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Flat knot 6.2081

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,1,0,1,1,-1,1,0,1,1,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.2081']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 12*K1*K2 + 3*K1 + 6*K2 + 5*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.2081']
Outer characteristic polynomial of the knot is: t^7+15t^5+51t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2081']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 1440*K1**4*K2 - 4416*K1**4 + 864*K1**3*K2*K3 + 192*K1**3*K3*K4 - 1696*K1**3*K3 + 384*K1**2*K2**3 - 4368*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 11496*K1**2*K2 - 1344*K1**2*K3**2 - 240*K1**2*K4**2 - 7944*K1**2 + 96*K1*K2**3*K3 - 1920*K1*K2**2*K3 - 192*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 9432*K1*K2*K3 + 2640*K1*K3*K4 + 600*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 480*K2**4 - 160*K2**3*K6 - 336*K2**2*K3**2 - 144*K2**2*K4**2 + 2208*K2**2*K4 - 7302*K2**2 + 864*K2*K3*K5 + 408*K2*K4*K6 - 3712*K3**2 - 1620*K4**2 - 432*K5**2 - 138*K6**2 + 7162
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2081']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.18883', 'vk6.18893', 'vk6.18961', 'vk6.18971', 'vk6.25588', 'vk6.25594', 'vk6.37618', 'vk6.37628', 'vk6.56413', 'vk6.56424', 'vk6.56450', 'vk6.56476']
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 O1O2U1U3O4O5U4U2O6O3U6U5
R3 orbit {'O1O2U1U3O4O5U4U2O6O3U6U5'}
R3 orbit length 1
Gauss code of -K O1O2U3U4O5O4U1U6O3O6U5U2
Gauss code of K* O1O2U3U4O3O5U6U2O6O4U1U5
Gauss code of -K* O1O2U3U2O4O5U1U5O3O6U4U6
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 1 -1 1 -1],[ 1 0 1 0 0 1 0],[-1 -1 0 -1 0 1 -1],[-1 0 1 0 -1 -1 -1],[ 1 0 0 1 0 1 0],[-1 -1 -1 1 -1 0 0],[ 1 0 1 1 0 0 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 -1 0 -1 -1],[-1 -1 0 1 -1 0 -1],[-1 1 -1 0 -1 -1 0],[ 1 0 1 1 0 0 0],[ 1 1 0 1 0 0 0],[ 1 1 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,1,0,1,1,-1,1,0,1,1,1,0,0,0,0]
Phi over symmetry [-1,-1,-1,1,1,1,-1,1,0,1,1,-1,1,0,1,1,1,0,0,0,0]
Phi of -K [-1,-1,-1,1,1,1,0,0,1,1,2,0,1,2,1,2,1,1,-1,1,-1]
Phi of K* [-1,-1,-1,1,1,1,-1,1,1,1,2,-1,1,2,1,2,1,1,0,0,0]
Phi of -K* [-1,-1,-1,1,1,1,0,0,0,1,1,0,1,0,1,1,1,0,-1,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+21z+39
Enhanced Jones-Krushkal polynomial w^3z^2+21w^2z+39w
Inner characteristic polynomial t^6+9t^4+21t^2+4
Outer characteristic polynomial t^7+15t^5+51t^3+10t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 12*K1*K2 + 3*K1 + 6*K2 + 5*K3 + 7
2-strand cable arrow polynomial -256*K1**6 + 1440*K1**4*K2 - 4416*K1**4 + 864*K1**3*K2*K3 + 192*K1**3*K3*K4 - 1696*K1**3*K3 + 384*K1**2*K2**3 - 4368*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 11496*K1**2*K2 - 1344*K1**2*K3**2 - 240*K1**2*K4**2 - 7944*K1**2 + 96*K1*K2**3*K3 - 1920*K1*K2**2*K3 - 192*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 9432*K1*K2*K3 + 2640*K1*K3*K4 + 600*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 480*K2**4 - 160*K2**3*K6 - 336*K2**2*K3**2 - 144*K2**2*K4**2 + 2208*K2**2*K4 - 7302*K2**2 + 864*K2*K3*K5 + 408*K2*K4*K6 - 3712*K3**2 - 1620*K4**2 - 432*K5**2 - 138*K6**2 + 7162
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice True
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