| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,1,4,2,0,1,1,0,0,1,1,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1355'] |
| 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+44t^5+68t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.725', '6.1355'] |
| 2-strand cable arrow polynomial of the knot is: -32*K1**4 - 1600*K1**2*K2**4 + 2720*K1**2*K2**3 - 6432*K1**2*K2**2 - 224*K1**2*K2*K4 + 4888*K1**2*K2 - 3052*K1**2 + 992*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3616*K1*K2*K3 + 64*K1*K3*K4 - 288*K2**6 + 160*K2**4*K4 - 1544*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 728*K2**2*K4 - 824*K2**2 + 8*K2*K3*K5 - 548*K3**2 - 106*K4**2 + 1880 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1355'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10512', 'vk6.10516', 'vk6.10585', 'vk6.10593', 'vk6.10774', 'vk6.10782', 'vk6.10889', 'vk6.10893', 'vk6.17676', 'vk6.17678', 'vk6.17725', 'vk6.17727', 'vk6.24286', 'vk6.24288', 'vk6.30197', 'vk6.30201', 'vk6.30272', 'vk6.30280', 'vk6.30401', 'vk6.30409', 'vk6.36512', 'vk6.36514', 'vk6.43616', 'vk6.43618', 'vk6.43720', 'vk6.43722', 'vk6.60352', 'vk6.60354', 'vk6.63457', 'vk6.63461', 'vk6.65419', 'vk6.65421'] |
| 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 | O1O2O3U2O4O5U1U4U5O6U3U6 |
| R3 orbit | {'O1O2O3U2O4O5U1U4U5O6U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1O4U5U6U3O5O6U2 |
| Gauss code of K* | O1O2O3U4O5O4U1U6U5O6U2U3 |
| Gauss code of -K* | O1O2O3U1U2O4U5U4U3O6O5U6 |
| 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 -1 1 0 2 1],[ 3 0 0 4 1 2 1],[ 1 0 0 1 0 0 1],[-1 -4 -1 0 -1 1 1],[ 0 -1 0 1 0 1 0],[-2 -2 0 -1 -1 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 0 -2],[-1 0 0 -1 0 -1 -1],[-1 1 1 0 -1 -1 -4],[ 0 1 0 1 0 0 -1],[ 1 0 1 1 0 0 0],[ 3 2 1 4 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,1,1,0,2,1,0,1,1,1,1,4,0,1,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,1,1,4,2,0,1,1,0,0,1,1,-1,0,1] |
| Phi of -K | [-3,-1,0,1,1,2,2,2,0,3,3,1,1,1,3,0,1,1,-1,0,1] |
| Phi of K* | [-2,-1,-1,0,1,3,0,1,1,3,3,1,0,1,0,1,1,3,1,2,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,1,1,4,2,0,1,1,0,0,1,1,-1,0,1] |
| 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 | -4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w |
| Inner characteristic polynomial | t^6+28t^4+31t^2 |
| Outer characteristic polynomial | t^7+44t^5+68t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -32*K1**4 - 1600*K1**2*K2**4 + 2720*K1**2*K2**3 - 6432*K1**2*K2**2 - 224*K1**2*K2*K4 + 4888*K1**2*K2 - 3052*K1**2 + 992*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3616*K1*K2*K3 + 64*K1*K3*K4 - 288*K2**6 + 160*K2**4*K4 - 1544*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 728*K2**2*K4 - 824*K2**2 + 8*K2*K3*K5 - 548*K3**2 - 106*K4**2 + 1880 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |