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Flat knot 6.765

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,1,2,4,0,1,0,1,1,1,3,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.765']
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+111t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.765']
2-strand cable arrow polynomial of the knot is: -1216*K1**2*K2**4 + 1952*K1**2*K2**3 - 6800*K1**2*K2**2 + 4800*K1**2*K2 - 2696*K1**2 + 1088*K1*K2**3*K3 - 640*K1*K2**2*K3 - 128*K1*K2**2*K5 + 4592*K1*K2*K3 - 288*K2**6 + 160*K2**4*K4 - 1504*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 968*K2**2*K4 - 1088*K2**2 + 16*K2*K3*K5 - 824*K3**2 - 112*K4**2 + 1870
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.765']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16792', 'vk6.16796', 'vk6.16825', 'vk6.16829', 'vk6.18156', 'vk6.18158', 'vk6.18492', 'vk6.18494', 'vk6.23208', 'vk6.23212', 'vk6.24615', 'vk6.25028', 'vk6.25030', 'vk6.35226', 'vk6.35252', 'vk6.36750', 'vk6.37170', 'vk6.37172', 'vk6.42708', 'vk6.42711', 'vk6.44332', 'vk6.44334', 'vk6.54984', 'vk6.55020', 'vk6.55963', 'vk6.55965', 'vk6.59377', 'vk6.59381', 'vk6.60501', 'vk6.65627', 'vk6.68174', 'vk6.68178']
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 O1O2O3O4U3O5U1U2U5O6U4U6
R3 orbit {'O1O2O3O4U3O5U1U2U5O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1O5U6U3U4O6U2
Gauss code of K* O1O2O3U4O5O4U1U2U6U5O6U3
Gauss code of -K* O1O2O3U1O4U5U4U2U3O6O5U6
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 -1 -1 2 2 1],[ 3 0 1 0 4 2 1],[ 1 -1 0 0 3 1 1],[ 1 0 0 0 1 0 1],[-2 -4 -3 -1 0 0 1],[-2 -2 -1 0 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 1 -1 -3 -4],[-2 0 0 0 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[ 1 1 0 1 0 0 0],[ 1 3 1 1 0 0 -1],[ 3 4 2 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,-1,1,3,4,0,0,1,2,1,1,1,0,0,1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,1,2,4,0,1,0,1,1,1,3,0,-1,0]
Phi of -K [-3,-1,-1,1,2,2,1,2,3,1,3,0,1,0,2,1,2,3,2,1,0]
Phi of K* [-2,-2,-1,1,1,3,0,1,2,3,3,2,0,2,1,1,1,3,0,1,2]
Phi of -K* [-3,-1,-1,1,2,2,0,1,1,2,4,0,1,0,1,1,1,3,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -4w^4z^2+6w^3z^2-16w^3z+23w^2z+7w
Inner characteristic polynomial t^6+36t^4+35t^2+1
Outer characteristic polynomial t^7+56t^5+111t^3+15t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial -1216*K1**2*K2**4 + 1952*K1**2*K2**3 - 6800*K1**2*K2**2 + 4800*K1**2*K2 - 2696*K1**2 + 1088*K1*K2**3*K3 - 640*K1*K2**2*K3 - 128*K1*K2**2*K5 + 4592*K1*K2*K3 - 288*K2**6 + 160*K2**4*K4 - 1504*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 968*K2**2*K4 - 1088*K2**2 + 16*K2*K3*K5 - 824*K3**2 - 112*K4**2 + 1870
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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