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Flat knot 6.534

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,3,2,4,2,2,2,2,2,1,3,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.534']
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^7+90t^5+187t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.534']
2-strand cable arrow polynomial of the knot is: -512*K1**2*K2**4 + 1184*K1**2*K2**3 - 5696*K1**2*K2**2 - 96*K1**2*K2*K4 + 4216*K1**2*K2 - 2320*K1**2 + 864*K1*K2**3*K3 - 416*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3680*K1*K2*K3 - 1216*K2**6 + 1088*K2**4*K4 - 4416*K2**4 - 128*K2**3*K6 - 256*K2**2*K3**2 - 112*K2**2*K4**2 + 2848*K2**2*K4 + 496*K2**2 + 40*K2*K3*K5 - 560*K3**2 - 288*K4**2 + 1726
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.534']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73998', 'vk6.74000', 'vk6.74002', 'vk6.74004', 'vk6.74526', 'vk6.74528', 'vk6.75996', 'vk6.76002', 'vk6.76739', 'vk6.76741', 'vk6.78974', 'vk6.78980', 'vk6.79526', 'vk6.79528', 'vk6.80970', 'vk6.80972', 'vk6.83028', 'vk6.83635', 'vk6.83638', 'vk6.83962', 'vk6.85200', 'vk6.85204', 'vk6.85209', 'vk6.85213', 'vk6.85285', 'vk6.85289', 'vk6.86559', 'vk6.86564', 'vk6.87481', 'vk6.89296', 'vk6.89300', 'vk6.89821']
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 O1O2O3O4O5U3U1U2O6U4U5U6
R3 orbit {'O1O2O3O4O5U3U1U2O6U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U1U2O6U4U5U3
Gauss code of K* O1O2O3U4O5O6O4U2U3U1U5U6
Gauss code of -K* O1O2O3U1O4O5O6U2U3U6U4U5
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 -3 -1 -2 1 3 2],[ 3 0 1 0 3 4 2],[ 1 -1 0 0 2 3 2],[ 2 0 0 0 1 2 2],[-1 -3 -2 -1 0 1 2],[-3 -4 -3 -2 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 1 -1 -3 -2 -4],[-2 -1 0 -2 -2 -2 -2],[-1 1 2 0 -2 -1 -3],[ 1 3 2 2 0 0 -1],[ 2 2 2 1 0 0 0],[ 3 4 2 3 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,-1,1,3,2,4,2,2,2,2,2,1,3,0,1,0]
Phi over symmetry [-3,-2,-1,1,2,3,-1,1,3,2,4,2,2,2,2,2,1,3,0,1,0]
Phi of -K [-3,-2,-1,1,2,3,1,1,1,3,2,1,2,2,3,0,1,1,-1,1,2]
Phi of K* [-3,-2,-1,1,2,3,2,1,1,3,2,-1,1,2,3,0,2,1,1,1,1]
Phi of -K* [-3,-2,-1,1,2,3,0,1,3,2,4,0,1,2,2,2,2,3,2,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -6w^4z^2+8w^3z^2-16w^3z+23w^2z+7w
Inner characteristic polynomial t^6+62t^4+69t^2+1
Outer characteristic polynomial t^7+90t^5+187t^3+19t
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -512*K1**2*K2**4 + 1184*K1**2*K2**3 - 5696*K1**2*K2**2 - 96*K1**2*K2*K4 + 4216*K1**2*K2 - 2320*K1**2 + 864*K1*K2**3*K3 - 416*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3680*K1*K2*K3 - 1216*K2**6 + 1088*K2**4*K4 - 4416*K2**4 - 128*K2**3*K6 - 256*K2**2*K3**2 - 112*K2**2*K4**2 + 2848*K2**2*K4 + 496*K2**2 + 40*K2*K3*K5 - 560*K3**2 - 288*K4**2 + 1726
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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