| Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,0,1,0,0,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1997', '7.45001', '7.45082'] |
| Arrow polynomial of the knot is: -8*K1**2 + 4*K2 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.668', '6.711', '6.777', '6.803', '6.828', '6.1015', '6.1032', '6.1055', '6.1082', '6.1132', '6.1264', '6.1288', '6.1333', '6.1391', '6.1395', '6.1396', '6.1400', '6.1404', '6.1405', '6.1419', '6.1471', '6.1473', '6.1536', '6.1563', '6.1611', '6.1618', '6.1623', '6.1627', '6.1629', '6.1631', '6.1695', '6.1700', '6.1731', '6.1740', '6.1767', '6.1773', '6.1790', '6.1792', '6.1796', '6.1848', '6.1899', '6.1901', '6.1937', '6.1954', '6.1955', '6.1958', '6.1964', '6.1975', '6.1997', '6.1998', '6.1999', '6.2002', '6.2003', '6.2004', '6.2005', '6.2007', '6.2008', '6.2009', '6.2010', '6.2011', '6.2013', '6.2018', '6.2019', '6.2021', '6.2034', '6.2039', '6.2043', '6.2046', '6.2050', '6.2051', '6.2057', '6.2063'] |
| Outer characteristic polynomial of the knot is: t^7+18t^5+31t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1997', '7.45082'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 2304*K1**4*K2 - 7536*K1**4 + 512*K1**3*K2*K3 - 704*K1**3*K3 + 1568*K1**2*K2**3 - 8288*K1**2*K2**2 - 480*K1**2*K2*K4 + 10336*K1**2*K2 - 48*K1**2*K3**2 - 672*K1**2 - 736*K1*K2**2*K3 + 5104*K1*K2*K3 + 80*K1*K3*K4 - 976*K2**4 + 712*K2**2*K4 - 2088*K2**2 - 608*K3**2 - 60*K4**2 + 2410 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1997'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.501', 'vk6.592', 'vk6.619', 'vk6.1001', 'vk6.1098', 'vk6.1122', 'vk6.1670', 'vk6.1842', 'vk6.2167', 'vk6.2184', 'vk6.2274', 'vk6.2311', 'vk6.2796', 'vk6.2893', 'vk6.3069', 'vk6.3197', 'vk6.5241', 'vk6.6496', 'vk6.8878', 'vk6.9793', 'vk6.20812', 'vk6.21047', 'vk6.22207', 'vk6.22469', 'vk6.28492', 'vk6.29773', 'vk6.39864', 'vk6.40280', 'vk6.46422', 'vk6.46932', 'vk6.49129', 'vk6.58834'] |
| 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 | O1O2U1O3U2O4U5U6O5O6U3U4 |
| R3 orbit | {'O1O2U1O3U2O4U5U6O5O6U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2U3U4O5O6U5U6O3U1O4U2 |
| Gauss code of K* | O1O2U1U2O3O4U5U6O5U3O6U4 |
| Gauss code of -K* | O1O2U3O4U5O6U4U6O3O5U1U2 |
| 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 0 0 1 -1 1],[ 1 0 1 1 0 1 1],[ 0 -1 0 1 1 0 0],[ 0 -1 -1 0 1 -1 1],[-1 0 -1 -1 0 -2 0],[ 1 -1 0 1 2 0 1],[-1 -1 0 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[-1 0 0 -1 -1 0 -2],[ 0 0 1 0 1 -1 0],[ 0 1 1 -1 0 -1 -1],[ 1 1 0 1 1 0 1],[ 1 1 2 0 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,1,1,0,0,1,1,1,1,1,0,2,-1,1,0,1,1,-1] |
| Phi over symmetry | [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,0,1,0,0,1,0,0] |
| Phi of -K | [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,0,1,0,0,1,0,0] |
| Phi of K* | [-1,-1,0,0,1,1,0,0,0,0,2,0,1,1,1,-1,0,0,1,0,-1] |
| Phi of -K* | [-1,-1,0,0,1,1,-1,0,1,1,2,1,1,1,0,1,0,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+14t^4+19t^2+4 |
| Outer characteristic polynomial | t^7+18t^5+31t^3+8t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -256*K1**6 - 576*K1**4*K2**2 + 2304*K1**4*K2 - 7536*K1**4 + 512*K1**3*K2*K3 - 704*K1**3*K3 + 1568*K1**2*K2**3 - 8288*K1**2*K2**2 - 480*K1**2*K2*K4 + 10336*K1**2*K2 - 48*K1**2*K3**2 - 672*K1**2 - 736*K1*K2**2*K3 + 5104*K1*K2*K3 + 80*K1*K3*K4 - 976*K2**4 + 712*K2**2*K4 - 2088*K2**2 - 608*K3**2 - 60*K4**2 + 2410 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | True |