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Flat knot 6.543

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,0,2,1,0,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.543']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794']
Outer characteristic polynomial of the knot is: t^7+53t^5+50t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.543']
2-strand cable arrow polynomial of the knot is: -704*K1**4*K2**2 + 1920*K1**4*K2 - 3792*K1**4 - 384*K1**3*K2**2*K3 + 1120*K1**3*K2*K3 - 800*K1**3*K3 - 704*K1**2*K2**4 + 3872*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 12080*K1**2*K2**2 - 1344*K1**2*K2*K4 + 11616*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5680*K1**2 + 2304*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 320*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8880*K1*K2*K3 - 64*K1*K2*K4*K5 + 1184*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**6 + 160*K2**4*K4 - 3160*K2**4 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 384*K2**2*K4**2 + 2448*K2**2*K4 - 3196*K2**2 - 96*K2*K3**2*K4 + 448*K2*K3*K5 + 176*K2*K4*K6 - 1912*K3**2 - 670*K4**2 - 64*K5**2 - 12*K6**2 + 4732
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.543']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4686', 'vk6.4989', 'vk6.6164', 'vk6.6637', 'vk6.8159', 'vk6.8575', 'vk6.9547', 'vk6.9890', 'vk6.20697', 'vk6.22135', 'vk6.28222', 'vk6.29645', 'vk6.39678', 'vk6.41917', 'vk6.46258', 'vk6.47863', 'vk6.48718', 'vk6.48933', 'vk6.49498', 'vk6.49707', 'vk6.50740', 'vk6.50945', 'vk6.51217', 'vk6.51414', 'vk6.57624', 'vk6.58780', 'vk6.62300', 'vk6.63231', 'vk6.67090', 'vk6.67952', 'vk6.69694', 'vk6.70375']
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 O1O2O3O4O5U3U4U2O6U5U1U6
R3 orbit {'O1O2O3O4O5U3U4U2O6U5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U1O6U4U2U3
Gauss code of K* O1O2O3U4O5O6O4U6U3U1U2U5
Gauss code of -K* O1O2O3U1O4O5O6U3U5U6U4U2
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 -2 0 2 2],[ 1 0 -1 -2 0 3 2],[ 1 1 0 -1 1 3 1],[ 2 2 1 0 1 2 1],[ 0 0 -1 -1 0 1 1],[-2 -3 -3 -2 -1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 1 -1 -3 -3 -2],[-2 -1 0 -1 -1 -2 -1],[ 0 1 1 0 -1 0 -1],[ 1 3 1 1 0 1 -1],[ 1 3 2 0 -1 0 -2],[ 2 2 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,-1,1,3,3,2,1,1,2,1,1,0,1,-1,1,2]
Phi over symmetry [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,0,2,1,0,1,-1,-1,0]
Phi of -K [-2,-1,-1,0,2,2,-1,0,1,2,3,1,1,0,1,0,0,2,1,1,-1]
Phi of K* [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,0,2,1,0,1,-1,-1,0]
Phi of -K* [-2,-1,-1,0,2,2,1,2,1,1,2,1,1,1,3,0,2,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+39t^4+21t^2+1
Outer characteristic polynomial t^7+53t^5+50t^3+8t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -704*K1**4*K2**2 + 1920*K1**4*K2 - 3792*K1**4 - 384*K1**3*K2**2*K3 + 1120*K1**3*K2*K3 - 800*K1**3*K3 - 704*K1**2*K2**4 + 3872*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 12080*K1**2*K2**2 - 1344*K1**2*K2*K4 + 11616*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5680*K1**2 + 2304*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 320*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8880*K1*K2*K3 - 64*K1*K2*K4*K5 + 1184*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**6 + 160*K2**4*K4 - 3160*K2**4 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 384*K2**2*K4**2 + 2448*K2**2*K4 - 3196*K2**2 - 96*K2*K3**2*K4 + 448*K2*K3*K5 + 176*K2*K4*K6 - 1912*K3**2 - 670*K4**2 - 64*K5**2 - 12*K6**2 + 4732
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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