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Flat knot 6.1002

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,0,1,2,1,0,2,3,1,-1,0,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1002']
Arrow polynomial of the knot is: -2*K1**2 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['4.6', '4.8', '6.780', '6.804', '6.914', '6.931', '6.946', '6.960', '6.1002', '6.1016', '6.1019', '6.1051', '6.1058', '6.1078', '6.1102', '6.1115', '6.1217', '6.1294', '6.1306', '6.1317', '6.1321', '6.1324', '6.1336', '6.1377', '6.1416', '6.1420', '6.1427', '6.1429', '6.1434', '6.1436', '6.1437', '6.1439', '6.1441', '6.1444', '6.1450', '6.1451', '6.1458', '6.1459', '6.1477', '6.1482', '6.1490', '6.1503', '6.1504', '6.1511', '6.1521', '6.1547', '6.1560', '6.1561', '6.1562', '6.1597', '6.1598', '6.1600', '6.1601', '6.1608', '6.1620', '6.1622', '6.1624', '6.1634', '6.1635', '6.1637', '6.1638', '6.1713', '6.1725', '6.1758', '6.1846', '6.1933', '6.1944', '6.1949', '6.1950', '6.1951']
Outer characteristic polynomial of the knot is: t^7+39t^5+71t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1002']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K2*K3 + 1024*K1**2*K2**3 - 4160*K1**2*K2**2 - 480*K1**2*K2*K4 + 5168*K1**2*K2 - 64*K1**2*K3**2 - 4136*K1**2 - 1184*K1*K2**2*K3 + 4320*K1*K2*K3 + 728*K1*K3*K4 - 888*K2**4 + 1160*K2**2*K4 - 2600*K2**2 - 1184*K3**2 - 422*K4**2 + 2748
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1002']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4224', 'vk6.4303', 'vk6.5489', 'vk6.5601', 'vk6.7592', 'vk6.7685', 'vk6.9093', 'vk6.9172', 'vk6.18380', 'vk6.18720', 'vk6.24837', 'vk6.25296', 'vk6.37029', 'vk6.37479', 'vk6.44194', 'vk6.44515', 'vk6.48544', 'vk6.48599', 'vk6.49245', 'vk6.49359', 'vk6.50335', 'vk6.50392', 'vk6.51074', 'vk6.51105', 'vk6.56153', 'vk6.56382', 'vk6.60682', 'vk6.61033', 'vk6.65821', 'vk6.66075', 'vk6.68814', 'vk6.69024']
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 O1O2O3O4U2U1O5U4O6U5U6U3
R3 orbit {'O1O2O3O4U2U1O5U4O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6O5U1O6U4U3
Gauss code of K* O1O2O3U4U5U3U6O5O4U1O6U2
Gauss code of -K* O1O2O3U2O4U3O5O6U4U1U6U5
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 -2 2 1 0 1],[ 2 0 0 3 2 1 0],[ 2 0 0 2 1 1 0],[-2 -3 -2 0 -1 0 1],[-1 -2 -1 1 0 1 1],[ 0 -1 -1 0 -1 0 1],[-1 0 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 1 -1 0 -2 -3],[-1 -1 0 -1 -1 0 0],[-1 1 1 0 1 -1 -2],[ 0 0 1 -1 0 -1 -1],[ 2 2 0 1 1 0 0],[ 2 3 0 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,-1,1,0,2,3,1,1,0,0,-1,1,2,1,1,0]
Phi over symmetry [-2,-2,0,1,1,2,0,1,0,1,2,1,0,2,3,1,-1,0,-1,-1,1]
Phi of -K [-2,-2,0,1,1,2,0,1,1,3,1,1,2,3,2,2,0,2,-1,0,2]
Phi of K* [-2,-1,-1,0,2,2,0,2,2,1,2,1,2,1,2,0,3,3,1,1,0]
Phi of -K* [-2,-2,0,1,1,2,0,1,0,1,2,1,0,2,3,1,-1,0,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2-4w^3z+28w^2z+21w
Inner characteristic polynomial t^6+25t^4+16t^2+1
Outer characteristic polynomial t^7+39t^5+71t^3+13t
Flat arrow polynomial -2*K1**2 + K2 + 2
2-strand cable arrow polynomial -64*K1**4 + 32*K1**3*K2*K3 + 1024*K1**2*K2**3 - 4160*K1**2*K2**2 - 480*K1**2*K2*K4 + 5168*K1**2*K2 - 64*K1**2*K3**2 - 4136*K1**2 - 1184*K1*K2**2*K3 + 4320*K1*K2*K3 + 728*K1*K3*K4 - 888*K2**4 + 1160*K2**2*K4 - 2600*K2**2 - 1184*K3**2 - 422*K4**2 + 2748
Genus of based matrix 2
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice False
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