| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,3,1,0,1,0,0,1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1024', '7.35701', '7.35817'] |
| Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078'] |
| Outer characteristic polynomial of the knot is: t^7+28t^5+44t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1024', '7.35817', '7.36025'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 704*K1**4*K2**2 + 3808*K1**4*K2 - 8144*K1**4 + 1632*K1**3*K2*K3 - 1568*K1**3*K3 - 576*K1**2*K2**4 + 4000*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 14864*K1**2*K2**2 - 1568*K1**2*K2*K4 + 13000*K1**2*K2 - 1008*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 1720*K1**2 + 1632*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 480*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9544*K1*K2*K3 - 32*K1*K2*K4*K5 + 1056*K1*K3*K4 + 152*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 3552*K2**4 - 32*K2**3*K6 - 784*K2**2*K3**2 - 96*K2**2*K4**2 + 2512*K2**2*K4 - 1868*K2**2 + 464*K2*K3*K5 + 64*K2*K4*K6 - 1300*K3**2 - 408*K4**2 - 76*K5**2 - 12*K6**2 + 3326 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1024'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.508', 'vk6.600', 'vk6.621', 'vk6.1007', 'vk6.1105', 'vk6.1128', 'vk6.1664', 'vk6.1838', 'vk6.2174', 'vk6.2182', 'vk6.2283', 'vk6.2309', 'vk6.2787', 'vk6.2888', 'vk6.3065', 'vk6.3190', 'vk6.5264', 'vk6.6521', 'vk6.8901', 'vk6.9818', 'vk6.20819', 'vk6.21049', 'vk6.22215', 'vk6.22471', 'vk6.28498', 'vk6.29780', 'vk6.39871', 'vk6.40274', 'vk6.46426', 'vk6.46913', 'vk6.49136', 'vk6.58830'] |
| 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 | O1O2O3O4U3U4O5U2O6U1U6U5 |
| R3 orbit | {'O1O2O3O4U3U4O5U2O6U1U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6U4O6U3O5U1U2 |
| Gauss code of K* | O1O2O3U1U4U5U6O5O6U3O4U2 |
| Gauss code of -K* | O1O2O3U2O4U1O5O6U5U6U4U3 |
| 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 -1 -1 1 2 1],[ 2 0 0 -1 1 3 1],[ 1 0 0 -1 1 1 0],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[-2 -3 -1 0 0 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 0 -1 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[ 1 0 0 1 0 1 1],[ 1 1 0 1 -1 0 0],[ 2 3 1 1 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,0,0,1,3,0,0,0,1,1,1,1,-1,-1,0] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,1,3,1,0,1,0,0,1,1,0,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,1,2,2,2,1,1,1,2,2,1,2,3,0,1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,1,1,2,3,1,0,1,1,2,2,2,2,-1,1,2] |
| Phi of -K* | [-2,-1,-1,1,1,2,-1,0,1,1,3,1,0,1,0,0,1,1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+24z+33 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+24w^2z+33w |
| Inner characteristic polynomial | t^6+16t^4+22t^2+1 |
| Outer characteristic polynomial | t^7+28t^5+44t^3+7t |
| Flat arrow polynomial | 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7 |
| 2-strand cable arrow polynomial | -256*K1**6 - 704*K1**4*K2**2 + 3808*K1**4*K2 - 8144*K1**4 + 1632*K1**3*K2*K3 - 1568*K1**3*K3 - 576*K1**2*K2**4 + 4000*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 14864*K1**2*K2**2 - 1568*K1**2*K2*K4 + 13000*K1**2*K2 - 1008*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 1720*K1**2 + 1632*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 480*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9544*K1*K2*K3 - 32*K1*K2*K4*K5 + 1056*K1*K3*K4 + 152*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 3552*K2**4 - 32*K2**3*K6 - 784*K2**2*K3**2 - 96*K2**2*K4**2 + 2512*K2**2*K4 - 1868*K2**2 + 464*K2*K3*K5 + 64*K2*K4*K6 - 1300*K3**2 - 408*K4**2 - 76*K5**2 - 12*K6**2 + 3326 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}]] |
| If K is slice | True |