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Flat knot 6.856

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,3,3,3,1,2,3,1,0,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.856']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+54t^5+78t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.856']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 384*K1**4*K2**2 + 960*K1**4*K2 - 3248*K1**4 + 864*K1**3*K2*K3 - 544*K1**3*K3 - 320*K1**2*K2**4 + 704*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 6864*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 9688*K1**2*K2 - 912*K1**2*K3**2 - 32*K1**2*K4**2 - 5480*K1**2 + 1120*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1152*K1*K2**2*K3 - 288*K1*K2**2*K5 + 192*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8640*K1*K2*K3 + 1296*K1*K3*K4 + 152*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1744*K2**4 - 1520*K2**2*K3**2 - 88*K2**2*K4**2 + 2056*K2**2*K4 - 4652*K2**2 - 96*K2*K3**2*K4 + 1120*K2*K3*K5 + 64*K2*K4*K6 - 96*K3**4 + 88*K3**2*K6 - 2488*K3**2 - 776*K4**2 - 248*K5**2 - 28*K6**2 + 5134
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.856']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11038', 'vk6.11118', 'vk6.12204', 'vk6.12313', 'vk6.16431', 'vk6.19224', 'vk6.19335', 'vk6.19517', 'vk6.19628', 'vk6.22738', 'vk6.22839', 'vk6.26036', 'vk6.26099', 'vk6.26420', 'vk6.26521', 'vk6.30607', 'vk6.30704', 'vk6.31917', 'vk6.34778', 'vk6.38103', 'vk6.38137', 'vk6.42395', 'vk6.44627', 'vk6.44761', 'vk6.51835', 'vk6.52705', 'vk6.52801', 'vk6.56569', 'vk6.56641', 'vk6.64723', 'vk6.66269', 'vk6.66307']
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 O1O2O3O4U1U4O5O6U2U6U3U5
R3 orbit {'O1O2O3O4U1U4O5O6U2U6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U6U3O6O5U1U4
Gauss code of K* O1O2O3O4U5U1U3U6O5O6U4U2
Gauss code of -K* O1O2O3O4U3U1O5O6U5U2U4U6
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 -2 1 1 2 1],[ 3 0 2 3 1 2 1],[ 2 -2 0 2 0 3 1],[-1 -3 -2 0 0 1 0],[-1 -1 0 0 0 0 0],[-2 -2 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 0 0 -1 -3 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[-1 1 0 0 0 -2 -3],[ 2 3 0 1 2 0 -2],[ 3 2 1 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,0,0,1,3,2,0,0,0,1,0,1,1,2,3,2]
Phi over symmetry [-3,-2,1,1,1,2,-1,1,3,3,3,1,2,3,1,0,0,0,0,1,1]
Phi of -K [-3,-2,1,1,1,2,-1,1,3,3,3,1,2,3,1,0,0,0,0,1,1]
Phi of K* [-2,-1,-1,-1,2,3,0,1,1,1,3,0,0,1,1,0,2,3,3,3,-1]
Phi of -K* [-3,-2,1,1,1,2,2,1,1,3,2,0,1,2,3,0,0,0,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+34t^4+28t^2+1
Outer characteristic polynomial t^7+54t^5+78t^3+11t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**6 - 384*K1**4*K2**2 + 960*K1**4*K2 - 3248*K1**4 + 864*K1**3*K2*K3 - 544*K1**3*K3 - 320*K1**2*K2**4 + 704*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 6864*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 9688*K1**2*K2 - 912*K1**2*K3**2 - 32*K1**2*K4**2 - 5480*K1**2 + 1120*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1152*K1*K2**2*K3 - 288*K1*K2**2*K5 + 192*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8640*K1*K2*K3 + 1296*K1*K3*K4 + 152*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1744*K2**4 - 1520*K2**2*K3**2 - 88*K2**2*K4**2 + 2056*K2**2*K4 - 4652*K2**2 - 96*K2*K3**2*K4 + 1120*K2*K3*K5 + 64*K2*K4*K6 - 96*K3**4 + 88*K3**2*K6 - 2488*K3**2 - 776*K4**2 - 248*K5**2 - 28*K6**2 + 5134
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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