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Flat knot 6.1134

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,2,0,0,1,1,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.1134']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 4*K1*K2 - 4*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.570', '6.808', '6.1005', '6.1045', '6.1134', '6.1538', '6.1819']
Outer characteristic polynomial of the knot is: t^7+38t^5+57t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.967', '6.1032', '6.1134']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**4 + 1536*K1**4*K2**3 - 1920*K1**4*K2**2 + 1664*K1**4*K2 - 2144*K1**4 + 192*K1**3*K2*K3 + 64*K1**3*K3*K4 + 1280*K1**2*K2**5 - 4224*K1**2*K2**4 + 3072*K1**2*K2**3 - 4640*K1**2*K2**2 + 4384*K1**2*K2 - 224*K1**2*K3**2 - 128*K1**2*K4**2 - 2240*K1**2 + 1600*K1*K2**3*K3 + 2464*K1*K2*K3 + 496*K1*K3*K4 + 96*K1*K4*K5 - 704*K2**6 + 128*K2**4*K4 - 640*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 208*K2**2*K4 - 776*K2**2 + 16*K2*K3*K5 - 648*K3**2 - 248*K4**2 - 24*K5**2 + 2046
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1134']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16964', 'vk6.17205', 'vk6.20846', 'vk6.22249', 'vk6.23364', 'vk6.23661', 'vk6.28307', 'vk6.35417', 'vk6.35840', 'vk6.39917', 'vk6.42015', 'vk6.43163', 'vk6.46467', 'vk6.55124', 'vk6.55382', 'vk6.57657', 'vk6.58843', 'vk6.59834', 'vk6.68398', 'vk6.69713']
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 O1O2O3O4U5U4O6U2U3O5U1U6
R3 orbit {'O1O2O3O4U5U4O6U2U3O5U1U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2U3O4O5U1O6O3U6U4U5U2
Gauss code of -K* O1O2U3O4O5U1O6O3U6U4U5U2
Diagrammatic symmetry type r
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -1 1 1 -2 2],[ 1 0 0 2 1 -2 2],[ 1 0 0 1 0 -1 1],[-1 -2 -1 0 0 -2 0],[-1 -1 0 0 0 -1 -1],[ 2 2 1 2 1 0 2],[-2 -2 -1 0 1 -2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 -1 -2 -2],[ 1 1 0 1 0 0 -1],[ 1 2 1 2 0 0 -2],[ 2 2 1 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,0,1,2,2,0,0,1,1,1,2,2,0,1,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,2,2,0,0,1,1,1,2,2,0,1,2]
Phi of -K [-2,-1,-1,1,1,2,-1,0,1,2,2,0,0,1,1,1,2,2,0,1,2]
Phi of K* [-2,-1,-1,1,1,2,1,2,1,2,2,0,0,1,1,1,2,2,0,-1,0]
Phi of -K* [-2,-1,-1,1,1,2,1,2,1,2,2,0,0,1,1,1,2,2,0,-1,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z+17
Enhanced Jones-Krushkal polynomial 4w^4z-12w^3z+16w^2z+17w
Inner characteristic polynomial t^6+26t^4+21t^2
Outer characteristic polynomial t^7+38t^5+57t^3
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 4*K1*K2 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial -512*K1**4*K2**4 + 1536*K1**4*K2**3 - 1920*K1**4*K2**2 + 1664*K1**4*K2 - 2144*K1**4 + 192*K1**3*K2*K3 + 64*K1**3*K3*K4 + 1280*K1**2*K2**5 - 4224*K1**2*K2**4 + 3072*K1**2*K2**3 - 4640*K1**2*K2**2 + 4384*K1**2*K2 - 224*K1**2*K3**2 - 128*K1**2*K4**2 - 2240*K1**2 + 1600*K1*K2**3*K3 + 2464*K1*K2*K3 + 496*K1*K3*K4 + 96*K1*K4*K5 - 704*K2**6 + 128*K2**4*K4 - 640*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 208*K2**2*K4 - 776*K2**2 + 16*K2*K3*K5 - 648*K3**2 - 248*K4**2 - 24*K5**2 + 2046
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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