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Flat knot 6.133

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-2,0,1,2,2,1,2,2,2,0,0,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.133']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.133', '6.517', '6.545', '6.1198', '6.1251', '6.1906']
Outer characteristic polynomial of the knot is: t^7+49t^5+115t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.133']
2-strand cable arrow polynomial of the knot is: 768*K1**2*K2**5 - 2816*K1**2*K2**4 + 2592*K1**2*K2**3 - 2976*K1**2*K2**2 - 96*K1**2*K2*K4 + 1840*K1**2*K2 - 1184*K1**2 + 896*K1*K2**5*K3 - 1408*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3328*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 2240*K1*K2*K3 + 192*K1*K3*K4 - 128*K2**8 + 256*K2**6*K4 - 1984*K2**6 - 128*K2**5*K6 - 1536*K2**4*K3**2 - 64*K2**4*K4**2 + 1664*K2**4*K4 - 1632*K2**4 + 672*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1504*K2**2*K3**2 - 256*K2**2*K4**2 + 1200*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 + 288*K2**2 + 544*K2*K3*K5 + 56*K2*K4*K6 - 464*K3**2 - 178*K4**2 - 32*K5**2 - 8*K6**2 + 920
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.133']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4762', 'vk6.5093', 'vk6.6321', 'vk6.6757', 'vk6.8279', 'vk6.8731', 'vk6.9655', 'vk6.9968', 'vk6.20719', 'vk6.22167', 'vk6.28281', 'vk6.29699', 'vk6.39732', 'vk6.41984', 'vk6.46294', 'vk6.47879', 'vk6.48796', 'vk6.49009', 'vk6.49625', 'vk6.49827', 'vk6.50824', 'vk6.51040', 'vk6.51297', 'vk6.51492', 'vk6.57652', 'vk6.58800', 'vk6.62329', 'vk6.63265', 'vk6.67116', 'vk6.67981', 'vk6.69710', 'vk6.70389']
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 O1O2O3O4O5O6U4U5U6U1U3U2
R3 orbit {'O1O2O3O4O5O6U4U5U6U1U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U4U6U1U2U3
Gauss code of K* O1O2O3O4O5O6U4U6U5U1U2U3
Gauss code of -K* O1O2O3O4O5O6U4U5U6U2U1U3
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 1 -2 0 2],[ 2 0 2 1 -2 0 2],[-1 -2 0 0 -2 0 2],[-1 -1 0 0 -2 0 2],[ 2 2 2 2 0 1 2],[ 0 0 0 0 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 -2 -2 -1 -2 -2],[-1 2 0 0 0 -1 -2],[-1 2 0 0 0 -2 -2],[ 0 1 0 0 0 0 -1],[ 2 2 1 2 0 0 -2],[ 2 2 2 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,2,2,1,2,2,0,0,1,2,0,2,2,0,1,2]
Phi over symmetry [-2,-2,0,1,1,2,-2,0,1,2,2,1,2,2,2,0,0,1,0,2,2]
Phi of -K [-2,-2,0,1,1,2,-2,1,1,1,2,2,1,2,2,1,1,1,0,-1,-1]
Phi of K* [-2,-1,-1,0,2,2,-1,-1,1,2,2,0,1,1,1,1,1,2,1,2,2]
Phi of -K* [-2,-2,0,1,1,2,-2,0,1,2,2,1,2,2,2,0,0,1,0,2,2]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+8z+5
Enhanced Jones-Krushkal polynomial -6w^4z^2+9w^3z^2-10w^3z+18w^2z+5w
Inner characteristic polynomial t^6+35t^4+38t^2
Outer characteristic polynomial t^7+49t^5+115t^3+6t
Flat arrow polynomial -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
2-strand cable arrow polynomial 768*K1**2*K2**5 - 2816*K1**2*K2**4 + 2592*K1**2*K2**3 - 2976*K1**2*K2**2 - 96*K1**2*K2*K4 + 1840*K1**2*K2 - 1184*K1**2 + 896*K1*K2**5*K3 - 1408*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3328*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 2240*K1*K2*K3 + 192*K1*K3*K4 - 128*K2**8 + 256*K2**6*K4 - 1984*K2**6 - 128*K2**5*K6 - 1536*K2**4*K3**2 - 64*K2**4*K4**2 + 1664*K2**4*K4 - 1632*K2**4 + 672*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1504*K2**2*K3**2 - 256*K2**2*K4**2 + 1200*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 + 288*K2**2 + 544*K2*K3*K5 + 56*K2*K4*K6 - 464*K3**2 - 178*K4**2 - 32*K5**2 - 8*K6**2 + 920
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}]]
If K is slice False
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