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Flat knot 6.941

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,1,1,1,0,0,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.941', '7.16076', '7.42682']
Arrow polynomial of the knot is: 16*K1**3 - 12*K1**2 - 12*K1*K2 - 6*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.141', '6.846', '6.918', '6.941', '6.2064', '6.2066']
Outer characteristic polynomial of the knot is: t^7+29t^5+75t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.941', '6.1413', '7.16076', '7.42682']
2-strand cable arrow polynomial of the knot is: -4096*K1**4*K2**2 + 6656*K1**4*K2 - 7744*K1**4 + 3072*K1**3*K2*K3 - 1024*K1**3*K3 - 6656*K1**2*K2**4 + 11648*K1**2*K2**3 + 1536*K1**2*K2**2*K4 - 19904*K1**2*K2**2 - 1984*K1**2*K2*K4 + 11136*K1**2*K2 - 448*K1**2*K3**2 - 64*K1**2 + 6272*K1*K2**3*K3 - 4608*K1*K2**2*K3 - 1408*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 9024*K1*K2*K3 + 320*K1*K3*K4 + 32*K1*K4*K5 - 2560*K2**6 + 2560*K2**4*K4 - 5328*K2**4 - 448*K2**3*K6 - 1024*K2**2*K3**2 - 272*K2**2*K4**2 + 3408*K2**2*K4 + 628*K2**2 + 288*K2*K3*K5 + 48*K2*K4*K6 - 624*K3**2 - 132*K4**2 - 16*K5**2 - 4*K6**2 + 2098
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.941']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.518', 'vk6.520', 'vk6.643', 'vk6.647', 'vk6.1159', 'vk6.1681', 'vk6.1881', 'vk6.1885', 'vk6.2198', 'vk6.2202', 'vk6.2331', 'vk6.3090', 'vk6.3094', 'vk6.3210', 'vk6.22563', 'vk6.28587', 'vk6.42235', 'vk6.46959', 'vk6.46962', 'vk6.58999']
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 O1O2O3O4U5U6O5O6U3U4U1U2
R3 orbit {'O1O2O3O4U5U6O5O6U3U4U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U4U1U2O5O6U5U6
Gauss code of K* O1O2O3O4U3U4U1U2O5O6U5U6
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 -1 1 -1 1 -1 1],[ 1 0 1 -1 1 0 2],[-1 -1 0 -1 1 -2 0],[ 1 1 1 0 1 0 2],[-1 -1 -1 -1 0 -2 0],[ 1 0 2 0 2 0 1],[-1 -2 0 -2 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 0 -1 -1 -2],[-1 -1 0 0 -1 -1 -2],[-1 0 0 0 -2 -2 -1],[ 1 1 1 2 0 1 0],[ 1 1 1 2 -1 0 0],[ 1 2 2 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,2,2,2,1,-1,0,0]
Phi over symmetry [-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,1,1,1,0,0,0,0,-1]
Phi of -K [-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,1,1,1,0,0,0,0,-1]
Phi of K* [-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,1,1,1,0,0,0,0,-1]
Phi of -K* [-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,2,2,2,1,-1,0,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+27z+27
Enhanced Jones-Krushkal polynomial 7w^3z^2+27w^2z+27w
Inner characteristic polynomial t^6+23t^4+55t^2+1
Outer characteristic polynomial t^7+29t^5+75t^3+7t
Flat arrow polynomial 16*K1**3 - 12*K1**2 - 12*K1*K2 - 6*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -4096*K1**4*K2**2 + 6656*K1**4*K2 - 7744*K1**4 + 3072*K1**3*K2*K3 - 1024*K1**3*K3 - 6656*K1**2*K2**4 + 11648*K1**2*K2**3 + 1536*K1**2*K2**2*K4 - 19904*K1**2*K2**2 - 1984*K1**2*K2*K4 + 11136*K1**2*K2 - 448*K1**2*K3**2 - 64*K1**2 + 6272*K1*K2**3*K3 - 4608*K1*K2**2*K3 - 1408*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 9024*K1*K2*K3 + 320*K1*K3*K4 + 32*K1*K4*K5 - 2560*K2**6 + 2560*K2**4*K4 - 5328*K2**4 - 448*K2**3*K6 - 1024*K2**2*K3**2 - 272*K2**2*K4**2 + 3408*K2**2*K4 + 628*K2**2 + 288*K2*K3*K5 + 48*K2*K4*K6 - 624*K3**2 - 132*K4**2 - 16*K5**2 - 4*K6**2 + 2098
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice True
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