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Flat knot 6.236

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,2,1,1,1,1,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.236', '7.23618']
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^7+48t^5+62t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.236', '7.23618']
2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 1856*K1**4*K2 - 1920*K1**4 + 704*K1**3*K2*K3 - 384*K1**3*K3 - 1152*K1**2*K2**4 + 3136*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7328*K1**2*K2**2 - 448*K1**2*K2*K4 + 4880*K1**2*K2 - 1296*K1**2 + 1408*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 128*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 4208*K1*K2*K3 + 144*K1*K3*K4 - 192*K2**6 + 192*K2**4*K4 - 2160*K2**4 - 480*K2**2*K3**2 - 48*K2**2*K4**2 + 1488*K2**2*K4 - 512*K2**2 + 96*K2*K3*K5 - 544*K3**2 - 148*K4**2 + 1378
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.236']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.309', 'vk6.346', 'vk6.697', 'vk6.743', 'vk6.1487', 'vk6.1940', 'vk6.1976', 'vk6.2453', 'vk6.2632', 'vk6.3103', 'vk6.18266', 'vk6.18601', 'vk6.24750', 'vk6.25155', 'vk6.36877', 'vk6.37338', 'vk6.44097', 'vk6.56061', 'vk6.60616', 'vk6.65728']
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 O1O2O3O4O5U3O6U4U5U6U1U2
R3 orbit {'O1O2O3O4O5U3O6U4U5U6U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U5U6U1U2O6U3
Gauss code of K* O1O2O3O4O5U4U5U6U1U2O6U3
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 -1 1 -2 -1 1 2],[ 1 0 1 -2 -1 1 2],[-1 -1 0 -2 -1 1 2],[ 2 2 2 0 1 2 2],[ 1 1 1 -1 0 1 2],[-1 -1 -1 -2 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -2 -2 -2 -2],[-1 1 0 -1 -1 -1 -2],[-1 2 1 0 -1 -1 -2],[ 1 2 1 1 0 1 -1],[ 1 2 1 1 -1 0 -2],[ 2 2 2 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,2,2,2,2,1,1,1,2,1,1,2,-1,1,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,1,2,1,1,1,1,1,1,1,-1,-1,0]
Phi of -K [-2,-1,-1,1,1,2,-1,0,1,1,2,1,1,1,1,1,1,1,-1,-1,0]
Phi of K* [-2,-1,-1,1,1,2,-1,0,1,1,2,1,1,1,1,1,1,1,-1,-1,0]
Phi of -K* [-2,-1,-1,1,1,2,1,2,2,2,2,1,1,1,2,1,1,2,-1,1,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+36t^4+20t^2+1
Outer characteristic polynomial t^7+48t^5+62t^3+5t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial -1024*K1**4*K2**2 + 1856*K1**4*K2 - 1920*K1**4 + 704*K1**3*K2*K3 - 384*K1**3*K3 - 1152*K1**2*K2**4 + 3136*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7328*K1**2*K2**2 - 448*K1**2*K2*K4 + 4880*K1**2*K2 - 1296*K1**2 + 1408*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 128*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 4208*K1*K2*K3 + 144*K1*K3*K4 - 192*K2**6 + 192*K2**4*K4 - 2160*K2**4 - 480*K2**2*K3**2 - 48*K2**2*K4**2 + 1488*K2**2*K4 - 512*K2**2 + 96*K2*K3*K5 - 544*K3**2 - 148*K4**2 + 1378
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}]]
If K is slice True
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