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Flat knot 6.1825

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,2,1,0,0,0,0,1,-1,0,2,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1825']
Arrow polynomial of the knot is: -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.763', '6.1515', '6.1741', '6.1825']
Outer characteristic polynomial of the knot is: t^7+23t^5+70t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1825']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 128*K1**4*K2**2 + 1088*K1**4*K2 - 4224*K1**4 + 768*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1216*K1**3*K3 - 3072*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 9296*K1**2*K2 - 2496*K1**2*K3**2 - 192*K1**2*K3*K5 - 352*K1**2*K4**2 - 7096*K1**2 - 512*K1*K2**2*K3 + 64*K1*K2*K3**3 - 256*K1*K2*K3*K4 + 8528*K1*K2*K3 - 64*K1*K3**2*K5 + 3728*K1*K3*K4 + 672*K1*K4*K5 - 80*K2**4 - 384*K2**2*K3**2 - 32*K2**2*K4**2 + 976*K2**2*K4 - 5928*K2**2 + 688*K2*K3*K5 + 64*K2*K4*K6 - 128*K3**4 + 96*K3**2*K6 - 3696*K3**2 - 1588*K4**2 - 408*K5**2 - 32*K6**2 + 6618
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1825']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4775', 'vk6.5112', 'vk6.6344', 'vk6.6774', 'vk6.8302', 'vk6.8754', 'vk6.9676', 'vk6.9987', 'vk6.21021', 'vk6.22445', 'vk6.28472', 'vk6.40241', 'vk6.42169', 'vk6.46743', 'vk6.48799', 'vk6.49016', 'vk6.49836', 'vk6.51503', 'vk6.58975', 'vk6.69811']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3U1U3O4U2O5O6U5U4U6
R3 orbit {'O1O2O3U1U3O4U2O5O6U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U6O4O6U2O5U1U3
Gauss code of K* O1O2O3U4U5U6O4O6U2O5U1U3
Gauss code of -K* Same
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 -2 0 1 0 -1 2],[ 2 0 2 1 1 0 0],[ 0 -2 0 0 1 0 1],[-1 -1 0 0 0 0 0],[ 0 -1 -1 0 0 0 2],[ 1 0 0 0 0 0 1],[-2 0 -1 0 -2 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -1 0],[-1 0 0 0 0 0 -1],[ 0 1 0 0 1 0 -2],[ 0 2 0 -1 0 0 -1],[ 1 1 0 0 0 0 0],[ 2 0 1 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,2,1,0,0,0,0,1,-1,0,2,0,1,0]
Phi over symmetry [-2,-1,0,0,1,2,0,1,2,1,0,0,0,0,1,-1,0,2,0,1,0]
Phi of -K [-2,-1,0,0,1,2,1,0,1,2,4,1,1,2,2,-1,1,1,1,0,1]
Phi of K* [-2,-1,0,0,1,2,1,0,1,2,4,1,1,2,2,-1,1,1,1,0,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,2,1,0,0,0,0,1,-1,0,2,0,1,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 20z+41
Enhanced Jones-Krushkal polynomial 20w^2z+41w
Inner characteristic polynomial t^6+13t^4+22t^2+1
Outer characteristic polynomial t^7+23t^5+70t^3+7t
Flat arrow polynomial -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -256*K1**6 - 128*K1**4*K2**2 + 1088*K1**4*K2 - 4224*K1**4 + 768*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1216*K1**3*K3 - 3072*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 9296*K1**2*K2 - 2496*K1**2*K3**2 - 192*K1**2*K3*K5 - 352*K1**2*K4**2 - 7096*K1**2 - 512*K1*K2**2*K3 + 64*K1*K2*K3**3 - 256*K1*K2*K3*K4 + 8528*K1*K2*K3 - 64*K1*K3**2*K5 + 3728*K1*K3*K4 + 672*K1*K4*K5 - 80*K2**4 - 384*K2**2*K3**2 - 32*K2**2*K4**2 + 976*K2**2*K4 - 5928*K2**2 + 688*K2*K3*K5 + 64*K2*K4*K6 - 128*K3**4 + 96*K3**2*K6 - 3696*K3**2 - 1588*K4**2 - 408*K5**2 - 32*K6**2 + 6618
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice True
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