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Flat knot 6.2036

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,2,2,0,1,1,1,-1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.2036']
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+14t^5+26t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2036']
2-strand cable arrow polynomial of the knot is: -832*K1**6 + 1600*K1**4*K2 - 5536*K1**4 - 640*K1**3*K3 - 1728*K1**2*K2**2 + 7600*K1**2*K2 - 32*K1**2*K3**2 - 1908*K1**2 + 1952*K1*K2*K3 + 80*K1*K3*K4 - 16*K2**4 + 32*K2**2*K4 - 2544*K2**2 - 556*K3**2 - 44*K4**2 + 2570
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2036']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4139', 'vk6.4172', 'vk6.5381', 'vk6.5414', 'vk6.7499', 'vk6.7528', 'vk6.9004', 'vk6.9037', 'vk6.12424', 'vk6.12457', 'vk6.13353', 'vk6.13572', 'vk6.13605', 'vk6.14261', 'vk6.14708', 'vk6.14729', 'vk6.15188', 'vk6.15868', 'vk6.15887', 'vk6.30825', 'vk6.30858', 'vk6.32013', 'vk6.32046', 'vk6.33077', 'vk6.33110', 'vk6.33842', 'vk6.34304', 'vk6.48497', 'vk6.50276', 'vk6.53541', 'vk6.53933', 'vk6.54264']
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 O1O2U1O3U4U2O5O4U5O6U3U6
R3 orbit {'O1O2U1O3U4U2O5O4U5O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2U3U4O3U5O6O5U1U6O4U2
Gauss code of K* O1O2U3U1O3U4O5O4U6U2O6U5
Gauss code of -K* O1O2U1O3U4U3O5O4U2O6U5U6
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 1 0 0 -1 1],[ 1 0 1 1 0 0 0],[-1 -1 0 0 -1 -1 1],[ 0 -1 0 0 1 -1 1],[ 0 0 1 -1 0 -1 1],[ 1 0 1 1 1 0 0],[-1 0 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 -1 0 0],[ 0 0 1 0 1 -1 -1],[ 0 1 1 -1 0 0 -1],[ 1 1 0 1 0 0 0],[ 1 1 0 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,0,0,-1,1,1,0,1,0]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,2,2,0,1,1,1,-1,0,1,0,0,0]
Phi of -K [-1,-1,0,0,1,1,0,0,0,1,2,0,1,1,2,-1,1,0,0,0,-1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,2,2,0,1,1,1,-1,0,1,0,0,0]
Phi of -K* [-1,-1,0,0,1,1,0,0,1,0,1,1,1,0,1,-1,1,1,1,0,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+10t^4+16t^2
Outer characteristic polynomial t^7+14t^5+26t^3+2t
Flat arrow polynomial -4*K1**2 + 2*K2 + 3
2-strand cable arrow polynomial -832*K1**6 + 1600*K1**4*K2 - 5536*K1**4 - 640*K1**3*K3 - 1728*K1**2*K2**2 + 7600*K1**2*K2 - 32*K1**2*K3**2 - 1908*K1**2 + 1952*K1*K2*K3 + 80*K1*K3*K4 - 16*K2**4 + 32*K2**2*K4 - 2544*K2**2 - 556*K3**2 - 44*K4**2 + 2570
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {3, 4}, {1, 2}]]
If K is slice True
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