0] |
| Phi of -K | [-3,-1,0,1,1,2,1,0,2,2,4,-1,0,1,2,-1,1,0,1,0,0] |
| Phi of K* | [-2,-1,-1,0,1,3,0,0,0,2,4,-1,-1,0,2,1,1,2,-1,0,1] |
| Phi of -K* | [-3,-1,0,1,1,2,1,3,2,2,1,2,1,2,1,0,2,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | 4w^4z-8w^3z+4w^3+9w^2z+7w |
| Inner characteristic polynomial | t^6+40t^4+8t^2 |
| Outer characteristic polynomial | t^7+56t^5+85t^3 |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -96*K1**4 + 96*K1**2*K2**3 - 592*K1**2*K2**2 + 744*K1**2*K2 - 700*K1**2 + 96*K1*K2**3*K3 + 832*K1*K2*K3 + 32*K1*K3*K4 - 288*K2**6 + 160*K2**4*K4 - 568*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 360*K2**2*K4 - 176*K2**2 + 136*K2*K3*K5 - 300*K3**2 - 46*K4**2 - 24*K5**2 + 564 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |