| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1569'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+29t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1569'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 896*K1**4*K2**2 + 3200*K1**4*K2 - 4944*K1**4 - 384*K1**3*K2**2*K3 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 + 384*K1**2*K2**5 - 960*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2752*K1**2*K2**3 + 608*K1**2*K2**2*K4 - 7312*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 7616*K1**2*K2 - 1072*K1**2*K3**2 - 96*K1**2*K3*K5 - 336*K1**2*K4**2 - 3000*K1**2 + 1056*K1*K2**3*K3 - 896*K1*K2**2*K3 - 480*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6360*K1*K2*K3 + 1560*K1*K3*K4 + 384*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1208*K2**4 - 32*K2**3*K6 - 416*K2**2*K3**2 - 112*K2**2*K4**2 + 1032*K2**2*K4 - 2518*K2**2 + 360*K2*K3*K5 + 24*K2*K4*K6 - 1504*K3**2 - 598*K4**2 - 160*K5**2 - 2*K6**2 + 3348 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1569'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16351', 'vk6.16357', 'vk6.16392', 'vk6.16398', 'vk6.19180', 'vk6.19181', 'vk6.19474', 'vk6.19475', 'vk6.22777', 'vk6.22781', 'vk6.25981', 'vk6.25982', 'vk6.26372', 'vk6.26373', 'vk6.34642', 'vk6.34648', 'vk6.34730', 'vk6.34738', 'vk6.38074', 'vk6.38075', 'vk6.42358', 'vk6.42361', 'vk6.44558', 'vk6.44559', 'vk6.54616', 'vk6.54618', 'vk6.56525', 'vk6.56526', 'vk6.59153', 'vk6.59157', 'vk6.66254', 'vk6.66255'] |
| 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 | O1O2O3U4O5U1U3O6O4U6U5U2 |
| R3 orbit | {'O1O2O3U4O5U1U3O6O4U6U5U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U4U5O6O5U1U3O4U6 |
| Gauss code of K* | O1O2O3U4U3U5O6U2O4O5U1U6 |
| Gauss code of -K* | O1O2O3U4U3O5O6U2O4U5U1U6 |
| 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 0 1 -1],[ 2 0 2 1 1 1 -1],[-1 -2 0 0 0 0 -1],[-1 -1 0 0 -1 0 -1],[ 0 -1 0 1 0 0 -1],[-1 -1 0 0 0 0 0],[ 1 1 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 0 0 1 0 -1 -1],[ 1 0 1 1 1 0 1],[ 2 1 2 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,0,0,1,0,0,1,2,1,1,1,1,1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,2,1,1,2,2,0,1,1,2,1,0,1,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,1,2,0,1,1,1,1,2,2,0,1,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+6w^3z^2-2w^3z+23w^2z+27w |
| Inner characteristic polynomial | t^6+12t^4+12t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+29t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -64*K1**6 - 896*K1**4*K2**2 + 3200*K1**4*K2 - 4944*K1**4 - 384*K1**3*K2**2*K3 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 + 384*K1**2*K2**5 - 960*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2752*K1**2*K2**3 + 608*K1**2*K2**2*K4 - 7312*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 7616*K1**2*K2 - 1072*K1**2*K3**2 - 96*K1**2*K3*K5 - 336*K1**2*K4**2 - 3000*K1**2 + 1056*K1*K2**3*K3 - 896*K1*K2**2*K3 - 480*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6360*K1*K2*K3 + 1560*K1*K3*K4 + 384*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1208*K2**4 - 32*K2**3*K6 - 416*K2**2*K3**2 - 112*K2**2*K4**2 + 1032*K2**2*K4 - 2518*K2**2 + 360*K2*K3*K5 + 24*K2*K4*K6 - 1504*K3**2 - 598*K4**2 - 160*K5**2 - 2*K6**2 + 3348 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |