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Flat knot 6.760

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,2,2,4,1,0,0,1,1,1,2,0,0,-2]
Flat knots (up to 7 crossings) with same phi are :['6.760']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^7+58t^5+53t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.760']
2-strand cable arrow polynomial of the knot is: -1216*K1**4*K2**2 + 1920*K1**4*K2 - 2160*K1**4 + 896*K1**3*K2*K3 - 800*K1**3*K3 + 2656*K1**2*K2**3 - 8432*K1**2*K2**2 - 544*K1**2*K2*K4 + 7744*K1**2*K2 - 272*K1**2*K3**2 - 4312*K1**2 + 352*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6896*K1*K2*K3 + 440*K1*K3*K4 + 88*K1*K4*K5 - 1648*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 1304*K2**2*K4 - 2774*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 - 1588*K3**2 - 328*K4**2 - 52*K5**2 - 2*K6**2 + 3446
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.760']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4197', 'vk6.4278', 'vk6.5451', 'vk6.5565', 'vk6.7558', 'vk6.7645', 'vk6.9062', 'vk6.9143', 'vk6.18238', 'vk6.18573', 'vk6.24710', 'vk6.25123', 'vk6.36829', 'vk6.37292', 'vk6.44069', 'vk6.44408', 'vk6.48509', 'vk6.48590', 'vk6.49201', 'vk6.49309', 'vk6.50294', 'vk6.50370', 'vk6.51059', 'vk6.51092', 'vk6.56041', 'vk6.56315', 'vk6.60594', 'vk6.60933', 'vk6.65703', 'vk6.65997', 'vk6.68748', 'vk6.68956']
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 O1O2O3O4U2O5U6U5U1O6U3U4
R3 orbit {'O1O2O3O4U2O5U6U5U1O6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2O5U4U6U5O6U3
Gauss code of K* O1O2O3U1O4O5U3U6U4U5O6U2
Gauss code of -K* O1O2O3U2O4U5U6U4U1O5O6U3
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 -2 1 3 1 -2],[ 1 0 0 1 2 0 0],[ 2 0 0 1 2 0 2],[-1 -1 -1 0 1 1 -2],[-3 -2 -2 -1 0 1 -4],[-1 0 0 -1 -1 0 -1],[ 2 0 -2 2 4 1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 1 -1 -2 -2 -4],[-1 -1 0 -1 0 0 -1],[-1 1 1 0 -1 -1 -2],[ 1 2 0 1 0 0 0],[ 2 2 0 1 0 0 2],[ 2 4 1 2 0 -2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,-1,1,2,2,4,1,0,0,1,1,1,2,0,0,-2]
Phi over symmetry [-3,-1,-1,1,2,2,-1,1,2,2,4,1,0,0,1,1,1,2,0,0,-2]
Phi of -K [-2,-2,-1,1,1,3,-2,1,2,3,3,1,1,2,1,1,2,2,-1,1,3]
Phi of K* [-3,-1,-1,1,2,2,1,3,2,1,3,1,1,1,2,2,2,3,1,1,-2]
Phi of -K* [-2,-2,-1,1,1,3,-2,0,1,2,4,0,0,1,2,0,1,2,-1,-1,1]
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+38t^4+33t^2+4
Outer characteristic polynomial t^7+58t^5+53t^3+7t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -1216*K1**4*K2**2 + 1920*K1**4*K2 - 2160*K1**4 + 896*K1**3*K2*K3 - 800*K1**3*K3 + 2656*K1**2*K2**3 - 8432*K1**2*K2**2 - 544*K1**2*K2*K4 + 7744*K1**2*K2 - 272*K1**2*K3**2 - 4312*K1**2 + 352*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6896*K1*K2*K3 + 440*K1*K3*K4 + 88*K1*K4*K5 - 1648*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 1304*K2**2*K4 - 2774*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 - 1588*K3**2 - 328*K4**2 - 52*K5**2 - 2*K6**2 + 3446
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{6}, {3, 5}, {1, 4}, {2}]]
If K is slice False
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