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Flat knot 6.1111

Min(phi) over symmetries of the knot is: [-2,-1,1,2,-1,1,3,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1111']
Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']
Outer characteristic polynomial of the knot is: t^5+27t^3+18t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1111']
2-strand cable arrow polynomial of the knot is: -512*K1**6 + 1376*K1**4*K2 - 4432*K1**4 - 288*K1**3*K3 - 2480*K1**2*K2**2 - 96*K1**2*K2*K4 + 6824*K1**2*K2 - 1424*K1**2*K3**2 - 512*K1**2*K4**2 - 3592*K1**2 - 384*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 4920*K1*K2*K3 - 96*K1*K2*K4*K5 + 2584*K1*K3*K4 + 688*K1*K4*K5 + 24*K1*K5*K6 - 112*K2**4 - 240*K2**2*K3**2 - 112*K2**2*K4**2 + 768*K2**2*K4 - 3780*K2**2 + 440*K2*K3*K5 + 136*K2*K4*K6 - 2220*K3**2 - 1176*K4**2 - 292*K5**2 - 28*K6**2 + 4302
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1111']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4452', 'vk6.4547', 'vk6.5838', 'vk6.5965', 'vk6.7888', 'vk6.8002', 'vk6.9317', 'vk6.9436', 'vk6.13421', 'vk6.13516', 'vk6.13705', 'vk6.14056', 'vk6.15029', 'vk6.15149', 'vk6.17786', 'vk6.17819', 'vk6.18842', 'vk6.19435', 'vk6.19728', 'vk6.24329', 'vk6.25435', 'vk6.25468', 'vk6.26607', 'vk6.33263', 'vk6.33322', 'vk6.37561', 'vk6.44884', 'vk6.48639', 'vk6.50535', 'vk6.53663', 'vk6.55807', 'vk6.65473']
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 O1O2O3O4U2U4O5U6U3O6U1U5
R3 orbit {'O1O2O3O4U2U4O5U6U3O6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4O6U2U6O5U1U3
Gauss code of K* O1O2U3O4O5U4O6O3U6U1U5U2
Gauss code of -K* O1O2U3O4O3U1O5O6U5U4U6U2
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 -2 1 1 2 -1],[ 1 0 -2 2 1 2 0],[ 2 2 0 2 1 1 1],[-1 -2 -2 0 0 0 -1],[-1 -1 -1 0 0 0 -1],[-2 -2 -1 0 0 0 -2],[ 1 0 -1 1 1 2 0]]
Primitive based matrix [[ 0 2 1 -1 -2],[-2 0 0 -2 -1],[-1 0 0 -2 -2],[ 1 2 2 0 -2],[ 2 1 2 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,1,2,0,2,1,2,2,2]
Phi over symmetry [-2,-1,1,2,-1,1,3,0,1,1]
Phi of -K [-2,-1,1,2,-1,1,3,0,1,1]
Phi of K* [-2,-1,1,2,1,1,3,0,1,-1]
Phi of -K* [-2,-1,1,2,2,2,1,2,2,0]
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^4+17t^2+4
Outer characteristic polynomial t^5+27t^3+18t
Flat arrow polynomial -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -512*K1**6 + 1376*K1**4*K2 - 4432*K1**4 - 288*K1**3*K3 - 2480*K1**2*K2**2 - 96*K1**2*K2*K4 + 6824*K1**2*K2 - 1424*K1**2*K3**2 - 512*K1**2*K4**2 - 3592*K1**2 - 384*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 4920*K1*K2*K3 - 96*K1*K2*K4*K5 + 2584*K1*K3*K4 + 688*K1*K4*K5 + 24*K1*K5*K6 - 112*K2**4 - 240*K2**2*K3**2 - 112*K2**2*K4**2 + 768*K2**2*K4 - 3780*K2**2 + 440*K2*K3*K5 + 136*K2*K4*K6 - 2220*K3**2 - 1176*K4**2 - 292*K5**2 - 28*K6**2 + 4302
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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