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Flat knot 6.1250

Min(phi) over symmetries of the knot is: [-2,-2,0,0,2,2,-1,0,0,2,2,1,1,2,2,0,1,2,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1250']
Arrow polynomial of the knot is: -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.113', '6.132', '6.220', '6.933', '6.1250', '6.1905']
Outer characteristic polynomial of the knot is: t^7+46t^5+93t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1250']
2-strand cable arrow polynomial of the knot is: -768*K2**8 + 512*K2**6*K4 - 3072*K2**6 - 64*K2**4*K4**2 + 2176*K2**4*K4 - 1664*K2**4 - 288*K2**2*K4**2 + 1984*K2**2*K4 + 1168*K2**2 + 16*K2*K4*K6 - 264*K4**2 + 262
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1250']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73184', 'vk6.73197', 'vk6.74301', 'vk6.74941', 'vk6.75094', 'vk6.75113', 'vk6.76510', 'vk6.76927', 'vk6.78032', 'vk6.78055', 'vk6.79351', 'vk6.79767', 'vk6.79960', 'vk6.80805', 'vk6.83781', 'vk6.85577', 'vk6.85739', 'vk6.87645', 'vk6.89621', 'vk6.90175']
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 O1O2O3O4U5U1U2O5O6U3U4U6
R3 orbit {'O1O2O3O4U5U1U2O5O6U3U4U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3U1U4O5O6O4U2U3U5U6
Gauss code of -K* O1O2O3U1U4O5O6O4U2U3U5U6
Diagrammatic symmetry type r
Flat genus of the diagram 2
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 0 2 -2 2],[ 2 0 1 1 2 1 2],[ 0 -1 0 0 1 0 2],[ 0 -1 0 0 1 0 2],[-2 -2 -1 -1 0 -2 1],[ 2 -1 0 0 2 0 2],[-2 -2 -2 -2 -1 -2 0]]
Primitive based matrix [[ 0 2 2 0 0 -2 -2],[-2 0 1 -1 -1 -2 -2],[-2 -1 0 -2 -2 -2 -2],[ 0 1 2 0 0 0 -1],[ 0 1 2 0 0 0 -1],[ 2 2 2 0 0 0 -1],[ 2 2 2 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,2,2,-1,1,1,2,2,2,2,2,2,0,0,1,0,1,1]
Phi over symmetry [-2,-2,0,0,2,2,-1,0,0,2,2,1,1,2,2,0,1,2,1,2,1]
Phi of -K [-2,-2,0,0,2,2,-1,1,1,2,2,2,2,2,2,0,0,1,0,1,1]
Phi of K* [-2,-2,0,0,2,2,-1,0,0,2,2,1,1,2,2,0,1,2,1,2,1]
Phi of -K* [-2,-2,0,0,2,2,-1,0,0,2,2,1,1,2,2,0,1,2,1,2,1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+8z+5
Enhanced Jones-Krushkal polynomial -8w^4z^2+11w^3z^2+8w^2z+5
Inner characteristic polynomial t^6+30t^4+13t^2
Outer characteristic polynomial t^7+46t^5+93t^3
Flat arrow polynomial -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
2-strand cable arrow polynomial -768*K2**8 + 512*K2**6*K4 - 3072*K2**6 - 64*K2**4*K4**2 + 2176*K2**4*K4 - 1664*K2**4 - 288*K2**2*K4**2 + 1984*K2**2*K4 + 1168*K2**2 + 16*K2*K4*K6 - 264*K4**2 + 262
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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