Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,1,3,4,0,0,1,3,0,0,1,0,-1,-1] |
Flat knots (up to 7 crossings) with same phi are :['6.1471', '7.31474'] |
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+59t^5+54t^3 |
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1471'] |
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 256*K1**4*K2**2 + 1280*K1**4*K2 - 4064*K1**4 + 192*K1**3*K2*K3 + 128*K1**2*K2**3 - 2048*K1**2*K2**2 + 5456*K1**2*K2 - 128*K1**2*K3**2 - 1464*K1**2 + 1472*K1*K2*K3 + 64*K1*K3*K4 - 64*K2**4 + 32*K2**2*K4 - 1936*K2**2 - 376*K3**2 - 24*K4**2 + 1990 |
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1471'] |
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4795', 'vk6.5132', 'vk6.6354', 'vk6.6794', 'vk6.8314', 'vk6.8760', 'vk6.9688', 'vk6.9999', 'vk6.21006', 'vk6.22428', 'vk6.28462', 'vk6.40226', 'vk6.42155', 'vk6.46728', 'vk6.48815', 'vk6.49036', 'vk6.49856', 'vk6.51511', 'vk6.58967', 'vk6.69803'] |
The R3 orbit of minmal crossing diagrams contains: |
The diagrammatic symmetry type of this knot is -. |
The reverse -K is |
The 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 | O1O2O3U2O4U1O5U6U5U4O6U3 |
R3 orbit | {'O1O2O3U2O4U1O5U6U5U4O6U3'} |
R3 orbit length | 1 |
Gauss code of -K | O1O2O3U1O4U5U6U4O6U3O5U2 |
Gauss code of K* | O1O2O3U1O4U5U6U4O6U3O5U2 |
Gauss code of -K* | Same |
Diagrammatic symmetry type | - |
Flat genus of the diagram | 2 |
If K is checkerboard colorable | False |
If K is almost classical | False |
Based matrix from Gauss code | [[ 0 -2 -1 2 2 1 -2],[ 2 0 0 3 1 0 1],[ 1 0 0 1 0 0 1],[-2 -3 -1 0 1 1 -4],[-2 -1 0 -1 0 0 -3],[-1 0 0 -1 0 0 -1],[ 2 -1 -1 4 3 1 0]] |
Primitive based matrix | [[ 0 2 2 1 -1 -2 -2],[-2 0 1 1 -1 -3 -4],[-2 -1 0 0 0 -1 -3],[-1 -1 0 0 0 0 -1],[ 1 1 0 0 0 0 1],[ 2 3 1 0 0 0 1],[ 2 4 3 1 -1 -1 0]] |
If based matrix primitive | True |
Phi of primitive based matrix | [-2,-2,-1,1,2,2,-1,-1,1,3,4,0,0,1,3,0,0,1,0,-1,-1] |
Phi over symmetry | [-2,-2,-1,1,2,2,-1,-1,1,3,4,0,0,1,3,0,0,1,0,-1,-1] |
Phi of -K | [-2,-2,-1,1,2,2,-1,1,3,1,3,2,2,0,1,2,2,3,2,1,-1] |
Phi of K* | [-2,-2,-1,1,2,2,-1,1,3,1,3,2,2,0,1,2,2,3,2,1,-1] |
Phi of -K* | [-2,-2,-1,1,2,2,-1,-1,1,3,4,0,0,1,3,0,0,1,0,-1,-1] |
Symmetry type of based matrix | - |
u-polynomial | 0 |
Normalized Jones-Krushkal polynomial | 17z+35 |
Enhanced Jones-Krushkal polynomial | 17w^2z+35w |
Inner characteristic polynomial | t^6+41t^4+36t^2 |
Outer characteristic polynomial | t^7+59t^5+54t^3 |
Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
2-strand cable arrow polynomial | -256*K1**6 - 256*K1**4*K2**2 + 1280*K1**4*K2 - 4064*K1**4 + 192*K1**3*K2*K3 + 128*K1**2*K2**3 - 2048*K1**2*K2**2 + 5456*K1**2*K2 - 128*K1**2*K3**2 - 1464*K1**2 + 1472*K1*K2*K3 + 64*K1*K3*K4 - 64*K2**4 + 32*K2**2*K4 - 1936*K2**2 - 376*K3**2 - 24*K4**2 + 1990 |
Genus of based matrix | 0 |
Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
If K is slice | True |