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Flat knot 6.796

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,3,3,1,1,2,1,1,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.796']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.573', '6.628', '6.703', '6.704', '6.723', '6.796', '6.1083', '6.1099', '6.1315']
Outer characteristic polynomial of the knot is: t^7+56t^5+79t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.796']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 896*K1**4*K2**2 + 1504*K1**4*K2 - 2800*K1**4 + 192*K1**3*K2*K3 - 224*K1**3*K3 - 1216*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3168*K1**2*K2**3 - 8160*K1**2*K2**2 - 544*K1**2*K2*K4 + 7424*K1**2*K2 - 112*K1**2*K3**2 - 3108*K1**2 + 1824*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 160*K1*K2**2*K5 + 5176*K1*K2*K3 + 400*K1*K3*K4 - 32*K2**6 + 160*K2**4*K4 - 2152*K2**4 - 688*K2**2*K3**2 - 104*K2**2*K4**2 + 1064*K2**2*K4 - 1432*K2**2 + 120*K2*K3*K5 - 908*K3**2 - 198*K4**2 + 2692
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.796']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19936', 'vk6.19981', 'vk6.21174', 'vk6.21250', 'vk6.26889', 'vk6.27000', 'vk6.28645', 'vk6.28726', 'vk6.38312', 'vk6.38406', 'vk6.40445', 'vk6.40589', 'vk6.45182', 'vk6.45296', 'vk6.47014', 'vk6.47078', 'vk6.56729', 'vk6.56787', 'vk6.57829', 'vk6.57921', 'vk6.61150', 'vk6.61279', 'vk6.62393', 'vk6.62472', 'vk6.66422', 'vk6.66483', 'vk6.67192', 'vk6.67276', 'vk6.69074', 'vk6.69141', 'vk6.69858', 'vk6.69900']
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 O1O2O3O4U5O6U3U4U2O5U1U6
R3 orbit {'O1O2O3O4U5O6U3U4U2O5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4O6U3U1U2O5U6
Gauss code of K* O1O2O3U4O5O6U5U3U1U2O4U6
Gauss code of -K* O1O2O3U4O5U2U3U1U6O4O6U5
Diagrammatic symmetry type c
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 0 -1 1 -2 3],[ 1 0 1 0 2 -2 3],[ 0 -1 0 -1 1 -2 2],[ 1 0 1 0 1 0 1],[-1 -2 -1 -1 0 -1 0],[ 2 2 2 0 1 0 3],[-3 -3 -2 -1 0 -3 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -2 -1 -3 -3],[-1 0 0 -1 -1 -2 -1],[ 0 2 1 0 -1 -1 -2],[ 1 1 1 1 0 0 0],[ 1 3 2 1 0 0 -2],[ 2 3 1 2 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,0,2,1,3,3,1,1,2,1,1,1,2,0,0,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,3,3,1,1,2,1,1,1,2,0,0,2]
Phi of -K [-2,-1,-1,0,1,3,-1,1,0,2,2,0,0,0,1,0,1,3,0,1,2]
Phi of K* [-3,-1,0,1,1,2,2,1,1,3,2,0,0,1,2,0,0,0,0,-1,1]
Phi of -K* [-2,-1,-1,0,1,3,0,2,2,1,3,0,1,1,1,1,2,3,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+40t^4+38t^2+1
Outer characteristic polynomial t^7+56t^5+79t^3+9t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 896*K1**4*K2**2 + 1504*K1**4*K2 - 2800*K1**4 + 192*K1**3*K2*K3 - 224*K1**3*K3 - 1216*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3168*K1**2*K2**3 - 8160*K1**2*K2**2 - 544*K1**2*K2*K4 + 7424*K1**2*K2 - 112*K1**2*K3**2 - 3108*K1**2 + 1824*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 160*K1*K2**2*K5 + 5176*K1*K2*K3 + 400*K1*K3*K4 - 32*K2**6 + 160*K2**4*K4 - 2152*K2**4 - 688*K2**2*K3**2 - 104*K2**2*K4**2 + 1064*K2**2*K4 - 1432*K2**2 + 120*K2*K3*K5 - 908*K3**2 - 198*K4**2 + 2692
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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