| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,1,2,3,-1,1,0,0,0,1,0,0,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1886'] |
| Arrow polynomial of the knot is: -10*K1**2 + 5*K2 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.947', '6.1027', '6.1399', '6.1430', '6.1433', '6.1442', '6.1465', '6.1469', '6.1476', '6.1505', '6.1529', '6.1606', '6.1612', '6.1613', '6.1616', '6.1649', '6.1694', '6.1736', '6.1768', '6.1771', '6.1774', '6.1884', '6.1886', '6.1887', '6.1889', '6.1960', '6.1962'] |
| Outer characteristic polynomial of the knot is: t^7+23t^5+43t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1886'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 768*K1**4*K2 - 3728*K1**4 + 224*K1**3*K2*K3 - 800*K1**3*K3 + 32*K1**2*K2**3 - 4016*K1**2*K2**2 - 32*K1**2*K2*K4 + 9208*K1**2*K2 - 368*K1**2*K3**2 - 4968*K1**2 - 128*K1*K2**2*K3 + 4984*K1*K2*K3 + 352*K1*K3*K4 - 88*K2**4 + 104*K2**2*K4 - 3856*K2**2 - 1400*K3**2 - 102*K4**2 + 3940 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1886'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16555', 'vk6.16648', 'vk6.18141', 'vk6.18475', 'vk6.22958', 'vk6.23079', 'vk6.24600', 'vk6.25011', 'vk6.34947', 'vk6.35068', 'vk6.36739', 'vk6.37156', 'vk6.42520', 'vk6.42631', 'vk6.44011', 'vk6.44321', 'vk6.54802', 'vk6.54886', 'vk6.55955', 'vk6.56253', 'vk6.59234', 'vk6.59312', 'vk6.60493', 'vk6.60857', 'vk6.64776', 'vk6.64841', 'vk6.65620', 'vk6.65925', 'vk6.68078', 'vk6.68143', 'vk6.68695', 'vk6.68904'] |
| 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 | O1O2O3U1U4O5U3O4U2O6U5U6 |
| R3 orbit | {'O1O2O3U1U4O5U3O4U2O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U2O6U1O5U6U3 |
| Gauss code of K* | O1O2U3O4U2O5U6O3O6U1U5U4 |
| Gauss code of -K* | O1O2U1O3U4O5U2O4O6U5U3U6 |
| 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 0 1 0 0 1],[ 2 0 2 1 1 2 0],[ 0 -2 0 1 -1 1 1],[-1 -1 -1 0 -1 0 1],[ 0 -1 1 1 0 0 0],[ 0 -2 -1 0 0 0 1],[-1 0 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 0 -1 0],[ 0 0 1 0 0 -1 -2],[ 0 1 0 0 0 1 -1],[ 0 1 1 1 -1 0 -2],[ 2 1 0 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,-1,0,1,1,1,1,0,1,0,0,1,2,-1,1,2] |
| Phi over symmetry | [-2,0,0,0,1,1,0,0,1,2,3,-1,1,0,0,0,1,0,0,1,-1] |
| Phi of -K | [-2,0,0,0,1,1,0,0,1,2,3,-1,1,0,0,0,1,0,0,1,-1] |
| Phi of K* | [-1,-1,0,0,0,2,-1,0,0,1,3,0,1,0,2,1,-1,0,0,0,1] |
| Phi of -K* | [-2,0,0,0,1,1,1,2,2,0,1,0,1,0,1,-1,1,0,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | z^2+20z+37 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+20w^2z+37w |
| Inner characteristic polynomial | t^6+17t^4+22t^2+1 |
| Outer characteristic polynomial | t^7+23t^5+43t^3+6t |
| Flat arrow polynomial | -10*K1**2 + 5*K2 + 6 |
| 2-strand cable arrow polynomial | -64*K1**6 - 64*K1**4*K2**2 + 768*K1**4*K2 - 3728*K1**4 + 224*K1**3*K2*K3 - 800*K1**3*K3 + 32*K1**2*K2**3 - 4016*K1**2*K2**2 - 32*K1**2*K2*K4 + 9208*K1**2*K2 - 368*K1**2*K3**2 - 4968*K1**2 - 128*K1*K2**2*K3 + 4984*K1*K2*K3 + 352*K1*K3*K4 - 88*K2**4 + 104*K2**2*K4 - 3856*K2**2 - 1400*K3**2 - 102*K4**2 + 3940 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |