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Flat knot 6.570

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,0,1,2,4,0,1,1,2,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.570', '7.17960']
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+98t^5+73t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.570']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 640*K1**4*K2 - 3264*K1**4 + 128*K1**3*K2*K3 - 192*K1**3*K3 - 512*K1**2*K2**4 + 2304*K1**2*K2**3 - 8096*K1**2*K2**2 - 128*K1**2*K2*K4 + 10912*K1**2*K2 - 32*K1**2*K3**2 - 5896*K1**2 + 768*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 + 6368*K1*K2*K3 + 160*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 2176*K2**4 - 448*K2**2*K3**2 - 48*K2**2*K4**2 + 1232*K2**2*K4 - 3360*K2**2 + 112*K2*K3*K5 - 1416*K3**2 - 152*K4**2 + 4502
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.570']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81555', 'vk6.81631', 'vk6.81836', 'vk6.82053', 'vk6.82217', 'vk6.82329', 'vk6.82546', 'vk6.83001', 'vk6.83133', 'vk6.83573', 'vk6.83935', 'vk6.84069', 'vk6.84531', 'vk6.84894', 'vk6.85907', 'vk6.86411', 'vk6.86463', 'vk6.88833', 'vk6.89773', 'vk6.89883']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U1O5U2O6U3U4U5U6
R3 orbit {'O1O2O3O4U1O5U2O6U3U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1U2O5U3O6U4
Gauss code of K* O1O2O3O4U5U6U1U2O5U3O6U4
Gauss code of -K* Same
Diagrammatic symmetry type -
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 -1 1 2 3],[ 3 0 1 2 3 3 2],[ 2 -1 0 1 2 3 3],[ 1 -2 -1 0 1 2 3],[-1 -3 -2 -1 0 1 2],[-2 -3 -3 -2 -1 0 1],[-3 -2 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 -2 -3 -3 -2],[-2 1 0 -1 -2 -3 -3],[-1 2 1 0 -1 -2 -3],[ 1 3 2 1 0 -1 -2],[ 2 3 3 2 1 0 -1],[ 3 2 3 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,1,2,3,3,2,1,2,3,3,1,2,3,1,2,1]
Phi over symmetry [-3,-2,-1,1,2,3,0,0,1,2,4,0,1,1,2,1,1,1,0,0,0]
Phi of -K [-3,-2,-1,1,2,3,0,0,1,2,4,0,1,1,2,1,1,1,0,0,0]
Phi of K* [-3,-2,-1,1,2,3,0,0,1,2,4,0,1,1,2,1,1,1,0,0,0]
Phi of -K* [-3,-2,-1,1,2,3,1,2,3,3,2,1,2,3,3,1,2,3,1,2,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+24z+33
Enhanced Jones-Krushkal polynomial 4w^3z^2+24w^2z+33w
Inner characteristic polynomial t^6+70t^4+45t^2+1
Outer characteristic polynomial t^7+98t^5+73t^3+7t
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 + 640*K1**4*K2 - 3264*K1**4 + 128*K1**3*K2*K3 - 192*K1**3*K3 - 512*K1**2*K2**4 + 2304*K1**2*K2**3 - 8096*K1**2*K2**2 - 128*K1**2*K2*K4 + 10912*K1**2*K2 - 32*K1**2*K3**2 - 5896*K1**2 + 768*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 + 6368*K1*K2*K3 + 160*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 2176*K2**4 - 448*K2**2*K3**2 - 48*K2**2*K4**2 + 1232*K2**2*K4 - 3360*K2**2 + 112*K2*K3*K5 - 1416*K3**2 - 152*K4**2 + 4502
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice True
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