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Flat knot 6.1068

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,2,2,2,0,1,1,1,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1068']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078']
Outer characteristic polynomial of the knot is: t^7+25t^5+41t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1068']
2-strand cable arrow polynomial of the knot is: -320*K1**6 - 192*K1**4*K2**2 + 1248*K1**4*K2 - 4480*K1**4 + 672*K1**3*K2*K3 - 1376*K1**3*K3 - 128*K1**2*K2**4 + 992*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7904*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 13168*K1**2*K2 - 800*K1**2*K3**2 - 32*K1**2*K4**2 - 7324*K1**2 + 768*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 224*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 9760*K1*K2*K3 + 1200*K1*K3*K4 + 72*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 1744*K2**4 - 64*K2**3*K6 - 704*K2**2*K3**2 - 128*K2**2*K4**2 + 1736*K2**2*K4 - 5372*K2**2 + 496*K2*K3*K5 + 48*K2*K4*K6 - 2636*K3**2 - 520*K4**2 - 48*K5**2 - 4*K6**2 + 5918
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1068']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16564', 'vk6.16655', 'vk6.18153', 'vk6.18487', 'vk6.22967', 'vk6.23086', 'vk6.24612', 'vk6.25023', 'vk6.34956', 'vk6.35075', 'vk6.36743', 'vk6.37160', 'vk6.42529', 'vk6.42638', 'vk6.44015', 'vk6.44325', 'vk6.54795', 'vk6.54881', 'vk6.55951', 'vk6.56249', 'vk6.59227', 'vk6.59306', 'vk6.60489', 'vk6.60853', 'vk6.64769', 'vk6.64832', 'vk6.65608', 'vk6.65913', 'vk6.68071', 'vk6.68134', 'vk6.68683', 'vk6.68892']
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 O1O2O3O4U5U3O6U1O5U6U4U2
R3 orbit {'O1O2O3O4U5U3O6U1O5U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U5O6U4O5U2U6
Gauss code of K* O1O2O3U4U3U5U2O6O5U1O4U6
Gauss code of -K* O1O2O3U4O5U3O6O4U2U6U1U5
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 -2 1 0 2 -1 0],[ 2 0 2 0 2 1 0],[-1 -2 0 0 1 -1 -1],[ 0 0 0 0 0 0 -1],[-2 -2 -1 0 0 -1 -1],[ 1 -1 1 0 1 0 0],[ 0 0 1 1 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -1 -2],[-1 1 0 0 -1 -1 -2],[ 0 0 0 0 -1 0 0],[ 0 1 1 1 0 0 0],[ 1 1 1 0 0 0 -1],[ 2 2 2 0 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,0,1,1,2,0,1,1,2,1,0,0,0,0,1]
Phi over symmetry [-2,-1,0,0,1,2,0,1,2,2,2,0,1,1,1,1,1,2,1,2,0]
Phi of -K [-2,-1,0,0,1,2,0,2,2,1,2,1,1,1,2,-1,0,1,1,2,0]
Phi of K* [-2,-1,0,0,1,2,0,1,2,2,2,0,1,1,1,1,1,2,1,2,0]
Phi of -K* [-2,-1,0,0,1,2,1,0,0,2,2,0,0,1,1,-1,0,0,1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+15t^4+15t^2+1
Outer characteristic polynomial t^7+25t^5+41t^3+7t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -320*K1**6 - 192*K1**4*K2**2 + 1248*K1**4*K2 - 4480*K1**4 + 672*K1**3*K2*K3 - 1376*K1**3*K3 - 128*K1**2*K2**4 + 992*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7904*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 13168*K1**2*K2 - 800*K1**2*K3**2 - 32*K1**2*K4**2 - 7324*K1**2 + 768*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 224*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 9760*K1*K2*K3 + 1200*K1*K3*K4 + 72*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 1744*K2**4 - 64*K2**3*K6 - 704*K2**2*K3**2 - 128*K2**2*K4**2 + 1736*K2**2*K4 - 5372*K2**2 + 496*K2*K3*K5 + 48*K2*K4*K6 - 2636*K3**2 - 520*K4**2 - 48*K5**2 - 4*K6**2 + 5918
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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