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Flat knot 6.618

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,3,2,1,1,2,1,1,0,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.618']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 2*K1*K2 - 2*K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.618', '6.640', '6.736', '6.787']
Outer characteristic polynomial of the knot is: t^7+52t^5+93t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.618']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1472*K1**4*K2**2 + 2976*K1**4*K2 - 4144*K1**4 - 128*K1**3*K2**2*K3 + 608*K1**3*K2*K3 - 512*K1**3*K3 - 896*K1**2*K2**4 + 4992*K1**2*K2**3 - 12240*K1**2*K2**2 - 544*K1**2*K2*K4 + 10616*K1**2*K2 - 48*K1**2*K3**2 - 3812*K1**2 + 1408*K1*K2**3*K3 - 2912*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7240*K1*K2*K3 + 232*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 3080*K2**4 - 576*K2**2*K3**2 - 8*K2**2*K4**2 + 2072*K2**2*K4 - 2184*K2**2 + 160*K2*K3*K5 - 980*K3**2 - 130*K4**2 + 3328
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.618']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19943', 'vk6.20057', 'vk6.21190', 'vk6.21339', 'vk6.26908', 'vk6.27122', 'vk6.28664', 'vk6.28810', 'vk6.38328', 'vk6.38515', 'vk6.40470', 'vk6.40714', 'vk6.45201', 'vk6.45415', 'vk6.47026', 'vk6.47158', 'vk6.56732', 'vk6.56863', 'vk6.57834', 'vk6.58004', 'vk6.61157', 'vk6.61392', 'vk6.62400', 'vk6.62553', 'vk6.66427', 'vk6.66570', 'vk6.67200', 'vk6.67363', 'vk6.69080', 'vk6.69222', 'vk6.69863', 'vk6.69964']
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 O1O2O3O4U5O6U1O5U3U4U6U2
R3 orbit {'O1O2O3O4U5O6U1O5U3U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U1U2O6U4O5U6
Gauss code of K* O1O2O3O4U5U4U1U2O6U3O5U6
Gauss code of -K* O1O2O3O4U5O6U2O5U3U4U1U6
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 1 -1 1 0 2],[ 3 0 3 0 1 3 2],[-1 -3 0 -2 0 0 1],[ 1 0 2 0 1 1 1],[-1 -1 0 -1 0 -1 0],[ 0 -3 0 -1 1 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -2 -1 -2],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 0 -2 -3],[ 0 2 1 0 0 -1 -3],[ 1 1 1 2 1 0 0],[ 3 2 1 3 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,2,1,2,0,1,1,1,0,2,3,1,3,0]
Phi over symmetry [-3,-1,0,1,1,2,0,3,1,3,2,1,1,2,1,1,0,2,0,0,1]
Phi of -K [-3,-1,0,1,1,2,2,0,1,3,3,0,0,1,2,1,0,0,0,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,0,2,3,0,1,0,1,0,1,3,0,0,2]
Phi of -K* [-3,-1,0,1,1,2,0,3,1,3,2,1,1,2,1,1,0,2,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+36t^4+52t^2+1
Outer characteristic polynomial t^7+52t^5+93t^3+7t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 2*K1*K2 - 2*K1 + 5*K2 + 6
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1472*K1**4*K2**2 + 2976*K1**4*K2 - 4144*K1**4 - 128*K1**3*K2**2*K3 + 608*K1**3*K2*K3 - 512*K1**3*K3 - 896*K1**2*K2**4 + 4992*K1**2*K2**3 - 12240*K1**2*K2**2 - 544*K1**2*K2*K4 + 10616*K1**2*K2 - 48*K1**2*K3**2 - 3812*K1**2 + 1408*K1*K2**3*K3 - 2912*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7240*K1*K2*K3 + 232*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 3080*K2**4 - 576*K2**2*K3**2 - 8*K2**2*K4**2 + 2072*K2**2*K4 - 2184*K2**2 + 160*K2*K3*K5 - 980*K3**2 - 130*K4**2 + 3328
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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