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Flat knot 6.1005

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,1,4,0,1,1,1,1,2,1,0,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1005']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 4*K1*K2 - 4*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.570', '6.808', '6.1005', '6.1045', '6.1134', '6.1538', '6.1819']
Outer characteristic polynomial of the knot is: t^7+31t^5+80t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1005']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 704*K1**4*K2 - 1248*K1**4 + 192*K1**3*K2*K3 - 96*K1**3*K3 - 896*K1**2*K2**4 + 3232*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8544*K1**2*K2**2 - 544*K1**2*K2*K4 + 8440*K1**2*K2 - 32*K1**2*K3**2 - 16*K1**2*K4**2 - 5432*K1**2 + 1152*K1*K2**3*K3 - 1440*K1*K2**2*K3 - 160*K1*K2**2*K5 + 6184*K1*K2*K3 + 424*K1*K3*K4 + 16*K1*K4*K5 - 192*K2**6 + 128*K2**4*K4 - 2096*K2**4 - 384*K2**2*K3**2 - 16*K2**2*K4**2 + 1392*K2**2*K4 - 2672*K2**2 + 80*K2*K3*K5 - 1308*K3**2 - 276*K4**2 - 4*K5**2 + 3730
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1005']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3622', 'vk6.3701', 'vk6.3892', 'vk6.4005', 'vk6.7044', 'vk6.7089', 'vk6.7264', 'vk6.7375', 'vk6.17699', 'vk6.17746', 'vk6.24246', 'vk6.24305', 'vk6.36541', 'vk6.36616', 'vk6.43647', 'vk6.43752', 'vk6.48250', 'vk6.48325', 'vk6.48408', 'vk6.48429', 'vk6.50006', 'vk6.50043', 'vk6.50126', 'vk6.50149', 'vk6.55723', 'vk6.55778', 'vk6.60295', 'vk6.60376', 'vk6.65431', 'vk6.65458', 'vk6.68559', 'vk6.68586']
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 O1O2O3O4U2U3O5U4O6U5U1U6
R3 orbit {'O1O2O3O4U2U3O5U4O6U5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U6O5U1O6U2U3
Gauss code of K* O1O2O3U2U4U5U6O4O5U1O6U3
Gauss code of -K* O1O2O3U1O4U3O5O6U4U5U6U2
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 0 1 0 2],[ 1 0 -2 0 1 1 2],[ 2 2 0 1 2 1 0],[ 0 0 -1 0 1 1 0],[-1 -1 -2 -1 0 1 1],[ 0 -1 -1 -1 -1 0 1],[-2 -2 0 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -2 0],[-1 1 0 -1 1 -1 -2],[ 0 0 1 0 1 0 -1],[ 0 1 -1 -1 0 -1 -1],[ 1 2 1 0 1 0 -2],[ 2 0 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,0,1,2,0,1,-1,1,2,-1,0,1,1,1,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,1,4,0,1,1,1,1,2,1,0,2,0]
Phi of -K [-2,-1,0,0,1,2,-1,1,1,1,4,0,1,1,1,1,2,1,0,2,0]
Phi of K* [-2,-1,0,0,1,2,0,1,2,1,4,2,0,1,1,-1,0,1,1,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,1,2,0,0,1,1,2,1,1,0,-1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+23z+31
Enhanced Jones-Krushkal polynomial 4w^3z^2-2w^3z+25w^2z+31w
Inner characteristic polynomial t^6+21t^4+26t^2
Outer characteristic polynomial t^7+31t^5+80t^3+9t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 4*K1*K2 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial -384*K1**4*K2**2 + 704*K1**4*K2 - 1248*K1**4 + 192*K1**3*K2*K3 - 96*K1**3*K3 - 896*K1**2*K2**4 + 3232*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8544*K1**2*K2**2 - 544*K1**2*K2*K4 + 8440*K1**2*K2 - 32*K1**2*K3**2 - 16*K1**2*K4**2 - 5432*K1**2 + 1152*K1*K2**3*K3 - 1440*K1*K2**2*K3 - 160*K1*K2**2*K5 + 6184*K1*K2*K3 + 424*K1*K3*K4 + 16*K1*K4*K5 - 192*K2**6 + 128*K2**4*K4 - 2096*K2**4 - 384*K2**2*K3**2 - 16*K2**2*K4**2 + 1392*K2**2*K4 - 2672*K2**2 + 80*K2*K3*K5 - 1308*K3**2 - 276*K4**2 - 4*K5**2 + 3730
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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