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Flat knot 6.1868

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,2,2,3,1,0,1,2,1,0,2,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1868']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+39t^5+150t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1868']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 768*K1**4*K2**2 + 1024*K1**4*K2 - 1312*K1**4 - 256*K1**3*K2**2*K3 + 1216*K1**3*K2*K3 - 384*K1**3*K3 - 1024*K1**2*K2**4 + 3392*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 12160*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 11216*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K4**2 - 7456*K1**2 + 2752*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 3328*K1*K2**2*K3 - 640*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 11088*K1*K2*K3 + 800*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 4480*K2**4 - 64*K2**3*K6 - 1344*K2**2*K3**2 - 160*K2**2*K4**2 + 3440*K2**2*K4 - 3948*K2**2 - 64*K2*K3**2*K4 + 512*K2*K3*K5 + 80*K2*K4*K6 - 2728*K3**2 - 520*K4**2 - 104*K5**2 - 4*K6**2 + 5590
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1868']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72566', 'vk6.72667', 'vk6.72983', 'vk6.73143', 'vk6.74218', 'vk6.74848', 'vk6.76415', 'vk6.76898', 'vk6.77853', 'vk6.77882', 'vk6.77999', 'vk6.79266', 'vk6.79742', 'vk6.80755', 'vk6.81155', 'vk6.82315', 'vk6.83990', 'vk6.86362', 'vk6.87283', 'vk6.88252']
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 O1O2O3U4U5O6U2O4O5U1U6U3
R3 orbit {'O1O2O3U4U5O6U2O4O5U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U3O5O6U2O4U5U6
Gauss code of K* O1O2O3U1U4U3O5O6U2O4U5U6
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 2 -1 1 0],[ 2 0 2 3 0 2 0],[ 0 -2 0 0 0 1 -1],[-2 -3 0 0 -2 0 -2],[ 1 0 0 2 0 1 1],[-1 -2 -1 0 -1 0 0],[ 0 0 1 2 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -2 -2 -3],[-1 0 0 -1 0 -1 -2],[ 0 0 1 0 -1 0 -2],[ 0 2 0 1 0 -1 0],[ 1 2 1 0 1 0 0],[ 2 3 2 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,0,2,2,3,1,0,1,2,1,0,2,1,0,0]
Phi over symmetry [-2,-1,0,0,1,2,0,0,2,2,3,1,0,1,2,1,0,2,1,0,0]
Phi of -K [-2,-1,0,0,1,2,1,0,2,1,1,1,0,1,1,1,0,2,1,0,1]
Phi of K* [-2,-1,0,0,1,2,1,0,2,1,1,1,0,1,1,1,0,2,1,0,1]
Phi of -K* [-2,-1,0,0,1,2,0,0,2,2,3,1,0,1,2,1,0,2,1,0,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+29t^4+88t^2+4
Outer characteristic polynomial t^7+39t^5+150t^3+8t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial 256*K1**4*K2**3 - 768*K1**4*K2**2 + 1024*K1**4*K2 - 1312*K1**4 - 256*K1**3*K2**2*K3 + 1216*K1**3*K2*K3 - 384*K1**3*K3 - 1024*K1**2*K2**4 + 3392*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 12160*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 11216*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K4**2 - 7456*K1**2 + 2752*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 3328*K1*K2**2*K3 - 640*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 11088*K1*K2*K3 + 800*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 4480*K2**4 - 64*K2**3*K6 - 1344*K2**2*K3**2 - 160*K2**2*K4**2 + 3440*K2**2*K4 - 3948*K2**2 - 64*K2*K3**2*K4 + 512*K2*K3*K5 + 80*K2*K4*K6 - 2728*K3**2 - 520*K4**2 - 104*K5**2 - 4*K6**2 + 5590
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
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