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Flat knot 6.831

Min(phi) over symmetries of the knot is: [-1,-1,1,1,0,0,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.831']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.206', '6.236', '6.575', '6.580', '6.613', '6.619', '6.810', '6.819', '6.831', '6.838', '6.957', '6.1018', '6.1028', '6.1046', '6.1073', '6.1279', '6.1507', '6.1532', '6.1556', '6.1639', '6.1688', '6.1924', '6.1931']
Outer characteristic polynomial of the knot is: t^5+10t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.5', '5.97', '6.831', '6.2072', '7.44282']
2-strand cable arrow polynomial of the knot is: 4480*K1**4*K2 - 10240*K1**4 + 1664*K1**3*K2*K3 - 2688*K1**3*K3 - 256*K1**2*K2**4 + 1152*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10176*K1**2*K2**2 - 1024*K1**2*K2*K4 + 14304*K1**2*K2 - 2240*K1**2*K3**2 - 64*K1**2*K4**2 - 3648*K1**2 + 512*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 10752*K1*K2*K3 + 2560*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1104*K2**4 - 640*K2**2*K3**2 - 48*K2**2*K4**2 + 1696*K2**2*K4 - 4968*K2**2 + 480*K2*K3*K5 + 32*K2*K4*K6 - 2784*K3**2 - 924*K4**2 - 128*K5**2 - 8*K6**2 + 5322
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.831']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13818', 'vk6.13830', 'vk6.13837', 'vk6.13862', 'vk6.14889', 'vk6.14912', 'vk6.14920', 'vk6.14935', 'vk6.34235', 'vk6.34248', 'vk6.53842', 'vk6.53845', 'vk6.54384', 'vk6.54386']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is a.
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4U3O5U1U5U6U4O6U2
R3 orbit {'O1O2O3O4U3O5U1U5U6U4O6U2'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* Same
Gauss code of -K* Same
Diagrammatic symmetry type a
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 3 1 -1],[ 3 0 3 0 3 1 2],[-1 -3 0 -1 2 0 -2],[ 1 0 1 0 1 0 0],[-3 -3 -2 -1 0 0 -3],[-1 -1 0 0 0 0 -1],[ 1 -2 2 0 3 1 0]]
Primitive based matrix [[ 0 1 1 -1 -1],[-1 0 0 0 -1],[-1 0 0 -1 -2],[ 1 0 1 0 0],[ 1 1 2 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,1,1,0,0,1,1,2,0]
Phi over symmetry [-1,-1,1,1,0,0,1,1,2,0]
Phi of -K [-1,-1,1,1,0,0,1,1,2,0]
Phi of K* [-1,-1,1,1,0,0,1,1,2,0]
Phi of -K* [-1,-1,1,1,0,0,1,1,2,0]
Symmetry type of based matrix a
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^4+6t^2+1
Outer characteristic polynomial t^5+10t^3+5t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial 4480*K1**4*K2 - 10240*K1**4 + 1664*K1**3*K2*K3 - 2688*K1**3*K3 - 256*K1**2*K2**4 + 1152*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10176*K1**2*K2**2 - 1024*K1**2*K2*K4 + 14304*K1**2*K2 - 2240*K1**2*K3**2 - 64*K1**2*K4**2 - 3648*K1**2 + 512*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 10752*K1*K2*K3 + 2560*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1104*K2**4 - 640*K2**2*K3**2 - 48*K2**2*K4**2 + 1696*K2**2*K4 - 4968*K2**2 + 480*K2*K3*K5 + 32*K2*K4*K6 - 2784*K3**2 - 924*K4**2 - 128*K5**2 - 8*K6**2 + 5322
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice True
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