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Flat knot 6.829

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,1,1,0,2,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.829']
Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384']
Outer characteristic polynomial of the knot is: t^7+51t^5+109t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.829']
2-strand cable arrow polynomial of the knot is: -1088*K1**4 - 320*K1**3*K3 + 1280*K1**2*K2**3 - 4576*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 8768*K1**2*K2 - 192*K1**2*K3**2 - 7264*K1**2 + 384*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 6032*K1*K2*K3 + 560*K1*K3*K4 - 1088*K2**4 - 448*K2**2*K3**2 - 8*K2**2*K4**2 + 952*K2**2*K4 - 4390*K2**2 + 208*K2*K3*K5 + 8*K2*K4*K6 - 1856*K3**2 - 276*K4**2 - 16*K5**2 - 2*K6**2 + 4802
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.829']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72419', 'vk6.72468', 'vk6.72488', 'vk6.72829', 'vk6.72849', 'vk6.72890', 'vk6.72906', 'vk6.74472', 'vk6.74478', 'vk6.75080', 'vk6.75081', 'vk6.76971', 'vk6.77779', 'vk6.77968', 'vk6.79468', 'vk6.79920', 'vk6.79926', 'vk6.80936', 'vk6.87235', 'vk6.89359']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4U2O5U4U6U3U1O6U5
R3 orbit {'O1O2O3O4U2O5U4U6U3U1O6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5O6U4U2U6U1O5U3
Gauss code of K* Same
Gauss code of -K* O1O2O3O4U5O6U4U2U6U1O5U3
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 0 -2 1 0 3 -2],[ 0 0 -2 1 1 2 -1],[ 2 2 0 2 1 2 1],[-1 -1 -2 0 0 1 -1],[ 0 -1 -1 0 0 1 0],[-3 -2 -2 -1 -1 0 -3],[ 2 1 -1 1 0 3 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 -1 -1 -2 -2 -3],[-1 1 0 0 -1 -2 -1],[ 0 1 0 0 -1 -1 0],[ 0 2 1 1 0 -2 -1],[ 2 2 2 1 2 0 1],[ 2 3 1 0 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,1,1,0,2,1,-1]
Phi over symmetry [-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,1,1,0,2,1,-1]
Phi of -K [-2,-2,0,0,1,3,-1,0,1,1,3,1,2,2,2,-1,0,1,1,2,1]
Phi of K* [-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,1,1,0,2,1,-1]
Phi of -K* [-2,-2,0,0,1,3,-1,0,1,1,3,1,2,2,2,-1,0,1,1,2,1]
Symmetry type of based matrix +
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial -4w^3z+21w^2z+35w
Inner characteristic polynomial t^6+33t^4+51t^2+4
Outer characteristic polynomial t^7+51t^5+109t^3+16t
Flat arrow polynomial -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -1088*K1**4 - 320*K1**3*K3 + 1280*K1**2*K2**3 - 4576*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 8768*K1**2*K2 - 192*K1**2*K3**2 - 7264*K1**2 + 384*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 6032*K1*K2*K3 + 560*K1*K3*K4 - 1088*K2**4 - 448*K2**2*K3**2 - 8*K2**2*K4**2 + 952*K2**2*K4 - 4390*K2**2 + 208*K2*K3*K5 + 8*K2*K4*K6 - 1856*K3**2 - 276*K4**2 - 16*K5**2 - 2*K6**2 + 4802
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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