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Flat knot 6.357

Min(phi) over symmetries of the knot is: [-4,-2,0,2,2,2,0,1,2,4,5,0,0,1,2,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.357']
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+88t^5+128t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.357']
2-strand cable arrow polynomial of the knot is: -128*K2**8 + 128*K2**6*K4 - 1088*K2**6 - 32*K2**4*K4**2 + 800*K2**4*K4 - 1504*K2**4 - 208*K2**2*K4**2 + 1592*K2**2*K4 + 384*K2**2 + 72*K2*K4*K6 - 336*K4**2 - 16*K6**2 + 334
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.357']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70473', 'vk6.70490', 'vk6.70528', 'vk6.70604', 'vk6.70651', 'vk6.70679', 'vk6.70758', 'vk6.70843', 'vk6.70933', 'vk6.70962', 'vk6.71009', 'vk6.71115', 'vk6.71163', 'vk6.71180', 'vk6.71247', 'vk6.71303', 'vk6.72397', 'vk6.72414', 'vk6.72756', 'vk6.73071', 'vk6.73618', 'vk6.74394', 'vk6.74925', 'vk6.75401', 'vk6.76490', 'vk6.76686', 'vk6.77734', 'vk6.78364', 'vk6.79438', 'vk6.79949', 'vk6.87186', 'vk6.90134']
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 O1O2O3O4O5U3U4O6U2U1U6U5
R3 orbit {'O1O2O3O4O5U3U4O6U2U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U5U4O6U2U3
Gauss code of K* O1O2O3O4U2U1U5U6U4O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U1U5U6U4U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -2 -2 0 4 2],[ 2 0 0 -1 1 5 2],[ 2 0 0 -1 1 4 1],[ 2 1 1 0 1 2 0],[ 0 -1 -1 -1 0 1 0],[-4 -5 -4 -2 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 4 2 0 -2 -2 -2],[-4 0 0 -1 -2 -4 -5],[-2 0 0 0 0 -1 -2],[ 0 1 0 0 -1 -1 -1],[ 2 2 0 1 0 1 1],[ 2 4 1 1 -1 0 0],[ 2 5 2 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,2,2,2,0,1,2,4,5,0,0,1,2,1,1,1,-1,-1,0]
Phi over symmetry [-4,-2,0,2,2,2,0,1,2,4,5,0,0,1,2,1,1,1,-1,-1,0]
Phi of -K [-2,-2,-2,0,2,4,-1,-1,1,4,4,0,1,2,1,1,3,2,2,3,2]
Phi of K* [-4,-2,0,2,2,2,2,3,1,2,4,2,2,3,4,1,1,1,0,-1,-1]
Phi of -K* [-2,-2,-2,0,2,4,-1,0,1,1,4,1,1,0,2,1,2,5,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^4+2t^2
Normalized Jones-Krushkal polynomial 5z^2+14z+9
Enhanced Jones-Krushkal polynomial -4w^4z^2+9w^3z^2+14w^2z+9
Inner characteristic polynomial t^6+56t^4+48t^2
Outer characteristic polynomial t^7+88t^5+128t^3
Flat arrow polynomial -8*K1**4 + 4*K1**2*K2 + 4*K1**2 + 1
2-strand cable arrow polynomial -128*K2**8 + 128*K2**6*K4 - 1088*K2**6 - 32*K2**4*K4**2 + 800*K2**4*K4 - 1504*K2**4 - 208*K2**2*K4**2 + 1592*K2**2*K4 + 384*K2**2 + 72*K2*K4*K6 - 336*K4**2 - 16*K6**2 + 334
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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