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Flat knot 6.683

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,1,2,3,1,1,1,1,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.683']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370']
Outer characteristic polynomial of the knot is: t^7+56t^5+38t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.683']
2-strand cable arrow polynomial of the knot is: 3136*K1**4*K2 - 6496*K1**4 + 512*K1**3*K2*K3 - 1248*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5632*K1**2*K2**2 - 320*K1**2*K2*K4 + 7712*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 864*K1**2 + 192*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3472*K1*K2*K3 + 192*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 432*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 256*K2**2*K4 - 1784*K2**2 - 448*K3**2 - 28*K4**2 + 1994
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.683']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20142', 'vk6.20152', 'vk6.20158', 'vk6.20168', 'vk6.21434', 'vk6.21446', 'vk6.27254', 'vk6.27276', 'vk6.27282', 'vk6.27296', 'vk6.28916', 'vk6.28938', 'vk6.28944', 'vk6.38675', 'vk6.38697', 'vk6.38707', 'vk6.38725', 'vk6.40887', 'vk6.40905', 'vk6.47263', 'vk6.47278', 'vk6.47296', 'vk6.56975', 'vk6.56977', 'vk6.56994', 'vk6.57002', 'vk6.58129', 'vk6.62678', 'vk6.62682', 'vk6.67468', 'vk6.70036', 'vk6.70048']
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 O1O2O3O4U3O5U2U1O6U4U6U5
R3 orbit {'O1O2O3O4U3O5U2U1O6U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1O6U4U3O5U2
Gauss code of K* O1O2O3U4U5U6U1O6U3O5O4U2
Gauss code of -K* O1O2O3U2O4O5U1O6U3U6U5U4
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 -2 -1 1 3 1],[ 2 0 0 0 3 3 1],[ 2 0 0 0 2 2 1],[ 1 0 0 0 1 1 1],[-1 -3 -2 -1 0 2 1],[-3 -3 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -2 -1 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -1 -2 -3],[ 1 1 1 1 0 0 0],[ 2 2 1 2 0 0 0],[ 2 3 1 3 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,2,1,2,3,1,1,1,1,1,2,3,0,0,0]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,1,2,3,1,1,1,1,1,2,3,0,0,0]
Phi of -K [-2,-2,-1,1,1,3,0,1,0,2,2,1,1,2,3,1,1,3,-1,0,2]
Phi of K* [-3,-1,-1,1,2,2,0,2,3,2,3,1,1,0,1,1,2,2,1,1,0]
Phi of -K* [-2,-2,-1,1,1,3,0,0,1,2,2,0,1,3,3,1,1,1,-1,0,2]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+36t^4+14t^2
Outer characteristic polynomial t^7+56t^5+38t^3+3t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial 3136*K1**4*K2 - 6496*K1**4 + 512*K1**3*K2*K3 - 1248*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5632*K1**2*K2**2 - 320*K1**2*K2*K4 + 7712*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 864*K1**2 + 192*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3472*K1*K2*K3 + 192*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 432*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 256*K2**2*K4 - 1784*K2**2 - 448*K3**2 - 28*K4**2 + 1994
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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