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Flat knot 6.1244

Min(phi) over symmetries of the knot is: [-1,0,0,1,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['4.11', '6.1244', '6.2058', '7.45752']
Arrow polynomial of the knot is: -16*K1**2 + 8*K2 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1244', '6.1401', '6.2079', '6.2084']
Outer characteristic polynomial of the knot is: t^5+5t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.11', '6.1244', '6.2058', '7.45752']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 3072*K1**4*K2 - 8480*K1**4 + 960*K1**3*K2*K3 - 1600*K1**3*K3 + 1344*K1**2*K2**3 - 11936*K1**2*K2**2 - 1088*K1**2*K2*K4 + 16320*K1**2*K2 - 352*K1**2*K3**2 - 5632*K1**2 - 1344*K1*K2**2*K3 + 10224*K1*K2*K3 + 912*K1*K3*K4 - 1440*K2**4 + 1744*K2**2*K4 - 5544*K2**2 - 2144*K3**2 - 544*K4**2 + 5782
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1244']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13956', 'vk6.14050', 'vk6.15027', 'vk6.15147', 'vk6.17454', 'vk6.17474', 'vk6.23967', 'vk6.23998', 'vk6.33760', 'vk6.33834', 'vk6.34302', 'vk6.36256', 'vk6.43419', 'vk6.53888', 'vk6.53921', 'vk6.54436', 'vk6.55591', 'vk6.60083', 'vk6.60096', 'vk6.65304']
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 O1O2O3O4U3U5U1O6O5U4U6U2
R3 orbit {'O1O2O3O4U3U5U1O6O5U4U6U2'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3U4U2O5O4O6U3U6U1U5
Gauss code of -K* O1O2O3U4U2O5O4O6U3U6U1U5
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 1 -1 1 0 0],[ 1 0 1 0 1 1 0],[-1 -1 0 -1 0 0 0],[ 1 0 1 0 1 1 1],[-1 -1 0 -1 0 -1 0],[ 0 -1 0 -1 1 0 0],[ 0 0 0 -1 0 0 0]]
Primitive based matrix [[ 0 1 0 0 -1],[-1 0 0 0 -1],[ 0 0 0 0 -1],[ 0 0 0 0 -1],[ 1 1 1 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,0,0,1,0,0,1,0,1,1]
Phi over symmetry [-1,0,0,1,0,0,1,0,1,1]
Phi of -K [-1,0,0,1,0,0,1,0,1,1]
Phi of K* [-1,0,0,1,1,1,1,0,0,0]
Phi of -K* [-1,0,0,1,1,1,1,0,0,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^4+3t^2
Outer characteristic polynomial t^5+5t^3+2t
Flat arrow polynomial -16*K1**2 + 8*K2 + 9
2-strand cable arrow polynomial -256*K1**4*K2**2 + 3072*K1**4*K2 - 8480*K1**4 + 960*K1**3*K2*K3 - 1600*K1**3*K3 + 1344*K1**2*K2**3 - 11936*K1**2*K2**2 - 1088*K1**2*K2*K4 + 16320*K1**2*K2 - 352*K1**2*K3**2 - 5632*K1**2 - 1344*K1*K2**2*K3 + 10224*K1*K2*K3 + 912*K1*K3*K4 - 1440*K2**4 + 1744*K2**2*K4 - 5544*K2**2 - 2144*K3**2 - 544*K4**2 + 5782
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
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