| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,1,3,1,0,1,1,1,1,3,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.686'] |
| Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863'] |
| Outer characteristic polynomial of the knot is: t^7+40t^5+34t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.686'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 256*K1**4*K2 - 1440*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**3*K3 + 192*K1**2*K2**3 - 1408*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 64*K1**2*K2*K4 + 4200*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 2732*K1**2 + 64*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2728*K1*K2*K3 + 472*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 320*K2**4 - 32*K2**3*K6 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 432*K2**2*K4 - 2030*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 904*K3**2 - 172*K4**2 - 12*K5**2 - 2*K6**2 + 2090 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.686'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4222', 'vk6.4302', 'vk6.5484', 'vk6.5595', 'vk6.7583', 'vk6.7676', 'vk6.9084', 'vk6.9164', 'vk6.11165', 'vk6.12253', 'vk6.12360', 'vk6.19375', 'vk6.19668', 'vk6.19787', 'vk6.26155', 'vk6.26224', 'vk6.26571', 'vk6.26669', 'vk6.30755', 'vk6.31302', 'vk6.31697', 'vk6.31960', 'vk6.32460', 'vk6.32875', 'vk6.38155', 'vk6.38200', 'vk6.39096', 'vk6.41352', 'vk6.44812', 'vk6.44945', 'vk6.45852', 'vk6.48532', 'vk6.49337', 'vk6.52294', 'vk6.53138', 'vk6.58444', 'vk6.62968', 'vk6.63587', 'vk6.66311', 'vk6.66340'] |
| 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 | O1O2O3O4U3O5U2U4O6U5U1U6 |
| R3 orbit | {'O1O2O3O4U3O5U2U4O6U5U1U6', 'O1O2O3O4U3U1O5U4O6U2U5U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U4U6O5U1U3O6U2 |
| Gauss code of K* | O1O2O3U2U4U5U6O5U1O4O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U4U6U5U2 |
| 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 -1 -2 -1 1 1 2],[ 1 0 -2 -1 1 2 2],[ 2 2 0 0 2 2 1],[ 1 1 0 0 1 1 0],[-1 -1 -2 -1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-2 -2 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 0 -2 -1],[-1 1 0 1 -1 -1 -2],[-1 1 -1 0 -1 -2 -2],[ 1 0 1 1 0 1 0],[ 1 2 1 2 -1 0 -2],[ 2 1 2 2 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,1,1,0,2,1,-1,1,1,2,1,2,2,-1,0,2] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,1,1,1,3,1,0,1,1,1,1,3,1,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,1,1,1,3,1,0,1,1,1,1,3,1,0,0] |
| Phi of K* | [-2,-1,-1,1,1,2,0,0,1,3,3,-1,0,1,1,1,1,1,-1,-1,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,2,2,2,1,1,1,1,0,1,2,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+28t^4+8t^2 |
| Outer characteristic polynomial | t^7+40t^5+34t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 + 256*K1**4*K2 - 1440*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**3*K3 + 192*K1**2*K2**3 - 1408*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 64*K1**2*K2*K4 + 4200*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 2732*K1**2 + 64*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2728*K1*K2*K3 + 472*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 320*K2**4 - 32*K2**3*K6 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 432*K2**2*K4 - 2030*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 904*K3**2 - 172*K4**2 - 12*K5**2 - 2*K6**2 + 2090 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |