| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,3,2,1,0,1,0,1,0,1,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1440'] |
| Arrow polynomial of the knot is: -4*K1**2 + 2*K2 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.5', '4.7', '4.10', '4.11', '6.142', '6.563', '6.606', '6.788', '6.892', '6.944', '6.949', '6.971', '6.1011', '6.1060', '6.1124', '6.1212', '6.1238', '6.1241', '6.1274', '6.1291', '6.1304', '6.1309', '6.1312', '6.1373', '6.1390', '6.1392', '6.1393', '6.1394', '6.1403', '6.1407', '6.1412', '6.1413', '6.1423', '6.1424', '6.1425', '6.1426', '6.1438', '6.1440', '6.1448', '6.1449', '6.1452', '6.1453', '6.1456', '6.1457', '6.1478', '6.1479', '6.1520', '6.1554', '6.1559', '6.1588', '6.1589', '6.1609', '6.1610', '6.1619', '6.1621', '6.1626', '6.1630', '6.1632', '6.1633', '6.1643', '6.1657', '6.1689', '6.1721', '6.1723', '6.1737', '6.1764', '6.1777', '6.1783', '6.1808', '6.1816', '6.1853', '6.1855', '6.1856', '6.1860', '6.1864', '6.1871', '6.1872', '6.1875', '6.1882', '6.1891', '6.1894', '6.1895', '6.1896', '6.1897', '6.1898', '6.1900', '6.1902', '6.1903', '6.1938', '6.1940', '6.1942', '6.1946', '6.1947', '6.1952', '6.1956', '6.1957', '6.1959', '6.1965', '6.1968', '6.1969', '6.1970', '6.1972', '6.1973', '6.1974', '6.2000', '6.2006', '6.2012', '6.2032', '6.2033', '6.2035', '6.2036', '6.2037', '6.2038', '6.2040', '6.2041', '6.2042', '6.2044', '6.2045', '6.2047', '6.2048', '6.2049', '6.2052', '6.2053', '6.2054', '6.2055', '6.2058', '6.2060', '6.2061', '6.2062', '6.2067', '6.2069', '6.2072', '6.2073', '6.2076', '6.2077', '6.2080'] |
| Outer characteristic polynomial of the knot is: t^7+35t^5+108t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1440'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 352*K1**4*K2 - 688*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 + 288*K1**2*K2**3 - 2416*K1**2*K2**2 - 32*K1**2*K2*K4 + 3256*K1**2*K2 - 16*K1**2*K3**2 - 1860*K1**2 - 224*K1*K2**2*K3 + 1720*K1*K2*K3 + 8*K1*K3*K4 - 112*K2**4 + 104*K2**2*K4 - 1224*K2**2 - 292*K3**2 - 4*K4**2 + 1234 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1440'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10134', 'vk6.10189', 'vk6.10334', 'vk6.10423', 'vk6.17662', 'vk6.17711', 'vk6.24225', 'vk6.24274', 'vk6.29913', 'vk6.29954', 'vk6.30019', 'vk6.30074', 'vk6.36493', 'vk6.36589', 'vk6.43586', 'vk6.43698', 'vk6.51622', 'vk6.51661', 'vk6.51712', 'vk6.51729', 'vk6.55700', 'vk6.55759', 'vk6.60266', 'vk6.60330', 'vk6.63335', 'vk6.63364', 'vk6.63386', 'vk6.63404', 'vk6.65408', 'vk6.65451', 'vk6.68546', 'vk6.68579'] |
| 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 | O1O2O3U4O5U1O4U6U5O6U2U3 |
| R3 orbit | {'O1O2O3U4O5U1O4U6U5O6U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2O4U5U4O6U3O5U6 |
| Gauss code of K* | O1O2U1O3O4U5U3U4O6U2O5U6 |
| Gauss code of -K* | O1O2U3O4O3U5O6U4O5U1U2U6 |
| 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 2 0 1 -1],[ 2 0 1 2 2 0 1],[ 0 -1 0 1 1 0 -1],[-2 -2 -1 0 -1 0 -3],[ 0 -2 -1 1 0 1 0],[-1 0 0 0 -1 0 -1],[ 1 -1 1 3 0 1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -3 -2],[-1 0 0 0 -1 -1 0],[ 0 1 0 0 1 -1 -1],[ 0 1 1 -1 0 0 -2],[ 1 3 1 1 0 0 -1],[ 2 2 0 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,1,3,2,0,1,1,0,-1,1,1,0,2,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,3,2,1,0,1,0,1,0,1,1,1,1] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,3,2,1,0,1,0,1,0,1,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,1,0,2,0,1,1,3,-1,1,0,0,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,0,2,1,0,1,3,1,0,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+22z+25 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+22w^2z+25w |
| Inner characteristic polynomial | t^6+25t^4+72t^2+1 |
| Outer characteristic polynomial | t^7+35t^5+108t^3+5t |
| Flat arrow polynomial | -4*K1**2 + 2*K2 + 3 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 352*K1**4*K2 - 688*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 + 288*K1**2*K2**3 - 2416*K1**2*K2**2 - 32*K1**2*K2*K4 + 3256*K1**2*K2 - 16*K1**2*K3**2 - 1860*K1**2 - 224*K1*K2**2*K3 + 1720*K1*K2*K3 + 8*K1*K3*K4 - 112*K2**4 + 104*K2**2*K4 - 1224*K2**2 - 292*K3**2 - 4*K4**2 + 1234 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {1, 5}, {3, 4}, {2}]] |
| If K is slice | False |