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Flat knot 6.779

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,2,3,1,1,1,0,1,0,0,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.779']
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+38t^5+73t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.779']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 448*K1**4*K2 - 768*K1**4 + 128*K1**2*K2**3 - 800*K1**2*K2**2 + 1200*K1**2*K2 - 64*K1**2*K3**2 - 600*K1**2 + 688*K1*K2*K3 + 144*K1*K3*K4 - 96*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 136*K2**2*K4 - 614*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 256*K3**2 - 108*K4**2 - 24*K5**2 - 2*K6**2 + 682
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.779']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4414', 'vk6.4509', 'vk6.5796', 'vk6.5923', 'vk6.6412', 'vk6.6843', 'vk6.7964', 'vk6.8373', 'vk6.9275', 'vk6.9394', 'vk6.17894', 'vk6.17959', 'vk6.18631', 'vk6.24397', 'vk6.25185', 'vk6.30038', 'vk6.30099', 'vk6.36906', 'vk6.37364', 'vk6.39840', 'vk6.43832', 'vk6.44127', 'vk6.44450', 'vk6.46402', 'vk6.47979', 'vk6.48621', 'vk6.49916', 'vk6.50601']
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 O1O2O3O4U3O5U4U6U1O6U2U5
R3 orbit {'O1O2O3O4U3O5U4U2U6U1O6U5', 'O1O2O3O4U3O5U4U6U1O6U2U5', 'O1O2O3U2O4O5U3U4U6U1O6U5'}
R3 orbit length 3
Gauss code of -K O1O2O3O4U5U3O6U4U6U1O5U2
Gauss code of K* Same
Gauss code of -K* O1O2O3U4O5U3U5U6U1O4O6U2
Diagrammatic symmetry type +
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 0 -1 0 3 -1],[ 1 0 0 -1 1 2 1],[ 0 0 0 -1 1 2 0],[ 1 1 1 0 1 1 1],[ 0 -1 -1 -1 0 1 0],[-3 -2 -2 -1 -1 0 -3],[ 1 -1 0 -1 0 3 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 -1 -2 -1 -2 -3],[ 0 1 0 -1 -1 -1 0],[ 0 2 1 0 -1 0 0],[ 1 1 1 1 0 1 1],[ 1 2 1 0 -1 0 1],[ 1 3 0 0 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,1,2,1,2,3,1,1,1,0,1,0,0,-1,-1,-1]
Phi over symmetry [-3,0,0,1,1,1,1,2,1,2,3,1,1,1,0,1,0,0,-1,-1,-1]
Phi of -K [-1,-1,-1,0,0,3,-1,-1,0,0,3,-1,0,1,2,1,1,1,1,2,1]
Phi of K* [-3,0,0,1,1,1,1,2,1,2,3,1,1,1,0,1,0,0,-1,-1,-1]
Phi of -K* [-1,-1,-1,0,0,3,-1,-1,0,0,3,-1,0,1,2,1,1,1,1,2,1]
Symmetry type of based matrix +
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 8z+17
Enhanced Jones-Krushkal polynomial 8w^2z+17w
Inner characteristic polynomial t^6+26t^4+29t^2
Outer characteristic polynomial t^7+38t^5+73t^3
Flat arrow polynomial -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -128*K1**4*K2**2 + 448*K1**4*K2 - 768*K1**4 + 128*K1**2*K2**3 - 800*K1**2*K2**2 + 1200*K1**2*K2 - 64*K1**2*K3**2 - 600*K1**2 + 688*K1*K2*K3 + 144*K1*K3*K4 - 96*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 136*K2**2*K4 - 614*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 256*K3**2 - 108*K4**2 - 24*K5**2 - 2*K6**2 + 682
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {5}, {3, 4}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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