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Flat knot 6.1690

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,2,3,1,1,0,1,1,1,0,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1690']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 4*K1*K2 - 4*K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1004', '6.1690']
Outer characteristic polynomial of the knot is: t^7+24t^5+39t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1690']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1216*K1**4*K2**2 + 2560*K1**4*K2 - 3104*K1**4 - 256*K1**3*K2**2*K3 + 480*K1**3*K2*K3 - 448*K1**3*K3 - 832*K1**2*K2**4 + 4992*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12544*K1**2*K2**2 - 576*K1**2*K2*K4 + 10656*K1**2*K2 - 32*K1**2*K3**2 - 4996*K1**2 + 1280*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7440*K1*K2*K3 + 200*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 3288*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 1960*K2**2*K4 - 2328*K2**2 + 80*K2*K3*K5 - 1140*K3**2 - 178*K4**2 + 3816
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1690']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19941', 'vk6.20053', 'vk6.21188', 'vk6.21337', 'vk6.26906', 'vk6.27118', 'vk6.28662', 'vk6.28808', 'vk6.38330', 'vk6.38511', 'vk6.40472', 'vk6.40712', 'vk6.45203', 'vk6.45411', 'vk6.47028', 'vk6.47156', 'vk6.56730', 'vk6.56867', 'vk6.57832', 'vk6.58006', 'vk6.61155', 'vk6.61396', 'vk6.62398', 'vk6.62555', 'vk6.66429', 'vk6.66574', 'vk6.67202', 'vk6.67365', 'vk6.69082', 'vk6.69226', 'vk6.69865', 'vk6.69966']
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 O1O2O3U4O5U3U1U2O4O6U5U6
R3 orbit {'O1O2O3U4O5U3U1U2O4O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4O6U2U3U1O5U6
Gauss code of K* O1O2O3U4U5O6O5U2U3U1O4U6
Gauss code of -K* O1O2O3U4O5U3U1U2O6O4U6U5
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 -1 1 0 -2 1 1],[ 1 0 1 0 -1 2 1],[-1 -1 0 0 -2 1 1],[ 0 0 0 0 0 0 1],[ 2 1 2 0 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 1 0 -2 -1],[-1 -1 -1 0 -1 -1 0],[ 0 0 0 1 0 0 0],[ 1 1 2 1 0 0 -1],[ 2 2 1 0 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,-1,0,1,2,-1,0,2,1,1,1,0,0,0,1]
Phi over symmetry [-2,-1,0,1,1,1,0,2,1,2,3,1,1,0,1,1,1,0,-1,-1,-1]
Phi of -K [-2,-1,0,1,1,1,0,2,1,2,3,1,1,0,1,1,1,0,-1,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,0,1,3,-1,1,0,2,1,1,1,1,2,0]
Phi of -K* [-2,-1,0,1,1,1,1,0,0,1,2,0,1,2,1,1,0,0,-1,-1,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+16t^4+18t^2+4
Outer characteristic polynomial t^7+24t^5+39t^3+14t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 4*K1*K2 - 4*K1 + 5*K2 + 6
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1216*K1**4*K2**2 + 2560*K1**4*K2 - 3104*K1**4 - 256*K1**3*K2**2*K3 + 480*K1**3*K2*K3 - 448*K1**3*K3 - 832*K1**2*K2**4 + 4992*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12544*K1**2*K2**2 - 576*K1**2*K2*K4 + 10656*K1**2*K2 - 32*K1**2*K3**2 - 4996*K1**2 + 1280*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7440*K1*K2*K3 + 200*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 3288*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 1960*K2**2*K4 - 2328*K2**2 + 80*K2*K3*K5 - 1140*K3**2 - 178*K4**2 + 3816
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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