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Flat knot 6.1552

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1552', '6.1814', '7.30914', '7.33303']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.813', '6.1105', '6.1107', '6.1552', '6.1687']
Outer characteristic polynomial of the knot is: t^7+26t^5+33t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1552', '6.1814', '7.33303']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 400*K1**4 - 256*K1**2*K2**4 + 928*K1**2*K2**3 - 4800*K1**2*K2**2 - 288*K1**2*K2*K4 + 5384*K1**2*K2 - 16*K1**2*K3**2 - 48*K1**2*K4**2 - 3600*K1**2 + 448*K1*K2**3*K3 - 416*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3512*K1*K2*K3 + 272*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 1128*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 928*K2**2*K4 - 1928*K2**2 + 16*K2*K3*K5 - 668*K3**2 - 270*K4**2 - 4*K5**2 + 2380
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1552']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71605', 'vk6.71609', 'vk6.71728', 'vk6.71732', 'vk6.72150', 'vk6.72153', 'vk6.72340', 'vk6.74044', 'vk6.74056', 'vk6.74612', 'vk6.76802', 'vk6.77220', 'vk6.77232', 'vk6.77532', 'vk6.77543', 'vk6.77673', 'vk6.79043', 'vk6.79054', 'vk6.79608', 'vk6.79622', 'vk6.80566', 'vk6.80574', 'vk6.81017', 'vk6.81026', 'vk6.81354', 'vk6.81364', 'vk6.81398', 'vk6.85416', 'vk6.85422', 'vk6.85494', 'vk6.87982', 'vk6.89312']
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 O1O2O3U4O5U3U1O4O6U2U5U6
R3 orbit {'O1O2O3U4O5U3U1O4O6U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U2O4O6U3U1O5U6
Gauss code of K* O1O2O3U4U1U5O6U2O5O4U6U3
Gauss code of -K* O1O2O3U1U4O5O6U2O4U6U3U5
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 -1 -1 0 -1 1 2],[ 1 0 0 0 0 2 2],[ 1 0 0 1 0 1 2],[ 0 0 -1 0 0 0 1],[ 1 0 0 0 0 1 1],[-1 -2 -1 0 -1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -1 -2],[ 0 1 0 0 0 -1 0],[ 1 1 1 0 0 0 0],[ 1 2 1 1 0 0 0],[ 1 2 2 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,1,1,1,2,2,0,1,1,2,0,1,0,0,0,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,0,1,0,0,0]
Phi of -K [-1,-1,-1,0,1,2,0,0,0,1,1,0,1,0,1,1,1,2,1,1,0]
Phi of K* [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,0,1,0,0,0]
Phi of -K* [-1,-1,-1,0,1,2,0,0,0,1,1,0,0,2,2,1,1,2,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+18t^4+20t^2
Outer characteristic polynomial t^7+26t^5+33t^3+4t
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial 96*K1**4*K2 - 400*K1**4 - 256*K1**2*K2**4 + 928*K1**2*K2**3 - 4800*K1**2*K2**2 - 288*K1**2*K2*K4 + 5384*K1**2*K2 - 16*K1**2*K3**2 - 48*K1**2*K4**2 - 3600*K1**2 + 448*K1*K2**3*K3 - 416*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3512*K1*K2*K3 + 272*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 1128*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 928*K2**2*K4 - 1928*K2**2 + 16*K2*K3*K5 - 668*K3**2 - 270*K4**2 - 4*K5**2 + 2380
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}]]
If K is slice False
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