| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,1,2,2,0,2,1,2,0,1,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1596'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1030', '6.1062', '6.1226', '6.1508', '6.1525', '6.1596', '6.1724', '6.1729', '6.1735', '6.1738', '6.1789', '6.1809', '6.1921'] |
| Outer characteristic polynomial of the knot is: t^7+32t^5+116t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1596'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1248*K1**4*K2 - 2784*K1**4 + 384*K1**3*K2*K3 - 1568*K1**3*K3 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4528*K1**2*K2**2 - 1280*K1**2*K2*K4 + 8816*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K4**2 - 6148*K1**2 + 384*K1*K2**3*K3 - 1152*K1*K2**2*K3 - 288*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 8016*K1*K2*K3 + 1864*K1*K3*K4 + 320*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 824*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 128*K2**2*K4**2 + 2160*K2**2*K4 - 5258*K2**2 + 880*K2*K3*K5 + 104*K2*K4*K6 - 2700*K3**2 - 1178*K4**2 - 312*K5**2 - 22*K6**2 + 5112 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1596'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11496', 'vk6.11808', 'vk6.12826', 'vk6.13153', 'vk6.17065', 'vk6.17308', 'vk6.20904', 'vk6.21057', 'vk6.22314', 'vk6.22485', 'vk6.23791', 'vk6.28382', 'vk6.31257', 'vk6.31616', 'vk6.32830', 'vk6.35575', 'vk6.36028', 'vk6.40032', 'vk6.40300', 'vk6.42084', 'vk6.43274', 'vk6.46564', 'vk6.46763', 'vk6.48020', 'vk6.52259', 'vk6.53416', 'vk6.57710', 'vk6.57715', 'vk6.58894', 'vk6.59952', 'vk6.64428', 'vk6.69758'] |
| 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 | O1O2O3U4O5U6U3O6O4U2U1U5 |
| R3 orbit | {'O1O2O3U4O5U6U3O6O4U2U1U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U3U2O5O6U1U6O4U5 |
| Gauss code of K* | O1O2O3U2U1U4O5U3O6O4U6U5 |
| Gauss code of -K* | O1O2O3U4U5O6O5U1O4U6U3U2 |
| 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 -1 -1 1 0 2 -1],[ 1 0 0 2 0 2 0],[ 1 0 0 2 0 1 0],[-1 -2 -2 0 0 -1 -1],[ 0 0 0 0 0 2 -1],[-2 -2 -1 1 -2 0 -2],[ 1 0 0 1 1 2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 1 -2 -1 -2 -2],[-1 -1 0 0 -2 -1 -2],[ 0 2 0 0 0 -1 0],[ 1 1 2 0 0 0 0],[ 1 2 1 1 0 0 0],[ 1 2 2 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,-1,2,1,2,2,0,2,1,2,0,1,0,0,0,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,2,1,2,2,0,2,1,2,0,1,0,0,0,0] |
| Phi of -K | [-1,-1,-1,0,1,2,0,0,0,1,1,0,1,0,1,1,0,2,1,0,2] |
| Phi of K* | [-2,-1,0,1,1,1,2,0,1,1,2,1,0,1,0,1,0,1,0,0,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,0,0,0,2,1,0,0,2,2,1,1,2,0,2,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 7z^2+28z+29 |
| Enhanced Jones-Krushkal polynomial | 7w^3z^2+28w^2z+29w |
| Inner characteristic polynomial | t^6+24t^4+71t^2+4 |
| Outer characteristic polynomial | t^7+32t^5+116t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 1248*K1**4*K2 - 2784*K1**4 + 384*K1**3*K2*K3 - 1568*K1**3*K3 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4528*K1**2*K2**2 - 1280*K1**2*K2*K4 + 8816*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K4**2 - 6148*K1**2 + 384*K1*K2**3*K3 - 1152*K1*K2**2*K3 - 288*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 8016*K1*K2*K3 + 1864*K1*K3*K4 + 320*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 824*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 128*K2**2*K4**2 + 2160*K2**2*K4 - 5258*K2**2 + 880*K2*K3*K5 + 104*K2*K4*K6 - 2700*K3**2 - 1178*K4**2 - 312*K5**2 - 22*K6**2 + 5112 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}]] |
| If K is slice | False |