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Flat knot 6.596

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,1,1,3,4,0,0,1,2,1,1,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.596']
Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 8*K1*K2 - 2*K1 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.356', '6.596']
Outer characteristic polynomial of the knot is: t^7+62t^5+92t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.596']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 128*K1**4*K2 - 272*K1**4 + 256*K1**3*K2**3*K3 + 672*K1**3*K2*K3 - 1216*K1**2*K2**4 + 736*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 3680*K1**2*K2**2 + 1920*K1**2*K2 - 544*K1**2*K3**2 - 1312*K1**2 + 1696*K1*K2**3*K3 + 128*K1*K2*K3**3 + 3656*K1*K2*K3 + 352*K1*K3*K4 - 192*K2**6 + 288*K2**4*K4 - 1256*K2**4 - 896*K2**2*K3**2 - 160*K2**2*K4**2 + 600*K2**2*K4 - 548*K2**2 + 184*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 - 1084*K3**2 - 254*K4**2 - 4*K5**2 - 4*K6**2 + 1484
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.596']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4712', 'vk6.5025', 'vk6.6231', 'vk6.6687', 'vk6.8213', 'vk6.8649', 'vk6.9589', 'vk6.9920', 'vk6.20304', 'vk6.21639', 'vk6.27600', 'vk6.29154', 'vk6.39022', 'vk6.41272', 'vk6.45790', 'vk6.47469', 'vk6.48752', 'vk6.48951', 'vk6.49547', 'vk6.49765', 'vk6.50766', 'vk6.50970', 'vk6.51241', 'vk6.51450', 'vk6.57159', 'vk6.58349', 'vk6.61785', 'vk6.62906', 'vk6.66776', 'vk6.67654', 'vk6.69424', 'vk6.70148']
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 O1O2O3O4U2O5U1O6U5U6U3U4
R3 orbit {'O1O2O3O4U2O5U1O6U5U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U5U6O5U4O6U3
Gauss code of K* O1O2O3O4U5U6U3U4O6U1O5U2
Gauss code of -K* O1O2O3O4U3O5U4O6U1U2U6U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -2 1 3 0 1],[ 3 0 0 3 4 1 1],[ 2 0 0 1 2 0 0],[-1 -3 -1 0 1 -1 1],[-3 -4 -2 -1 0 -1 1],[ 0 -1 0 1 1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 1 1 0 -2 -3],[-3 0 1 -1 -1 -2 -4],[-1 -1 0 -1 -1 0 -1],[-1 1 1 0 -1 -1 -3],[ 0 1 1 1 0 0 -1],[ 2 2 0 1 0 0 0],[ 3 4 1 3 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,2,3,-1,1,1,2,4,1,1,0,1,1,1,3,0,1,0]
Phi over symmetry [-3,-2,0,1,1,3,0,1,1,3,4,0,0,1,2,1,1,1,-1,-1,1]
Phi of -K [-3,-2,0,1,1,3,1,2,1,3,2,2,2,3,3,0,0,2,-1,1,3]
Phi of K* [-3,-1,-1,0,2,3,1,3,2,3,2,1,0,2,1,0,3,3,2,2,1]
Phi of -K* [-3,-2,0,1,1,3,0,1,1,3,4,0,0,1,2,1,1,1,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -12w^3z+17w^2z+11w
Inner characteristic polynomial t^6+38t^4+31t^2
Outer characteristic polynomial t^7+62t^5+92t^3
Flat arrow polynomial 8*K1**3 - 2*K1**2 - 8*K1*K2 - 2*K1 + K2 + 2*K3 + 2
2-strand cable arrow polynomial -384*K1**4*K2**2 + 128*K1**4*K2 - 272*K1**4 + 256*K1**3*K2**3*K3 + 672*K1**3*K2*K3 - 1216*K1**2*K2**4 + 736*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 3680*K1**2*K2**2 + 1920*K1**2*K2 - 544*K1**2*K3**2 - 1312*K1**2 + 1696*K1*K2**3*K3 + 128*K1*K2*K3**3 + 3656*K1*K2*K3 + 352*K1*K3*K4 - 192*K2**6 + 288*K2**4*K4 - 1256*K2**4 - 896*K2**2*K3**2 - 160*K2**2*K4**2 + 600*K2**2*K4 - 548*K2**2 + 184*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 - 1084*K3**2 - 254*K4**2 - 4*K5**2 - 4*K6**2 + 1484
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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