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Flat knot 6.576

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,2,2,4,1,1,1,2,1,1,1,0,-1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.576']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.576', '6.581', '6.622', '6.627', '6.983', '6.1017']
Outer characteristic polynomial of the knot is: t^7+67t^5+47t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.576']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 288*K1**4*K2 - 1168*K1**4 + 160*K1**3*K2*K3 - 288*K1**3*K3 + 512*K1**2*K2**3 - 3936*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 7208*K1**2*K2 - 144*K1**2*K3**2 - 5632*K1**2 + 192*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5688*K1*K2*K3 + 736*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 1016*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 40*K2**2*K4**2 + 1432*K2**2*K4 - 4050*K2**2 - 32*K2*K3**2*K4 + 232*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 1848*K3**2 - 498*K4**2 - 56*K5**2 - 6*K6**2 + 4136
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.576']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71392', 'vk6.71451', 'vk6.71914', 'vk6.71973', 'vk6.72436', 'vk6.72607', 'vk6.72726', 'vk6.72799', 'vk6.72862', 'vk6.73035', 'vk6.74236', 'vk6.74366', 'vk6.74445', 'vk6.74864', 'vk6.75060', 'vk6.76634', 'vk6.76912', 'vk6.77049', 'vk6.77413', 'vk6.77746', 'vk6.77797', 'vk6.79288', 'vk6.79404', 'vk6.79761', 'vk6.79820', 'vk6.79895', 'vk6.80850', 'vk6.80920', 'vk6.81389', 'vk6.85506', 'vk6.87207', 'vk6.89260']
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 O1O2O3O4U1O5U2O6U5U4U6U3
R3 orbit {'O1O2O3O4U1O5U2O6U5U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U1U6O5U3O6U4
Gauss code of K* O1O2O3O4U5U6U4U2O5U1O6U3
Gauss code of -K* O1O2O3O4U2O5U4O6U3U1U5U6
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 -3 -2 2 1 0 2],[ 3 0 1 3 2 1 1],[ 2 -1 0 3 2 1 2],[-2 -3 -3 0 -1 -1 2],[-1 -2 -2 1 0 0 2],[ 0 -1 -1 1 0 0 1],[-2 -1 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 2 -1 -1 -3 -3],[-2 -2 0 -2 -1 -2 -1],[-1 1 2 0 0 -2 -2],[ 0 1 1 0 0 -1 -1],[ 2 3 2 2 1 0 -1],[ 3 3 1 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,-2,1,1,3,3,2,1,2,1,0,2,2,1,1,1]
Phi over symmetry [-3,-2,0,1,2,2,0,2,2,2,4,1,1,1,2,1,1,1,0,-1,-2]
Phi of -K [-3,-2,0,1,2,2,0,2,2,2,4,1,1,1,2,1,1,1,0,-1,-2]
Phi of K* [-2,-2,-1,0,2,3,-2,-1,1,2,4,0,1,1,2,1,1,2,1,2,0]
Phi of -K* [-3,-2,0,1,2,2,1,1,2,1,3,1,2,2,3,0,1,1,2,1,-2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+45t^4+8t^2
Outer characteristic polynomial t^7+67t^5+47t^3+4t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -64*K1**4*K2**2 + 288*K1**4*K2 - 1168*K1**4 + 160*K1**3*K2*K3 - 288*K1**3*K3 + 512*K1**2*K2**3 - 3936*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 7208*K1**2*K2 - 144*K1**2*K3**2 - 5632*K1**2 + 192*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5688*K1*K2*K3 + 736*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 1016*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 40*K2**2*K4**2 + 1432*K2**2*K4 - 4050*K2**2 - 32*K2*K3**2*K4 + 232*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 1848*K3**2 - 498*K4**2 - 56*K5**2 - 6*K6**2 + 4136
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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