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Flat knot 6.953

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,2,2,1,1,1,1,1,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.953']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.324', '6.672', '6.953', '6.1196', '6.1215', '6.1216', '6.1699']
Outer characteristic polynomial of the knot is: t^7+29t^5+60t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.953']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 1536*K1**4*K2 - 3792*K1**4 + 704*K1**3*K2*K3 - 416*K1**3*K3 - 1920*K1**2*K2**4 + 4032*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 11296*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 11952*K1**2*K2 - 560*K1**2*K3**2 - 80*K1**2*K4**2 - 5420*K1**2 - 256*K1*K2**4*K3 + 2528*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2688*K1*K2**2*K3 - 192*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 8520*K1*K2*K3 + 1064*K1*K3*K4 + 96*K1*K4*K5 - 352*K2**6 + 448*K2**4*K4 - 2768*K2**4 - 1072*K2**2*K3**2 - 192*K2**2*K4**2 + 1984*K2**2*K4 - 3246*K2**2 + 328*K2*K3*K5 + 16*K2*K4*K6 - 1812*K3**2 - 464*K4**2 - 40*K5**2 - 2*K6**2 + 4574
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.953']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4698', 'vk6.5001', 'vk6.6188', 'vk6.6659', 'vk6.8181', 'vk6.8599', 'vk6.9563', 'vk6.9902', 'vk6.17394', 'vk6.20915', 'vk6.20979', 'vk6.22327', 'vk6.22403', 'vk6.23565', 'vk6.23902', 'vk6.28391', 'vk6.36162', 'vk6.40045', 'vk6.40172', 'vk6.42098', 'vk6.43077', 'vk6.43381', 'vk6.46573', 'vk6.46677', 'vk6.48738', 'vk6.49538', 'vk6.49741', 'vk6.51434', 'vk6.55552', 'vk6.58907', 'vk6.65290', 'vk6.69759']
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 O1O2O3O4U5U1O6O5U6U3U4U2
R3 orbit {'O1O2O3O4U5U1O6O5U6U3U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U2U5O6O5U4U6
Gauss code of K* O1O2O3O4U5U4U2U3O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U2U3U1U6
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 -2 1 0 2 0 -1],[ 2 0 2 0 1 2 -1],[-1 -2 0 -1 1 0 -1],[ 0 0 1 0 1 1 -1],[-2 -1 -1 -1 0 -1 -1],[ 0 -2 0 -1 1 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -1 -1 -1],[-1 1 0 0 -1 -1 -2],[ 0 1 0 0 -1 -1 -2],[ 0 1 1 1 0 -1 0],[ 1 1 1 1 1 0 1],[ 2 1 2 2 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,1,1,1,0,1,1,2,1,1,2,1,0,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,2,2,1,1,1,1,1,1,1,1,0,1,1]
Phi of -K [-2,-1,0,0,1,2,2,0,2,1,3,0,0,1,2,1,1,1,0,1,0]
Phi of K* [-2,-1,0,0,1,2,0,1,1,2,3,0,1,1,1,1,0,2,0,0,2]
Phi of -K* [-2,-1,0,0,1,2,-1,0,2,2,1,1,1,1,1,1,1,1,0,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+23z+31
Enhanced Jones-Krushkal polynomial 4w^3z^2-2w^3z+25w^2z+31w
Inner characteristic polynomial t^6+19t^4+14t^2
Outer characteristic polynomial t^7+29t^5+60t^3+9t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -256*K1**6 - 576*K1**4*K2**2 + 1536*K1**4*K2 - 3792*K1**4 + 704*K1**3*K2*K3 - 416*K1**3*K3 - 1920*K1**2*K2**4 + 4032*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 11296*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 11952*K1**2*K2 - 560*K1**2*K3**2 - 80*K1**2*K4**2 - 5420*K1**2 - 256*K1*K2**4*K3 + 2528*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2688*K1*K2**2*K3 - 192*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 8520*K1*K2*K3 + 1064*K1*K3*K4 + 96*K1*K4*K5 - 352*K2**6 + 448*K2**4*K4 - 2768*K2**4 - 1072*K2**2*K3**2 - 192*K2**2*K4**2 + 1984*K2**2*K4 - 3246*K2**2 + 328*K2*K3*K5 + 16*K2*K4*K6 - 1812*K3**2 - 464*K4**2 - 40*K5**2 - 2*K6**2 + 4574
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}]]
If K is slice False
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