| Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-2,0,2,3,3,0,3,1,2,2,1,2,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1188'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+45t^5+91t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1188'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4 + 448*K1**3*K2*K3 - 288*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1472*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 2744*K1**2*K2 - 1888*K1**2*K3**2 - 64*K1**2*K3*K5 - 4240*K1**2 + 288*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 - 32*K1*K2**2*K5 + 544*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 6272*K1*K2*K3 - 32*K1*K3**2*K5 + 2392*K1*K3*K4 + 192*K1*K4*K5 + 40*K1*K5*K6 - 144*K2**4 - 992*K2**2*K3**2 - 80*K2**2*K4**2 + 456*K2**2*K4 - 3236*K2**2 - 288*K2*K3**2*K4 + 784*K2*K3*K5 + 160*K2*K4*K6 - 320*K3**4 + 304*K3**2*K6 - 2816*K3**2 - 868*K4**2 - 264*K5**2 - 92*K6**2 + 3802 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1188'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4659', 'vk6.4948', 'vk6.6113', 'vk6.6602', 'vk6.8126', 'vk6.8530', 'vk6.9512', 'vk6.9869', 'vk6.20373', 'vk6.21714', 'vk6.27685', 'vk6.29229', 'vk6.39125', 'vk6.41379', 'vk6.45869', 'vk6.47530', 'vk6.48691', 'vk6.48896', 'vk6.49447', 'vk6.49668', 'vk6.50707', 'vk6.50908', 'vk6.51194', 'vk6.51397', 'vk6.57242', 'vk6.58467', 'vk6.61868', 'vk6.63003', 'vk6.66869', 'vk6.67737', 'vk6.69493', 'vk6.70215'] |
| 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 | O1O2O3O4U2U1U3O5O6U5U4U6 |
| R3 orbit | {'O1O2O3O4U2U1U3O5O6U5U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U6O5O6U2U4U3 |
| Gauss code of K* | O1O2O3U4U5O4O6O5U2U1U3U6 |
| Gauss code of -K* | O1O2O3U1U3O4O5O6U2U4U6U5 |
| 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 -2 -2 1 2 -1 2],[ 2 0 0 2 3 0 1],[ 2 0 0 1 2 0 1],[-1 -2 -1 0 1 0 1],[-2 -3 -2 -1 0 0 2],[ 1 0 0 0 0 0 1],[-2 -1 -1 -1 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -2 -2],[-2 0 2 -1 0 -2 -3],[-2 -2 0 -1 -1 -1 -1],[-1 1 1 0 0 -1 -2],[ 1 0 1 0 0 0 0],[ 2 2 1 1 0 0 0],[ 2 3 1 2 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,2,2,-2,1,0,2,3,1,1,1,1,0,1,2,0,0,0] |
| Phi over symmetry | [-2,-2,-1,1,2,2,-2,0,2,3,3,0,3,1,2,2,1,2,1,1,0] |
| Phi of -K | [-2,-2,-1,1,2,2,0,1,1,1,3,1,2,2,3,2,3,2,0,0,-2] |
| Phi of K* | [-2,-2,-1,1,2,2,-2,0,2,3,3,0,3,1,2,2,1,2,1,1,0] |
| Phi of -K* | [-2,-2,-1,1,2,2,0,0,1,1,2,0,2,1,3,0,1,0,1,1,-2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+25z+31 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+25w^2z+31w |
| Inner characteristic polynomial | t^6+27t^4+27t^2+1 |
| Outer characteristic polynomial | t^7+45t^5+91t^3+7t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -320*K1**4 + 448*K1**3*K2*K3 - 288*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1472*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 2744*K1**2*K2 - 1888*K1**2*K3**2 - 64*K1**2*K3*K5 - 4240*K1**2 + 288*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 - 32*K1*K2**2*K5 + 544*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 6272*K1*K2*K3 - 32*K1*K3**2*K5 + 2392*K1*K3*K4 + 192*K1*K4*K5 + 40*K1*K5*K6 - 144*K2**4 - 992*K2**2*K3**2 - 80*K2**2*K4**2 + 456*K2**2*K4 - 3236*K2**2 - 288*K2*K3**2*K4 + 784*K2*K3*K5 + 160*K2*K4*K6 - 320*K3**4 + 304*K3**2*K6 - 2816*K3**2 - 868*K4**2 - 264*K5**2 - 92*K6**2 + 3802 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}]] |
| If K is slice | True |