| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,4,2,2,1,2,1,2,0,0,1,2,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.718'] |
| 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+69t^5+74t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.718'] |
| 2-strand cable arrow polynomial of the knot is: -768*K1**4 - 128*K1**3*K3 + 384*K1**2*K2**5 - 3200*K1**2*K2**4 + 3232*K1**2*K2**3 - 3376*K1**2*K2**2 - 160*K1**2*K2*K4 + 3072*K1**2*K2 - 96*K1**2*K3**2 - 1892*K1**2 + 1856*K1*K2**3*K3 - 256*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1856*K1*K2*K3 + 160*K1*K3*K4 - 288*K2**6 + 64*K2**4*K4 - 1016*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 304*K2**2*K4 - 328*K2**2 + 16*K2*K3*K5 - 444*K3**2 - 70*K4**2 + 1340 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.718'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16896', 'vk6.17141', 'vk6.20204', 'vk6.21486', 'vk6.23280', 'vk6.23583', 'vk6.27380', 'vk6.29012', 'vk6.35278', 'vk6.35723', 'vk6.38807', 'vk6.40984', 'vk6.42797', 'vk6.43083', 'vk6.45558', 'vk6.47343', 'vk6.55041', 'vk6.55287', 'vk6.57045', 'vk6.58145', 'vk6.59425', 'vk6.59717', 'vk6.61542', 'vk6.62724', 'vk6.64884', 'vk6.65100', 'vk6.66663', 'vk6.67492', 'vk6.68197', 'vk6.68343', 'vk6.69308', 'vk6.70070'] |
| 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 | O1O2O3O4U1O5U2U3U4O6U5U6 |
| R3 orbit | {'O1O2O3O4U1O5U2U3U4O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6O5U1U2U3O6U4 |
| Gauss code of K* | O1O2O3U4O5O4U6U1U2U3O6U5 |
| Gauss code of -K* | O1O2O3U4O5U1U2U3U5O6O4U6 |
| 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 -3 -2 0 2 2 1],[ 3 0 1 2 3 3 0],[ 2 -1 0 1 2 3 1],[ 0 -2 -1 0 1 2 1],[-2 -3 -2 -1 0 1 1],[-2 -3 -3 -2 -1 0 1],[-1 0 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 1 1 -1 -2 -3],[-2 -1 0 1 -2 -3 -3],[-1 -1 -1 0 -1 -1 0],[ 0 1 2 1 0 -1 -2],[ 2 2 3 1 1 0 -1],[ 3 3 3 0 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,-1,-1,1,2,3,-1,2,3,3,1,1,0,1,2,1] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,4,2,2,1,2,1,2,0,0,1,2,2,1] |
| Phi of -K | [-3,-2,0,1,2,2,0,1,4,2,2,1,2,1,2,0,0,1,2,2,1] |
| Phi of K* | [-2,-2,-1,0,2,3,-1,2,0,1,2,2,1,2,2,0,2,4,1,1,0] |
| Phi of -K* | [-3,-2,0,1,2,2,1,2,0,3,3,1,1,2,3,1,1,2,-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 | -6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w |
| Inner characteristic polynomial | t^6+47t^4+17t^2 |
| Outer characteristic polynomial | t^7+69t^5+74t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -768*K1**4 - 128*K1**3*K3 + 384*K1**2*K2**5 - 3200*K1**2*K2**4 + 3232*K1**2*K2**3 - 3376*K1**2*K2**2 - 160*K1**2*K2*K4 + 3072*K1**2*K2 - 96*K1**2*K3**2 - 1892*K1**2 + 1856*K1*K2**3*K3 - 256*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1856*K1*K2*K3 + 160*K1*K3*K4 - 288*K2**6 + 64*K2**4*K4 - 1016*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 304*K2**2*K4 - 328*K2**2 + 16*K2*K3*K5 - 444*K3**2 - 70*K4**2 + 1340 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | False |