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Flat knot 6.220

Min(phi) over symmetries of the knot is: [-4,-2,-2,2,2,4,0,1,2,4,5,0,0,1,2,1,3,4,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.220']
Arrow polynomial of the knot is: -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.113', '6.132', '6.220', '6.933', '6.1250', '6.1905']
Outer characteristic polynomial of the knot is: t^7+126t^5+237t^3+25t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.220']
2-strand cable arrow polynomial of the knot is: -256*K2**8 + 256*K2**6*K4 - 2944*K2**6 - 192*K2**4*K4**2 + 3584*K2**4*K4 - 5760*K2**4 + 64*K2**3*K4*K6 - 448*K2**3*K6 - 768*K2**2*K4**2 + 4176*K2**2*K4 + 2192*K2**2 + 96*K2*K4*K6 - 616*K4**2 + 614
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.220']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73185', 'vk6.73200', 'vk6.74300', 'vk6.74940', 'vk6.75095', 'vk6.75116', 'vk6.76505', 'vk6.76924', 'vk6.78029', 'vk6.78054', 'vk6.79354', 'vk6.79772', 'vk6.79959', 'vk6.80806', 'vk6.83780', 'vk6.85571', 'vk6.85732', 'vk6.87646', 'vk6.89617', 'vk6.90176']
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 O1O2O3O4O5U3O6U1U2U6U4U5
R3 orbit {'O1O2O3O4O5U3O6U1U2U6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U2U6U4U5O6U3
Gauss code of K* O1O2O3O4O5U1U2U6U4U5O6U3
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 3
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 -2 -2 2 4 2],[ 4 0 1 0 4 5 2],[ 2 -1 0 0 3 4 1],[ 2 0 0 0 1 2 0],[-2 -4 -3 -1 0 1 0],[-4 -5 -4 -2 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 4 2 2 -2 -2 -4],[-4 0 0 -1 -2 -4 -5],[-2 0 0 0 0 -1 -2],[-2 1 0 0 -1 -3 -4],[ 2 2 0 1 0 0 0],[ 2 4 1 3 0 0 -1],[ 4 5 2 4 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-2,2,2,4,0,1,2,4,5,0,0,1,2,1,3,4,0,0,1]
Phi over symmetry [-4,-2,-2,2,2,4,0,1,2,4,5,0,0,1,2,1,3,4,0,0,1]
Phi of -K [-4,-2,-2,2,2,4,1,2,2,4,3,0,1,3,2,3,4,4,0,1,2]
Phi of K* [-4,-2,-2,2,2,4,1,2,2,4,3,0,1,3,2,3,4,4,0,1,2]
Phi of -K* [-4,-2,-2,2,2,4,0,1,2,4,5,0,0,1,2,1,3,4,0,0,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^3+19z^2+28z+13
Enhanced Jones-Krushkal polynomial 4w^4z^3+19w^3z^2+28w^2z+13
Inner characteristic polynomial t^6+78t^4+61t^2+1
Outer characteristic polynomial t^7+126t^5+237t^3+25t
Flat arrow polynomial -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
2-strand cable arrow polynomial -256*K2**8 + 256*K2**6*K4 - 2944*K2**6 - 192*K2**4*K4**2 + 3584*K2**4*K4 - 5760*K2**4 + 64*K2**3*K4*K6 - 448*K2**3*K6 - 768*K2**2*K4**2 + 4176*K2**2*K4 + 2192*K2**2 + 96*K2*K4*K6 - 616*K4**2 + 614
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice True
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