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Flat knot 6.557

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,3,2,3,0,1,1,1,1,1,2,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.557']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+59t^5+64t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.557']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 256*K1**4*K2 - 1312*K1**4 + 544*K1**3*K2*K3 - 288*K1**3*K3 + 320*K1**2*K2**3 - 3344*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 5808*K1**2*K2 - 768*K1**2*K3**2 - 4504*K1**2 + 192*K1*K2**3*K3 - 832*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5280*K1*K2*K3 + 1248*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 544*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 88*K2**2*K4**2 + 944*K2**2*K4 - 3556*K2**2 - 32*K2*K3**2*K4 + 632*K2*K3*K5 + 88*K2*K4*K6 + 40*K3**2*K6 - 2028*K3**2 - 584*K4**2 - 188*K5**2 - 36*K6**2 + 3742
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.557']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17102', 'vk6.17343', 'vk6.20592', 'vk6.22000', 'vk6.23493', 'vk6.23830', 'vk6.28055', 'vk6.29513', 'vk6.35634', 'vk6.36075', 'vk6.39465', 'vk6.41666', 'vk6.43002', 'vk6.43312', 'vk6.46049', 'vk6.47717', 'vk6.55241', 'vk6.55491', 'vk6.57463', 'vk6.58628', 'vk6.59643', 'vk6.59989', 'vk6.62134', 'vk6.63099', 'vk6.65035', 'vk6.65234', 'vk6.66988', 'vk6.67852', 'vk6.68304', 'vk6.68452', 'vk6.69604', 'vk6.70296']
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 O1O2O3O4O5U4U1U3O6U5U2U6
R3 orbit {'O1O2O3O4O5U4U1U3O6U5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U1O6U3U5U2
Gauss code of K* O1O2O3U4O5O6O4U2U6U3U1U5
Gauss code of -K* O1O2O3U1O4O5O6U3U6U4U2U5
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 0 0 -1 2 2],[ 3 0 3 1 0 3 2],[ 0 -3 0 -1 -1 2 2],[ 0 -1 1 0 0 2 1],[ 1 0 1 0 0 1 1],[-2 -3 -2 -2 -1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 1 -2 -2 -1 -3],[-2 -1 0 -1 -2 -1 -2],[ 0 2 1 0 1 0 -1],[ 0 2 2 -1 0 -1 -3],[ 1 1 1 0 1 0 0],[ 3 3 2 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-1,2,2,1,3,1,2,1,2,-1,0,1,1,3,0]
Phi over symmetry [-3,-1,0,0,2,2,0,1,3,2,3,0,1,1,1,1,1,2,2,2,-1]
Phi of -K [-3,-1,0,0,2,2,2,0,2,2,3,0,1,2,2,1,0,0,0,1,-1]
Phi of K* [-2,-2,0,0,1,3,-1,0,1,2,3,0,0,2,2,-1,0,0,1,2,2]
Phi of -K* [-3,-1,0,0,2,2,0,1,3,2,3,0,1,1,1,1,1,2,2,2,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+41t^4+16t^2
Outer characteristic polynomial t^7+59t^5+64t^3+4t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -128*K1**4*K2**2 + 256*K1**4*K2 - 1312*K1**4 + 544*K1**3*K2*K3 - 288*K1**3*K3 + 320*K1**2*K2**3 - 3344*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 5808*K1**2*K2 - 768*K1**2*K3**2 - 4504*K1**2 + 192*K1*K2**3*K3 - 832*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5280*K1*K2*K3 + 1248*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 544*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 88*K2**2*K4**2 + 944*K2**2*K4 - 3556*K2**2 - 32*K2*K3**2*K4 + 632*K2*K3*K5 + 88*K2*K4*K6 + 40*K3**2*K6 - 2028*K3**2 - 584*K4**2 - 188*K5**2 - 36*K6**2 + 3742
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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