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Flat knot 6.1858

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,0,2,1,0,1,0,0,1,0,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1858', '7.41753']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1*K2 - 4*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.134', '6.409', '6.424', '6.534', '6.942', '6.969', '6.1192', '6.1280', '6.1310', '6.1325', '6.1858', '6.1925']
Outer characteristic polynomial of the knot is: t^7+47t^5+146t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1858', '7.41753']
2-strand cable arrow polynomial of the knot is: -1920*K1**2*K2**4 + 1024*K1**2*K2**3 - 1376*K1**2*K2**2 + 672*K1**2*K2 - 160*K1**2 + 1152*K1*K2**3*K3 + 800*K1*K2*K3 - 704*K2**6 + 448*K2**4*K4 - 320*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 320*K2**2*K4 + 176*K2**2 - 96*K3**2 - 48*K4**2 + 174
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1858']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.497', 'vk6.588', 'vk6.994', 'vk6.1092', 'vk6.1825', 'vk6.2271', 'vk6.2511', 'vk6.2556', 'vk6.2774', 'vk6.2878', 'vk6.3049', 'vk6.3187', 'vk6.17650', 'vk6.17657', 'vk6.22179', 'vk6.24207', 'vk6.28299', 'vk6.29711', 'vk6.36465', 'vk6.39749', 'vk6.42011', 'vk6.43564', 'vk6.46311', 'vk6.47888', 'vk6.60257', 'vk6.68541']
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 O1O2O3U4U5O4U6O5O6U1U2U3
R3 orbit {'O1O2O3U4U5U6O4O5O6U3U1U2', 'O1O2O3U4U5O4U6O5O6U1U2U3'}
R3 orbit length 2
Gauss code of -K O1O2O3U1U2U3O4O5U4O6U5U6
Gauss code of K* O1O2O3U1U2U3O4O5U4O6U5U6
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 2 -1 0 1],[ 2 0 1 2 1 2 3],[ 0 -1 0 1 -1 0 1],[-2 -2 -1 0 -3 -2 -1],[ 1 -1 1 3 0 0 1],[ 0 -2 0 2 0 0 0],[-1 -3 -1 1 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -3 -2],[-1 1 0 -1 0 -1 -3],[ 0 1 1 0 0 -1 -1],[ 0 2 0 0 0 0 -2],[ 1 3 1 1 0 0 -1],[ 2 2 3 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,2,3,2,1,0,1,3,0,1,1,0,2,1]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,0,2,1,0,1,0,0,1,0,0,1,0]
Phi of -K [-2,-1,0,0,1,2,0,0,1,0,2,1,0,1,0,0,1,0,0,1,0]
Phi of K* [-2,-1,0,0,1,2,0,0,1,0,2,1,0,1,0,0,1,0,0,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,2,3,2,1,0,1,3,0,1,1,0,2,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial -z-1
Enhanced Jones-Krushkal polynomial -10w^3z+9w^2z-w
Inner characteristic polynomial t^6+37t^4+110t^2
Outer characteristic polynomial t^7+47t^5+146t^3
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -1920*K1**2*K2**4 + 1024*K1**2*K2**3 - 1376*K1**2*K2**2 + 672*K1**2*K2 - 160*K1**2 + 1152*K1*K2**3*K3 + 800*K1*K2*K3 - 704*K2**6 + 448*K2**4*K4 - 320*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 320*K2**2*K4 + 176*K2**2 - 96*K3**2 - 48*K4**2 + 174
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}]]
If K is slice True
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