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Flat knot 6.1116

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,1,2,1,0,1,1,1,1,0,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1116']
Arrow polynomial of the knot is: -6*K1**2 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.689', '6.691', '6.752', '6.754', '6.1106', '6.1116', '6.1126', '6.1335', '6.1379', '6.1386', '6.1409', '6.1415', '6.1417', '6.1418', '6.1421', '6.1422', '6.1428', '6.1431', '6.1432', '6.1435', '6.1443', '6.1445', '6.1446', '6.1447', '6.1454', '6.1455', '6.1460', '6.1462', '6.1464', '6.1466', '6.1472', '6.1474', '6.1475', '6.1501', '6.1516', '6.1518', '6.1566', '6.1570', '6.1590', '6.1599', '6.1602', '6.1603', '6.1604', '6.1605', '6.1614', '6.1615', '6.1625', '6.1628', '6.1730', '6.1780', '6.1883', '6.1885', '6.1888', '6.1890', '6.1941', '6.1943', '6.1945', '6.1948', '6.1961', '6.1963', '6.1966', '6.1967', '6.1971']
Outer characteristic polynomial of the knot is: t^7+24t^5+28t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1116', '6.1747']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 1984*K1**4 + 32*K1**3*K2*K3 - 448*K1**3*K3 - 720*K1**2*K2**2 + 3512*K1**2*K2 - 96*K1**2*K3**2 - 1452*K1**2 + 1432*K1*K2*K3 + 56*K1*K3*K4 - 24*K2**4 + 40*K2**2*K4 - 1336*K2**2 - 428*K3**2 - 22*K4**2 + 1340
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1116']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4359', 'vk6.4392', 'vk6.5681', 'vk6.5714', 'vk6.7750', 'vk6.7783', 'vk6.9232', 'vk6.9265', 'vk6.10475', 'vk6.10545', 'vk6.10640', 'vk6.10692', 'vk6.10725', 'vk6.10831', 'vk6.14597', 'vk6.15318', 'vk6.15445', 'vk6.16220', 'vk6.17993', 'vk6.24435', 'vk6.24665', 'vk6.30162', 'vk6.30232', 'vk6.30327', 'vk6.30458', 'vk6.30568', 'vk6.30663', 'vk6.33952', 'vk6.34357', 'vk6.34413', 'vk6.37257', 'vk6.43862', 'vk6.50452', 'vk6.50485', 'vk6.52686', 'vk6.52780', 'vk6.54199', 'vk6.60536', 'vk6.60878', 'vk6.65950']
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 O1O2O3O4U3U1O5U4U5O6U2U6
R3 orbit {'O1O2O3U2O4U5U3O6U4U1O5U6', 'O1O2O3O4U3U1O5U4U5O6U2U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U5U3O5U6U1O6U4U2
Gauss code of K* O1O2U3O4O3U5O6O5U2U6U1U4
Gauss code of -K* O1O2U1O3O4U3O5O6U4U6U2U5
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 0 -1 1 1 1],[ 2 0 2 0 2 1 1],[ 0 -2 0 -1 0 1 1],[ 1 0 1 0 1 1 0],[-1 -2 0 -1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 -1 0 -1],[ 0 0 1 1 0 -1 -2],[ 1 1 1 0 1 0 0],[ 2 2 1 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,1,1,0,1,1,2,0]
Phi over symmetry [-2,-1,0,1,1,1,0,2,1,1,2,1,0,1,1,1,1,0,0,0,-1]
Phi of -K [-2,-1,0,1,1,1,1,0,1,2,2,0,1,1,2,1,0,0,-1,0,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,1,0,2,2,0,0,1]
Phi of -K* [-2,-1,0,1,1,1,0,2,1,1,2,1,0,1,1,1,1,0,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial 2w^3z^2+15w^2z+23w
Inner characteristic polynomial t^6+16t^4+13t^2+1
Outer characteristic polynomial t^7+24t^5+28t^3+4t
Flat arrow polynomial -6*K1**2 + 3*K2 + 4
2-strand cable arrow polynomial 96*K1**4*K2 - 1984*K1**4 + 32*K1**3*K2*K3 - 448*K1**3*K3 - 720*K1**2*K2**2 + 3512*K1**2*K2 - 96*K1**2*K3**2 - 1452*K1**2 + 1432*K1*K2*K3 + 56*K1*K3*K4 - 24*K2**4 + 40*K2**2*K4 - 1336*K2**2 - 428*K3**2 - 22*K4**2 + 1340
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
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