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Flat knot 6.628

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,1,2,4,1,1,1,2,0,0,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.628']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.573', '6.628', '6.703', '6.704', '6.723', '6.796', '6.1083', '6.1099', '6.1315']
Outer characteristic polynomial of the knot is: t^7+79t^5+63t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.628']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 608*K1**4*K2 - 1328*K1**4 + 64*K1**3*K2*K3 - 224*K1**3*K3 - 256*K1**2*K2**4 + 1792*K1**2*K2**3 - 5568*K1**2*K2**2 - 96*K1**2*K2*K4 + 7224*K1**2*K2 - 48*K1**2*K3**2 - 4904*K1**2 + 512*K1*K2**3*K3 - 800*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4352*K1*K2*K3 + 272*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1192*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 640*K2**2*K4 - 2576*K2**2 + 40*K2*K3*K5 - 1064*K3**2 - 162*K4**2 - 8*K5**2 + 3296
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.628']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73267', 'vk6.73410', 'vk6.74018', 'vk6.74562', 'vk6.75171', 'vk6.75407', 'vk6.76040', 'vk6.76768', 'vk6.78132', 'vk6.78373', 'vk6.78999', 'vk6.79562', 'vk6.79969', 'vk6.80126', 'vk6.80529', 'vk6.80991', 'vk6.81877', 'vk6.82150', 'vk6.82179', 'vk6.82595', 'vk6.83587', 'vk6.83763', 'vk6.84037', 'vk6.84615', 'vk6.84946', 'vk6.85590', 'vk6.85714', 'vk6.85939', 'vk6.86737', 'vk6.87659', 'vk6.88940', 'vk6.89965']
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 O1O2O3O4U1O5U2U3O6U4U5U6
R3 orbit {'O1O2O3O4U1O5U2U3O6U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1O5U2U3O6U4
Gauss code of K* O1O2O3U4U5U6U1O4U2O5O6U3
Gauss code of -K* O1O2O3U1O4O5U2O6U3U4U5U6
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 -3 -2 0 1 2 2],[ 3 0 1 2 3 3 1],[ 2 -1 0 1 2 3 2],[ 0 -2 -1 0 1 2 2],[-1 -3 -2 -1 0 1 2],[-2 -3 -3 -2 -1 0 1],[-2 -1 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 1 -1 -2 -3 -3],[-2 -1 0 -2 -2 -2 -1],[-1 1 2 0 -1 -2 -3],[ 0 2 2 1 0 -1 -2],[ 2 3 2 2 1 0 -1],[ 3 3 1 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,-1,1,2,3,3,2,2,2,1,1,2,3,1,2,1]
Phi over symmetry [-3,-2,0,1,2,2,0,1,1,2,4,1,1,1,2,0,0,0,0,-1,-1]
Phi of -K [-3,-2,0,1,2,2,0,1,1,2,4,1,1,1,2,0,0,0,0,-1,-1]
Phi of K* [-2,-2,-1,0,2,3,-1,-1,0,2,4,0,0,1,2,0,1,1,1,1,0]
Phi of -K* [-3,-2,0,1,2,2,1,2,3,1,3,1,2,2,3,1,2,2,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+57t^4+30t^2
Outer characteristic polynomial t^7+79t^5+63t^3+5t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
2-strand cable arrow polynomial -320*K1**4*K2**2 + 608*K1**4*K2 - 1328*K1**4 + 64*K1**3*K2*K3 - 224*K1**3*K3 - 256*K1**2*K2**4 + 1792*K1**2*K2**3 - 5568*K1**2*K2**2 - 96*K1**2*K2*K4 + 7224*K1**2*K2 - 48*K1**2*K3**2 - 4904*K1**2 + 512*K1*K2**3*K3 - 800*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4352*K1*K2*K3 + 272*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1192*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 640*K2**2*K4 - 2576*K2**2 + 40*K2*K3*K5 - 1064*K3**2 - 162*K4**2 - 8*K5**2 + 3296
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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