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Flat knot 6.1951

Min(phi) over symmetries of the knot is: [-2,0,1,1,1,1,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['4.6', '6.1770', '6.1951', '7.43053', '7.44462']
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^5+13t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1770', '6.1951', '7.38625', '7.43063']
2-strand cable arrow polynomial of the knot is: -384*K1**6 + 1856*K1**4*K2 - 3584*K1**4 - 1088*K1**3*K3 + 160*K1**2*K2**3 - 2704*K1**2*K2**2 - 544*K1**2*K2*K4 + 6736*K1**2*K2 - 3708*K1**2 - 160*K1*K2**2*K3 + 4000*K1*K2*K3 + 816*K1*K3*K4 - 168*K2**4 + 576*K2**2*K4 - 3224*K2**2 - 1396*K3**2 - 498*K4**2 + 3312
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1951']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11031', 'vk6.11109', 'vk6.12201', 'vk6.12308', 'vk6.16416', 'vk6.19234', 'vk6.19338', 'vk6.19527', 'vk6.19633', 'vk6.22717', 'vk6.22816', 'vk6.26042', 'vk6.26102', 'vk6.26425', 'vk6.26526', 'vk6.30600', 'vk6.30695', 'vk6.31912', 'vk6.34769', 'vk6.38109', 'vk6.38134', 'vk6.42384', 'vk6.44629', 'vk6.44756', 'vk6.51840', 'vk6.52708', 'vk6.52804', 'vk6.56586', 'vk6.56646', 'vk6.64715', 'vk6.66287', 'vk6.66304']
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 O1O2U3O4O5U2O6U4U6O3U1U5
R3 orbit {'O1O2U3O4O5U2O6U4U6O3U1U5'}
R3 orbit length 1
Gauss code of -K O1O2U3O4O5U1U5O3U6U2O6U4
Gauss code of K* O1O2U3O4O5U4U6O3U1U5O6U2
Gauss code of -K* O1O2U3O4O5U4O6U1U5O3U6U2
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 -1 2 1],[ 1 0 0 0 0 2 1],[ 1 0 0 0 1 1 1],[ 0 0 0 0 0 1 0],[ 1 0 -1 0 0 2 1],[-2 -2 -1 -1 -2 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 2 0 -1 -1],[-2 0 -1 -1 -2],[ 0 1 0 0 0],[ 1 1 0 0 1],[ 1 2 0 -1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,0,1,1,1,1,2,0,0,-1]
Phi over symmetry [-2,0,1,1,1,1,2,0,0,-1]
Phi of -K [-1,-1,0,2,-1,1,2,1,1,1]
Phi of K* [-2,0,1,1,1,1,2,1,1,-1]
Phi of -K* [-1,-1,0,2,-1,0,2,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^4+7t^2+1
Outer characteristic polynomial t^5+13t^3+4t
Flat arrow polynomial -2*K1**2 + K2 + 2
2-strand cable arrow polynomial -384*K1**6 + 1856*K1**4*K2 - 3584*K1**4 - 1088*K1**3*K3 + 160*K1**2*K2**3 - 2704*K1**2*K2**2 - 544*K1**2*K2*K4 + 6736*K1**2*K2 - 3708*K1**2 - 160*K1*K2**2*K3 + 4000*K1*K2*K3 + 816*K1*K3*K4 - 168*K2**4 + 576*K2**2*K4 - 3224*K2**2 - 1396*K3**2 - 498*K4**2 + 3312
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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