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Flat knot 6.1135

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.1135', '6.1248']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+33t^5+55t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1127', '6.1135', '6.1248']
2-strand cable arrow polynomial of the knot is: -1312*K1**4 + 704*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1152*K1**3*K3 + 192*K1**2*K2**2*K4 - 1536*K1**2*K2**2 - 512*K1**2*K2*K4 + 5872*K1**2*K2 - 1536*K1**2*K3**2 - 192*K1**2*K3*K5 - 352*K1**2*K4**2 - 32*K1**2*K5**2 - 6000*K1**2 - 704*K1*K2**2*K3 - 576*K1*K2*K3*K4 + 6352*K1*K2*K3 + 2848*K1*K3*K4 + 688*K1*K4*K5 + 80*K1*K5*K6 - 64*K2**4 - 16*K2**2*K4**2 + 1040*K2**2*K4 - 4412*K2**2 + 256*K2*K3*K5 + 32*K2*K4*K6 - 2960*K3**2 - 1352*K4**2 - 256*K5**2 - 44*K6**2 + 4814
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1135']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16979', 'vk6.17220', 'vk6.20878', 'vk6.22287', 'vk6.23382', 'vk6.23686', 'vk6.28349', 'vk6.35437', 'vk6.35872', 'vk6.39981', 'vk6.42053', 'vk6.43175', 'vk6.46517', 'vk6.55140', 'vk6.55392', 'vk6.57677', 'vk6.58875', 'vk6.59864', 'vk6.68407', 'vk6.69737']
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 O1O2O3O4U5U4O6U3U2O5U1U6
R3 orbit {'O1O2O3O4U5U4O6U3U2O5U1U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2U3O4O5U1O6O3U6U5U4U2
Gauss code of -K* O1O2U3O4O5U1O6O3U6U5U4U2
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 0 0 1 -2 2],[ 1 0 1 1 1 -2 2],[ 0 -1 0 0 0 -2 1],[ 0 -1 0 0 0 -1 0],[-1 -1 0 0 0 -1 -1],[ 2 2 2 1 1 0 2],[-2 -2 -1 0 1 -2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 0 0 -1 -1],[ 0 0 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -2],[ 1 2 1 1 1 0 -2],[ 2 2 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Phi of -K [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Phi of K* [-2,-1,0,0,1,2,2,1,2,1,2,1,1,1,2,0,0,0,0,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,2,1,2,1,1,1,2,0,0,0,0,1,-1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+25z+31
Enhanced Jones-Krushkal polynomial 5w^3z^2+25w^2z+31w
Inner characteristic polynomial t^6+23t^4+23t^2+1
Outer characteristic polynomial t^7+33t^5+55t^3+7t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial -1312*K1**4 + 704*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1152*K1**3*K3 + 192*K1**2*K2**2*K4 - 1536*K1**2*K2**2 - 512*K1**2*K2*K4 + 5872*K1**2*K2 - 1536*K1**2*K3**2 - 192*K1**2*K3*K5 - 352*K1**2*K4**2 - 32*K1**2*K5**2 - 6000*K1**2 - 704*K1*K2**2*K3 - 576*K1*K2*K3*K4 + 6352*K1*K2*K3 + 2848*K1*K3*K4 + 688*K1*K4*K5 + 80*K1*K5*K6 - 64*K2**4 - 16*K2**2*K4**2 + 1040*K2**2*K4 - 4412*K2**2 + 256*K2*K3*K5 + 32*K2*K4*K6 - 2960*K3**2 - 1352*K4**2 - 256*K5**2 - 44*K6**2 + 4814
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}, {1, 4}, {3}, {2}]]
If K is slice False
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