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Flat knot 6.682

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,1,3,3,0,1,1,1,0,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.682']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.520', '6.682', '6.706', '6.748', '6.1331']
Outer characteristic polynomial of the knot is: t^7+39t^5+35t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.682']
2-strand cable arrow polynomial of the knot is: 3008*K1**4*K2 - 8864*K1**4 + 1600*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7296*K1**2*K2**2 - 608*K1**2*K2*K4 + 9760*K1**2*K2 - 1088*K1**2*K3**2 - 32*K1**2*K4**2 - 88*K1**2 + 256*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5088*K1*K2*K3 + 736*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 544*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 480*K2**2*K4 - 2256*K2**2 + 64*K2*K3*K5 - 856*K3**2 - 112*K4**2 + 2438
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.682']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13812', 'vk6.13816', 'vk6.13817', 'vk6.13828', 'vk6.13831', 'vk6.13835', 'vk6.13838', 'vk6.13845', 'vk6.13854', 'vk6.13856', 'vk6.13860', 'vk6.13861', 'vk6.14887', 'vk6.14888', 'vk6.14891', 'vk6.14897', 'vk6.14904', 'vk6.14910', 'vk6.14913', 'vk6.14914', 'vk6.14918', 'vk6.14921', 'vk6.14922', 'vk6.14933', 'vk6.34236', 'vk6.34240', 'vk6.53834', 'vk6.53836', 'vk6.53840', 'vk6.53844', 'vk6.54378', 'vk6.54383']
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 O1O2O3O4U3O5U1U5O6U4U6U2
R3 orbit {'O1O2O3O4U3O5U1U5O6U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U1O5U6U4O6U2
Gauss code of K* O1O2O3U4U3U5U1O5U6O4O6U2
Gauss code of -K* O1O2O3U2O4O5U4O6U3U6U1U5
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 1 1 1],[ 3 0 3 0 3 1 1],[-1 -3 0 -1 0 0 1],[ 1 0 1 0 1 0 1],[-1 -3 0 -1 0 0 1],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -1 -3],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 1 0 0 -1 -3],[ 1 1 1 0 1 0 0],[ 3 3 1 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,0,0,1,3,0,1,1,1,0,0,1,1,3,0]
Phi over symmetry [-3,-1,1,1,1,1,0,1,1,3,3,0,1,1,1,0,0,0,-1,-1,0]
Phi of -K [-3,-1,1,1,1,1,2,1,1,3,3,1,1,1,2,0,-1,0,-1,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,0,1,1,0,1,1,2,3,2]
Phi of -K* [-3,-1,1,1,1,1,0,1,1,3,3,0,1,1,1,0,0,0,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+25t^4+13t^2
Outer characteristic polynomial t^7+39t^5+35t^3+2t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial 3008*K1**4*K2 - 8864*K1**4 + 1600*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7296*K1**2*K2**2 - 608*K1**2*K2*K4 + 9760*K1**2*K2 - 1088*K1**2*K3**2 - 32*K1**2*K4**2 - 88*K1**2 + 256*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5088*K1*K2*K3 + 736*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 544*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 480*K2**2*K4 - 2256*K2**2 + 64*K2*K3*K5 - 856*K3**2 - 112*K4**2 + 2438
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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