| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-2,1,0,3,2,1,1,1,1,0,1,1,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1197'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+39t^5+79t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1197'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 480*K1**4*K2 - 2112*K1**4 + 128*K1**3*K2**3*K3 + 864*K1**3*K2*K3 - 448*K1**3*K3 - 448*K1**2*K2**4 + 672*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 5952*K1**2*K2**2 - 1024*K1**2*K2*K4 + 8072*K1**2*K2 - 640*K1**2*K3**2 - 96*K1**2*K4**2 - 5936*K1**2 + 1632*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 64*K1*K2*K3*K4 + 7848*K1*K2*K3 - 64*K1*K2*K4*K5 + 1688*K1*K3*K4 + 264*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1336*K2**4 - 1664*K2**2*K3**2 - 400*K2**2*K4**2 + 1368*K2**2*K4 - 3790*K2**2 - 224*K2*K3**2*K4 + 568*K2*K3*K5 + 192*K2*K4*K6 + 24*K3**2*K6 - 2516*K3**2 - 910*K4**2 - 156*K5**2 - 18*K6**2 + 4860 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1197'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4675', 'vk6.4968', 'vk6.6145', 'vk6.6628', 'vk6.8152', 'vk6.8562', 'vk6.9536', 'vk6.9885', 'vk6.20699', 'vk6.22137', 'vk6.28228', 'vk6.29651', 'vk6.39688', 'vk6.41927', 'vk6.46264', 'vk6.47869', 'vk6.48707', 'vk6.48912', 'vk6.49479', 'vk6.49698', 'vk6.50735', 'vk6.50936', 'vk6.51210', 'vk6.51411', 'vk6.57630', 'vk6.58790', 'vk6.62314', 'vk6.63253', 'vk6.67108', 'vk6.67970', 'vk6.69704', 'vk6.70385'] |
| 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 | O1O2O3O4U2U3U1O5O6U5U4U6 |
| R3 orbit | {'O1O2O3O4U2U3U1O5O6U5U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U6O5O6U4U2U3 |
| Gauss code of K* | O1O2O3U4U5O4O6O5U3U1U2U6 |
| Gauss code of -K* | O1O2O3U1U3O4O5O6U2U5U6U4 |
| 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 -2 0 2 -1 2],[ 1 0 -1 1 3 0 1],[ 2 1 0 1 2 0 1],[ 0 -1 -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 0 -1 -1 -2],[-2 0 2 -1 0 -3 -2],[-2 -2 0 -1 -1 -1 -1],[ 0 1 1 0 0 -1 -1],[ 1 0 1 0 0 0 0],[ 1 3 1 1 0 0 -1],[ 2 2 1 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,1,1,2,-2,1,0,3,2,1,1,1,1,0,1,1,0,0,1] |
| Phi over symmetry | [-2,-2,0,1,1,2,-2,1,0,3,2,1,1,1,1,0,1,1,0,0,1] |
| Phi of -K | [-2,-1,-1,0,2,2,0,1,1,2,3,0,0,0,2,1,3,2,1,1,-2] |
| Phi of K* | [-2,-2,0,1,1,2,-2,1,2,2,3,1,0,3,2,0,1,1,0,0,1] |
| Phi of -K* | [-2,-1,-1,0,2,2,0,1,1,1,2,0,0,1,0,1,1,3,1,1,-2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+25t^4+28t^2 |
| Outer characteristic polynomial | t^7+39t^5+79t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -320*K1**4*K2**2 + 480*K1**4*K2 - 2112*K1**4 + 128*K1**3*K2**3*K3 + 864*K1**3*K2*K3 - 448*K1**3*K3 - 448*K1**2*K2**4 + 672*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 5952*K1**2*K2**2 - 1024*K1**2*K2*K4 + 8072*K1**2*K2 - 640*K1**2*K3**2 - 96*K1**2*K4**2 - 5936*K1**2 + 1632*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 64*K1*K2*K3*K4 + 7848*K1*K2*K3 - 64*K1*K2*K4*K5 + 1688*K1*K3*K4 + 264*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1336*K2**4 - 1664*K2**2*K3**2 - 400*K2**2*K4**2 + 1368*K2**2*K4 - 3790*K2**2 - 224*K2*K3**2*K4 + 568*K2*K3*K5 + 192*K2*K4*K6 + 24*K3**2*K6 - 2516*K3**2 - 910*K4**2 - 156*K5**2 - 18*K6**2 + 4860 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}]] |
| If K is slice | False |