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Flat knot 6.1315

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,3,2,0,1,2,2,0,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1315']
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+44t^5+44t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1315']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 1664*K1**4*K2**2 + 3424*K1**4*K2 - 4672*K1**4 - 128*K1**3*K2**2*K3 + 640*K1**3*K2*K3 - 992*K1**3*K3 - 640*K1**2*K2**4 + 4544*K1**2*K2**3 - 10896*K1**2*K2**2 - 672*K1**2*K2*K4 + 10288*K1**2*K2 - 96*K1**2*K3**2 - 3660*K1**2 + 1056*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6568*K1*K2*K3 + 208*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 2440*K2**4 - 416*K2**2*K3**2 - 8*K2**2*K4**2 + 1504*K2**2*K4 - 2200*K2**2 + 104*K2*K3*K5 - 916*K3**2 - 106*K4**2 + 3248
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1315']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19930', 'vk6.19986', 'vk6.21164', 'vk6.21252', 'vk6.26877', 'vk6.27009', 'vk6.28639', 'vk6.28731', 'vk6.38310', 'vk6.38415', 'vk6.40441', 'vk6.40595', 'vk6.45178', 'vk6.45301', 'vk6.47012', 'vk6.47081', 'vk6.56716', 'vk6.56800', 'vk6.57803', 'vk6.57929', 'vk6.61130', 'vk6.61304', 'vk6.62379', 'vk6.62486', 'vk6.66413', 'vk6.66508', 'vk6.67177', 'vk6.67293', 'vk6.69063', 'vk6.69154', 'vk6.69849', 'vk6.69909']
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 O1O2O3U4O5O6U1U6O4U2U3U5
R3 orbit {'O1O2O3U4O5O6U1U6O4U2U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1U2O5U6U3O6O4U5
Gauss code of K* O1O2O3U4U1U2O5U3U6O4O6U5
Gauss code of -K* O1O2O3U4O5O6U5U1O4U2U3U6
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 -1 1 0 2 1],[ 3 0 1 2 2 3 1],[ 1 -1 0 1 1 1 0],[-1 -2 -1 0 -1 0 0],[ 0 -2 -1 1 0 2 1],[-2 -3 -1 0 -2 0 0],[-1 -1 0 0 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -2 -1 -3],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -2],[ 0 2 1 1 0 -1 -2],[ 1 1 0 1 1 0 -1],[ 3 3 1 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,0,2,1,3,0,1,0,1,1,1,2,1,2,1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,2,3,2,0,1,2,2,0,0,0,0,1,1]
Phi of -K [-3,-1,0,1,1,2,1,1,2,3,2,0,1,2,2,0,0,0,0,1,1]
Phi of K* [-2,-1,-1,0,1,3,1,1,0,2,2,0,0,1,2,0,2,3,0,1,1]
Phi of -K* [-3,-1,0,1,1,2,1,2,1,2,3,1,0,1,1,1,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+28t^4+23t^2
Outer characteristic polynomial t^7+44t^5+44t^3+3t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 1664*K1**4*K2**2 + 3424*K1**4*K2 - 4672*K1**4 - 128*K1**3*K2**2*K3 + 640*K1**3*K2*K3 - 992*K1**3*K3 - 640*K1**2*K2**4 + 4544*K1**2*K2**3 - 10896*K1**2*K2**2 - 672*K1**2*K2*K4 + 10288*K1**2*K2 - 96*K1**2*K3**2 - 3660*K1**2 + 1056*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6568*K1*K2*K3 + 208*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 2440*K2**4 - 416*K2**2*K3**2 - 8*K2**2*K4**2 + 1504*K2**2*K4 - 2200*K2**2 + 104*K2*K3*K5 - 916*K3**2 - 106*K4**2 + 3248
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}]]
If K is slice False
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