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Flat knot 6.1310

Min(phi) over symmetries of the knot is: [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['5.90', '6.1310']
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^6+39t^4+74t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.90', '6.1310']
2-strand cable arrow polynomial of the knot is: -4224*K1**2*K2**4 + 2560*K1**2*K2**3 - 3872*K1**2*K2**2 + 2080*K1**2*K2 - 608*K1**2 + 2432*K1*K2**3*K3 + 1952*K1*K2*K3 - 704*K2**6 + 384*K2**4*K4 - 512*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 256*K2**2*K4 + 208*K2**2 - 160*K3**2 - 16*K4**2 + 398
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1310']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3247', 'vk6.3266', 'vk6.3369', 'vk6.3402', 'vk6.3462', 'vk6.3510', 'vk6.17646', 'vk6.17653', 'vk6.24199', 'vk6.24216', 'vk6.36456', 'vk6.36472', 'vk6.43556', 'vk6.43573', 'vk6.48103', 'vk6.48138', 'vk6.48178', 'vk6.48192', 'vk6.60251', 'vk6.68538']
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 O1O2O3U4O5O4U6U5O6U1U2U3
R3 orbit {'O1O2O3U4O5O4U6U5O6U1U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U2U3O4U5U4O6O5U6
Gauss code of K* O1O2O3U1U2U3O4U5U4O6O5U6
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 3 0 1],[ 0 -1 0 1 1 0 -1],[-2 -2 -1 0 -1 0 -3],[-1 -3 -1 1 0 0 -1],[ 0 0 0 0 0 0 0],[ 1 -1 1 3 1 0 0]]
Primitive based matrix [[ 0 2 1 0 -1 -2],[-2 0 -1 -1 -3 -2],[-1 1 0 -1 -1 -3],[ 0 1 1 0 -1 -1],[ 1 3 1 1 0 -1],[ 2 2 3 1 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,0,1,2,1,1,3,2,1,1,3,1,1,1]
Phi over symmetry [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0]
Phi of -K [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0]
Phi of K* [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0]
Phi of -K* [-2,-1,0,1,2,1,1,3,2,1,1,3,1,1,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial -z-1
Enhanced Jones-Krushkal polynomial -16w^3z+15w^2z-w
Inner characteristic polynomial t^5+29t^3+54t
Outer characteristic polynomial t^6+39t^4+74t^2
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -4224*K1**2*K2**4 + 2560*K1**2*K2**3 - 3872*K1**2*K2**2 + 2080*K1**2*K2 - 608*K1**2 + 2432*K1*K2**3*K3 + 1952*K1*K2*K3 - 704*K2**6 + 384*K2**4*K4 - 512*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 256*K2**2*K4 + 208*K2**2 - 160*K3**2 - 16*K4**2 + 398
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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