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Flat knot 6.2074

Min(phi) over symmetries of the knot is: [-1,0,0,0,0,1,0,0,0,1,1,-1,0,0,1,0,0,0,-1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.2074']
Arrow polynomial of the knot is: -20*K1**2 + 10*K2 + 11
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.2074']
Outer characteristic polynomial of the knot is: t^7+9t^5+17t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2074']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 2304*K1**4*K2 - 7168*K1**4 + 896*K1**3*K2*K3 - 1216*K1**3*K3 + 128*K1**2*K2**3 - 8800*K1**2*K2**2 - 320*K1**2*K2*K4 + 14960*K1**2*K2 - 384*K1**2*K3**2 - 6224*K1**2 - 256*K1*K2**2*K3 + 8048*K1*K2*K3 + 352*K1*K3*K4 - 368*K2**4 + 448*K2**2*K4 - 5504*K2**2 - 1808*K3**2 - 164*K4**2 + 5586
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2074']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3571', 'vk6.3595', 'vk6.3822', 'vk6.3853', 'vk6.6984', 'vk6.7015', 'vk6.7206', 'vk6.7234', 'vk6.15336', 'vk6.15463', 'vk6.33981', 'vk6.34027', 'vk6.34438', 'vk6.48235', 'vk6.48386', 'vk6.49963', 'vk6.49989', 'vk6.53985', 'vk6.54041', 'vk6.54486']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2U1U3O4O3U5U4O6O5U2U6
R3 orbit {'O1O2U1U3O4O3U5U4O6O5U2U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2U3U1O3O4U5U4O6O5U2U6
Gauss code of -K* O1O2U3U1O3O4U5U4O6O5U2U6
Diagrammatic symmetry type r
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 0 1 0 0 0],[ 1 0 1 1 0 1 1],[ 0 -1 0 1 0 -1 0],[-1 -1 -1 0 0 0 0],[ 0 0 0 0 0 0 -1],[ 0 -1 1 0 0 0 0],[ 0 -1 0 0 1 0 0]]
Primitive based matrix [[ 0 1 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 0 0 0 1 0 0 -1],[ 0 0 -1 0 0 0 0],[ 0 0 0 0 0 1 -1],[ 0 1 0 0 -1 0 -1],[ 1 1 1 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,0,0,0,0,1,0,0,0,1,1,-1,0,0,1,0,0,0,-1,1,1]
Phi over symmetry [-1,0,0,0,0,1,0,0,0,1,1,-1,0,0,1,0,0,0,-1,1,1]
Phi of -K [-1,0,0,0,0,1,0,0,0,1,1,-1,0,0,1,0,0,0,-1,1,1]
Phi of K* [-1,0,0,0,0,1,0,1,1,1,1,-1,0,0,0,0,0,0,-1,1,0]
Phi of -K* [-1,0,0,0,0,1,0,1,1,1,1,-1,0,0,0,0,0,0,-1,1,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+7t^4+11t^2+4
Outer characteristic polynomial t^7+9t^5+17t^3+8t
Flat arrow polynomial -20*K1**2 + 10*K2 + 11
2-strand cable arrow polynomial -512*K1**4*K2**2 + 2304*K1**4*K2 - 7168*K1**4 + 896*K1**3*K2*K3 - 1216*K1**3*K3 + 128*K1**2*K2**3 - 8800*K1**2*K2**2 - 320*K1**2*K2*K4 + 14960*K1**2*K2 - 384*K1**2*K3**2 - 6224*K1**2 - 256*K1*K2**2*K3 + 8048*K1*K2*K3 + 352*K1*K3*K4 - 368*K2**4 + 448*K2**2*K4 - 5504*K2**2 - 1808*K3**2 - 164*K4**2 + 5586
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {1, 3}, {2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice False
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