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Flat knot 6.899

Min(phi) over symmetries of the knot is: [0]
Flat knots (up to 7 crossings) with same phi are :['6.129', '6.899', '6.1258', '7.13893', '7.14277', '7.20990', '7.25000', '7.25725', '7.28256', '7.28266', '7.31466', '7.36145', '7.36268', '7.44910', '7.45069', '7.45098', '7.45148', '7.45357', '7.45690', '7.45856', '7.46147', '7.46161']
Arrow polynomial of the knot is: -8*K1**2 - 8*K1*K2 + 4*K1 + 4*K2 + 4*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.899', '6.912', '6.1806']
Outer characteristic polynomial of the knot is: t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.129', '6.899', '6.1258', '7.13893', '7.14277', '7.20990', '7.25000', '7.25725', '7.28256', '7.28266', '7.31466', '7.36145', '7.36268', '7.44910', '7.45069', '7.45098', '7.45148', '7.45357', '7.45690', '7.45856', '7.46147', '7.46161']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 640*K1**4*K2**2 + 1088*K1**4*K2 - 2496*K1**4 + 448*K1**3*K2*K3 - 2816*K1**2*K2**2 + 3360*K1**2*K2 - 1600*K1**2*K3**2 - 768*K1**2*K4**2 + 3328*K1*K2*K3 + 1472*K1*K3*K4 + 512*K1*K4*K5 - 672*K2**4 - 1024*K2**2*K3**2 - 512*K2**2*K4**2 + 704*K2**2*K4 - 672*K2**2 + 608*K2*K3*K5 + 320*K2*K4*K6 - 592*K3**2 - 312*K4**2 - 112*K5**2 - 32*K6**2 + 1174
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.899']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17', 'vk6.26', 'vk6.37', 'vk6.145', 'vk6.158', 'vk6.160', 'vk6.175', 'vk6.1202', 'vk6.1205', 'vk6.1296', 'vk6.1306', 'vk6.1313', 'vk6.2357', 'vk6.2394', 'vk6.2395', 'vk6.2958', 'vk6.3536', 'vk6.3537', 'vk6.6912', 'vk6.6913', 'vk6.6944', 'vk6.6945', 'vk6.15377', 'vk6.15390', 'vk6.15496', 'vk6.33449', 'vk6.33453', 'vk6.33504', 'vk6.33508', 'vk6.33610', 'vk6.49926', 'vk6.53755']
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 O1O2O3O4U2U4O5O6U5U6U1U3
R3 orbit {'O1O2O3O4U2U4O5O6U5U6U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U4U5U6O5O6U1U3
Gauss code of K* O1O2O3O4U3U5U4U6O5O6U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U5U1U6U2
Diagrammatic symmetry type c
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 -2 2 1 -1 1],[ 1 0 -1 2 1 -1 1],[ 2 1 0 2 1 0 0],[-2 -2 -2 0 0 -1 1],[-1 -1 -1 0 0 0 0],[ 1 1 0 1 0 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix [[0 0],[0 0]]
If based matrix primitive False
Phi of primitive based matrix [0]
Phi over symmetry [0]
Phi of -K [0]
Phi of K* [0]
Phi of -K* [0]
Symmetry type of based matrix a
u-polynomial 0
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial 13w^2z+27w
Inner characteristic polynomial t
Outer characteristic polynomial t^2
Flat arrow polynomial -8*K1**2 - 8*K1*K2 + 4*K1 + 4*K2 + 4*K3 + 5
2-strand cable arrow polynomial -768*K1**6 - 640*K1**4*K2**2 + 1088*K1**4*K2 - 2496*K1**4 + 448*K1**3*K2*K3 - 2816*K1**2*K2**2 + 3360*K1**2*K2 - 1600*K1**2*K3**2 - 768*K1**2*K4**2 + 3328*K1*K2*K3 + 1472*K1*K3*K4 + 512*K1*K4*K5 - 672*K2**4 - 1024*K2**2*K3**2 - 512*K2**2*K4**2 + 704*K2**2*K4 - 672*K2**2 + 608*K2*K3*K5 + 320*K2*K4*K6 - 592*K3**2 - 312*K4**2 - 112*K5**2 - 32*K6**2 + 1174
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice True
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