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Flat knot 6.332

Min(phi) over symmetries of the knot is: [-3,-1,1,3,0,2,3,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.332']
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+38t^3+26t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.332']
2-strand cable arrow polynomial of the knot is: -384*K1**6 + 384*K1**4*K2 - 1440*K1**4 - 720*K1**2*K2**2 + 1720*K1**2*K2 - 544*K1**2*K3**2 - 144*K1**2*K4**2 - 784*K1**2 + 1352*K1*K2*K3 + 1096*K1*K3*K4 + 368*K1*K4*K5 + 32*K1*K5*K6 - 48*K2**4 - 48*K2**2*K3**2 - 16*K2**2*K4**2 + 152*K2**2*K4 - 916*K2**2 + 176*K2*K3*K5 + 64*K2*K4*K6 - 824*K3**2 - 612*K4**2 - 240*K5**2 - 44*K6**2 + 1418
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.332']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10952', 'vk6.10953', 'vk6.10984', 'vk6.10985', 'vk6.12118', 'vk6.12119', 'vk6.12150', 'vk6.12151', 'vk6.13781', 'vk6.13808', 'vk6.14235', 'vk6.14243', 'vk6.14682', 'vk6.14690', 'vk6.14852', 'vk6.14883', 'vk6.15838', 'vk6.15846', 'vk6.31817', 'vk6.31818', 'vk6.33621', 'vk6.33638', 'vk6.33652', 'vk6.33671', 'vk6.51781', 'vk6.51782', 'vk6.52643', 'vk6.52644', 'vk6.53807', 'vk6.53818', 'vk6.54245', 'vk6.54253']
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 O1O2O3O4O5U2U5O6U3U6U1U4
R3 orbit {'O1O2O3O4O5U2U5O6U3U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U5U6U3O6U1U4
Gauss code of K* O1O2O3O4U3U5U1U4U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U1U4U6U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -3 -1 3 1 1],[ 1 0 -2 0 3 1 1],[ 3 2 0 2 3 1 1],[ 1 0 -2 0 2 0 1],[-3 -3 -3 -2 0 0 0],[-1 -1 -1 0 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 -1 -3],[-3 0 0 -2 -3],[-1 0 0 0 -1],[ 1 2 0 0 -2],[ 3 3 1 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-1,1,3,0,2,3,0,1,2]
Phi over symmetry [-3,-1,1,3,0,2,3,0,1,2]
Phi of -K [-3,-1,1,3,0,3,3,2,2,2]
Phi of K* [-3,-1,1,3,2,2,3,2,3,0]
Phi of -K* [-3,-1,1,3,2,1,3,0,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial 13w^2z+27w
Inner characteristic polynomial t^4+18t^2+4
Outer characteristic polynomial t^5+38t^3+26t
Flat arrow polynomial -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -384*K1**6 + 384*K1**4*K2 - 1440*K1**4 - 720*K1**2*K2**2 + 1720*K1**2*K2 - 544*K1**2*K3**2 - 144*K1**2*K4**2 - 784*K1**2 + 1352*K1*K2*K3 + 1096*K1*K3*K4 + 368*K1*K4*K5 + 32*K1*K5*K6 - 48*K2**4 - 48*K2**2*K3**2 - 16*K2**2*K4**2 + 152*K2**2*K4 - 916*K2**2 + 176*K2*K3*K5 + 64*K2*K4*K6 - 824*K3**2 - 612*K4**2 - 240*K5**2 - 44*K6**2 + 1418
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}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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