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Flat knot 6.477

Min(phi) over symmetries of the knot is: [-4,-2,0,2,2,2,1,1,2,4,4,0,1,2,3,1,0,2,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.477']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**2*K2 + 4*K1**2 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.61', '6.177', '6.254', '6.357', '6.477']
Outer characteristic polynomial of the knot is: t^7+90t^5+197t^3+25t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.477']
2-strand cable arrow polynomial of the knot is: -128*K2**8 + 256*K2**6*K4 - 1856*K2**6 - 288*K2**4*K4**2 + 2848*K2**4*K4 - 5376*K2**4 + 96*K2**3*K4*K6 - 352*K2**3*K6 - 816*K2**2*K4**2 + 3568*K2**2*K4 + 1936*K2**2 + 112*K2*K4*K6 - 528*K4**2 + 526
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.477']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73182', 'vk6.73196', 'vk6.73198', 'vk6.74299', 'vk6.74304', 'vk6.74939', 'vk6.74946', 'vk6.75087', 'vk6.75093', 'vk6.75110', 'vk6.75114', 'vk6.76506', 'vk6.76511', 'vk6.76925', 'vk6.78023', 'vk6.78031', 'vk6.78049', 'vk6.78056', 'vk6.79353', 'vk6.79771', 'vk6.79777', 'vk6.79957', 'vk6.80807', 'vk6.80811', 'vk6.83776', 'vk6.85574', 'vk6.85734', 'vk6.85737', 'vk6.87640', 'vk6.87644', 'vk6.89615', 'vk6.90174']
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 O1O2O3O4O5U1U2U5O6U3U4U6
R3 orbit {'O1O2O3O4O5U1U2U5O6U3U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U2U3O6U1U4U5
Gauss code of K* O1O2O3U4O5O6O4U1U2U5U6U3
Gauss code of -K* O1O2O3U1O4O5O6U4U2U3U5U6
Diagrammatic symmetry type c
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 0 2 2 2],[ 4 0 1 3 4 2 2],[ 2 -1 0 2 3 1 2],[ 0 -3 -2 0 1 0 2],[-2 -4 -3 -1 0 0 1],[-2 -2 -1 0 0 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 2 2 2 0 -2 -4],[-2 0 1 0 -1 -3 -4],[-2 -1 0 0 -2 -2 -2],[-2 0 0 0 0 -1 -2],[ 0 1 2 0 0 -2 -3],[ 2 3 2 1 2 0 -1],[ 4 4 2 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,0,2,4,-1,0,1,3,4,0,2,2,2,0,1,2,2,3,1]
Phi over symmetry [-4,-2,0,2,2,2,1,1,2,4,4,0,1,2,3,1,0,2,-1,0,0]
Phi of -K [-4,-2,0,2,2,2,1,1,2,4,4,0,1,2,3,1,0,2,-1,0,0]
Phi of K* [-2,-2,-2,0,2,4,-1,0,0,2,4,0,1,1,2,2,3,4,0,1,1]
Phi of -K* [-4,-2,0,2,2,2,1,3,2,2,4,2,1,2,3,0,2,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
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+58t^4+53t^2+1
Outer characteristic polynomial t^7+90t^5+197t^3+25t
Flat arrow polynomial -8*K1**4 + 4*K1**2*K2 + 4*K1**2 + 1
2-strand cable arrow polynomial -128*K2**8 + 256*K2**6*K4 - 1856*K2**6 - 288*K2**4*K4**2 + 2848*K2**4*K4 - 5376*K2**4 + 96*K2**3*K4*K6 - 352*K2**3*K6 - 816*K2**2*K4**2 + 3568*K2**2*K4 + 1936*K2**2 + 112*K2*K4*K6 - 528*K4**2 + 526
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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