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Flat knot 6.520

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,2,2,4,1,1,0,1,1,1,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.520']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.520', '6.682', '6.706', '6.748', '6.1331']
Outer characteristic polynomial of the knot is: t^7+68t^5+24t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.520']
2-strand cable arrow polynomial of the knot is: -112*K1**4 + 384*K1**2*K2**5 - 1152*K1**2*K2**4 + 832*K1**2*K2**3 - 1088*K1**2*K2**2 + 1088*K1**2*K2 - 16*K1**2*K3**2 - 16*K1**2*K4**2 - 872*K1**2 + 480*K1*K2**3*K3 + 568*K1*K2*K3 + 120*K1*K3*K4 + 32*K1*K4*K5 - 288*K2**6 + 64*K2**4*K4 - 128*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 80*K2**2*K4 - 216*K2**2 + 8*K2*K3*K5 - 180*K3**2 - 84*K4**2 - 12*K5**2 + 578
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.520']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4638', 'vk6.4909', 'vk6.6068', 'vk6.6573', 'vk6.8095', 'vk6.8479', 'vk6.9471', 'vk6.9844', 'vk6.20283', 'vk6.21612', 'vk6.27559', 'vk6.29121', 'vk6.38968', 'vk6.41213', 'vk6.45743', 'vk6.47436', 'vk6.48672', 'vk6.48857', 'vk6.49402', 'vk6.49643', 'vk6.50680', 'vk6.50857', 'vk6.51153', 'vk6.51374', 'vk6.57124', 'vk6.58314', 'vk6.61722', 'vk6.62862', 'vk6.66749', 'vk6.67631', 'vk6.69407', 'vk6.70129', 'vk6.82014', 'vk6.82750', 'vk6.85455', 'vk6.86351', 'vk6.86922', 'vk6.87138', 'vk6.88675', 'vk6.88776']
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 O1O2O3O4O5U2U3U4O6U5U1U6
R3 orbit {'O1O2O3O4O5U2U3U4O6U5U1U6', 'O1O2O3O4U1O5U3U4O6U2U5U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U6U5U1O6U2U3U4
Gauss code of K* O1O2O3U4O5O6O4U6U1U2U3U5
Gauss code of -K* O1O2O3U1O4O5O6U3U4U5U6U2
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 1 2 2],[ 1 0 -3 -1 1 3 2],[ 3 3 0 1 2 3 1],[ 1 1 -1 0 1 2 1],[-1 -1 -2 -1 0 1 1],[-2 -3 -3 -2 -1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 -1 -2 -3 -3],[-2 -1 0 -1 -1 -2 -1],[-1 1 1 0 -1 -1 -2],[ 1 2 1 1 0 1 -1],[ 1 3 2 1 -1 0 -3],[ 3 3 1 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,1,2,3,3,1,1,2,1,1,1,2,-1,1,3]
Phi over symmetry [-3,-1,-1,1,2,2,-1,1,2,2,4,1,1,0,1,1,1,2,0,0,-1]
Phi of -K [-3,-1,-1,1,2,2,-1,1,2,2,4,1,1,0,1,1,1,2,0,0,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,0,1,2,4,0,0,1,2,1,1,2,-1,-1,1]
Phi of -K* [-3,-1,-1,1,2,2,1,3,2,1,3,1,1,1,2,1,2,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 4z+9
Enhanced Jones-Krushkal polynomial 4w^4z-8w^3z+8w^2z+9w
Inner characteristic polynomial t^6+48t^4
Outer characteristic polynomial t^7+68t^5+24t^3
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial -112*K1**4 + 384*K1**2*K2**5 - 1152*K1**2*K2**4 + 832*K1**2*K2**3 - 1088*K1**2*K2**2 + 1088*K1**2*K2 - 16*K1**2*K3**2 - 16*K1**2*K4**2 - 872*K1**2 + 480*K1*K2**3*K3 + 568*K1*K2*K3 + 120*K1*K3*K4 + 32*K1*K4*K5 - 288*K2**6 + 64*K2**4*K4 - 128*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 80*K2**2*K4 - 216*K2**2 + 8*K2*K3*K5 - 180*K3**2 - 84*K4**2 - 12*K5**2 + 578
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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