| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,1,0,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.381'] |
| 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+42t^5+49t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.381'] |
| 2-strand cable arrow polynomial of the knot is: -32*K1**4 - 192*K1**2*K2**4 + 480*K1**2*K2**3 - 2496*K1**2*K2**2 - 128*K1**2*K2*K4 + 2720*K1**2*K2 - 1900*K1**2 + 224*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1992*K1*K2*K3 + 72*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 488*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 448*K2**2*K4 - 1064*K2**2 + 8*K2*K3*K5 - 380*K3**2 - 90*K4**2 + 1200 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.381'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10505', 'vk6.10509', 'vk6.10570', 'vk6.10578', 'vk6.10757', 'vk6.10765', 'vk6.10880', 'vk6.10884', 'vk6.17688', 'vk6.17690', 'vk6.17737', 'vk6.17739', 'vk6.24298', 'vk6.24300', 'vk6.24777', 'vk6.25236', 'vk6.30190', 'vk6.30194', 'vk6.30257', 'vk6.30265', 'vk6.30384', 'vk6.30392', 'vk6.30638', 'vk6.30732', 'vk6.36526', 'vk6.36967', 'vk6.43632', 'vk6.43634', 'vk6.43738', 'vk6.43740', 'vk6.52724', 'vk6.52830', 'vk6.60362', 'vk6.60364', 'vk6.60632', 'vk6.60969', 'vk6.63450', 'vk6.63454', 'vk6.65423', 'vk6.65769'] |
| 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 | O1O2O3O4O5U4U1O6U5U6U2U3 |
| R3 orbit | {'O1O2O3O4U5U2O6U4U6U1O5U3', 'O1O2O3O4O5U4U1O6U5U6U2U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U4U6U1O6U5U2 |
| Gauss code of K* | O1O2O3O4U5U3U4U6U1O6O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U4U6U1U2U5 |
| 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 0 2 -1 1 1],[ 3 0 2 3 0 2 1],[ 0 -2 0 1 -1 0 1],[-2 -3 -1 0 -1 0 1],[ 1 0 1 1 0 1 1],[-1 -2 0 0 -1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 1 0 -1 -1 -3],[-1 -1 0 -1 -1 -1 -1],[-1 0 1 0 0 -1 -2],[ 0 1 1 0 0 -1 -2],[ 1 1 1 1 1 0 0],[ 3 3 1 2 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,-1,0,1,1,3,1,1,1,1,0,1,2,1,2,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,1,0,1,-1,-1,0] |
| Phi of -K | [-3,-1,0,1,1,2,2,1,2,3,2,0,1,1,2,1,0,1,-1,1,2] |
| Phi of K* | [-2,-1,-1,0,1,3,1,2,1,2,2,1,1,1,2,0,1,3,0,1,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,1,0,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+26t^4+16t^2+1 |
| Outer characteristic polynomial | t^7+42t^5+49t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -32*K1**4 - 192*K1**2*K2**4 + 480*K1**2*K2**3 - 2496*K1**2*K2**2 - 128*K1**2*K2*K4 + 2720*K1**2*K2 - 1900*K1**2 + 224*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1992*K1*K2*K3 + 72*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 488*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 448*K2**2*K4 - 1064*K2**2 + 8*K2*K3*K5 - 380*K3**2 - 90*K4**2 + 1200 |
| 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 |