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Flat knot 6.969

Min(phi) over symmetries of the knot is: [-2,-1,0,1,2,-1,1,1,2,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['5.21', '6.969']
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^6+23t^4+26t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.969']
2-strand cable arrow polynomial of the knot is: -2688*K1**2*K2**4 + 1792*K1**2*K2**3 - 3488*K1**2*K2**2 + 1888*K1**2*K2 - 608*K1**2 + 1664*K1*K2**3*K3 + 1760*K1*K2*K3 - 192*K2**6 + 128*K2**4*K4 - 640*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 192*K2**2*K4 + 144*K2**2 - 160*K3**2 - 16*K4**2 + 398
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.969']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3248', 'vk6.3267', 'vk6.3371', 'vk6.3403', 'vk6.3460', 'vk6.3509', 'vk6.17648', 'vk6.17655', 'vk6.24201', 'vk6.24218', 'vk6.36461', 'vk6.36478', 'vk6.43558', 'vk6.43575', 'vk6.48104', 'vk6.48139', 'vk6.48181', 'vk6.48193', 'vk6.60246', 'vk6.68536']
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 O1O2O3O4U5U4O6O5U6U1U2U3
R3 orbit {'O1O2O3O4U5U4O6O5U6U1U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3U4U5O6O5U1U6
Gauss code of K* O1O2O3O4U2U3U4U5O6O5U1U6
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 2 1 0 -1],[ 2 0 1 2 1 2 -1],[ 0 -1 0 1 1 0 -1],[-2 -2 -1 0 1 -2 -1],[-1 -1 -1 -1 0 -1 -1],[ 0 -2 0 2 1 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -2],[-2 0 1 -1 -1 -2],[-1 -1 0 -1 -1 -1],[ 0 1 1 0 -1 -1],[ 1 1 1 1 0 1],[ 2 2 1 1 -1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,0,1,2,-1,1,1,2,1,1,1,1,1,-1]
Phi over symmetry [-2,-1,0,1,2,-1,1,1,2,1,1,1,1,1,-1]
Phi of -K [-2,-1,0,1,2,2,1,2,2,0,1,2,0,1,2]
Phi of K* [-2,-1,0,1,2,2,1,2,2,0,1,2,0,1,2]
Phi of -K* [-2,-1,0,1,2,-1,1,1,2,1,1,1,1,1,-1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial -z-1
Enhanced Jones-Krushkal polynomial -16w^3z+15w^2z-w
Inner characteristic polynomial t^5+13t^3+6t
Outer characteristic polynomial t^6+23t^4+26t^2
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -2688*K1**2*K2**4 + 1792*K1**2*K2**3 - 3488*K1**2*K2**2 + 1888*K1**2*K2 - 608*K1**2 + 1664*K1*K2**3*K3 + 1760*K1*K2*K3 - 192*K2**6 + 128*K2**4*K4 - 640*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 192*K2**2*K4 + 144*K2**2 - 160*K3**2 - 16*K4**2 + 398
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}]]
If K is slice True
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