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Flat knot 6.1258

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**2 + 4*K2 + 4*K3 + 2*K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1258']
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: -128*K1**6 - 128*K1**4*K2**2 + 1024*K1**4*K2 - 2848*K1**4 + 192*K1**3*K2*K3 + 128*K1**3*K3*K4 + 64*K1**3*K4*K5 + 128*K1**2*K2**3 - 2016*K1**2*K2**2 + 4080*K1**2*K2 - 1088*K1**2*K3**2 - 704*K1**2*K4**2 - 128*K1**2*K5**2 - 2800*K1**2 + 3072*K1*K2*K3 + 2176*K1*K3*K4 + 848*K1*K4*K5 + 112*K1*K5*K6 - 128*K2**4 - 96*K2**2*K3**2 - 96*K2**2*K4**2 + 336*K2**2*K4 - 2608*K2**2 + 480*K2*K3*K5 + 192*K2*K4*K6 - 32*K3**4 - 128*K3**2*K4**2 + 64*K3**2*K6 - 1872*K3**2 + 128*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1128*K4**2 - 496*K5**2 - 128*K6**2 - 32*K7**2 - 4*K8**2 + 3578
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1258']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4794', 'vk6.5129', 'vk6.6353', 'vk6.6787', 'vk6.8321', 'vk6.8761', 'vk6.9691', 'vk6.10000', 'vk6.21015', 'vk6.22439', 'vk6.28466', 'vk6.40235', 'vk6.42165', 'vk6.46733', 'vk6.48814', 'vk6.49033', 'vk6.49849', 'vk6.51512', 'vk6.58963', 'vk6.69793']
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 O1O2O3O4U5U4U2O5O6U3U1U6
R3 orbit {'O1O2O3O4U5U4U2O5O6U3U1U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3U1U4O5O6O4U6U3U5U2
Gauss code of -K* O1O2O3U1U4O5O6O4U6U3U5U2
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 0 0 1 -2 2],[ 1 0 0 1 1 -1 2],[ 0 0 0 0 0 -1 1],[ 0 -1 0 0 1 -1 1],[-1 -1 0 -1 0 -1 0],[ 2 1 1 1 1 0 2],[-2 -2 -1 -1 0 -2 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 16z+33
Enhanced Jones-Krushkal polynomial 16w^2z+33w
Inner characteristic polynomial t
Outer characteristic polynomial t^2
Flat arrow polynomial -8*K1**2 - 8*K1*K2 + 4*K1 - 4*K2**2 + 4*K2 + 4*K3 + 2*K4 + 7
2-strand cable arrow polynomial -128*K1**6 - 128*K1**4*K2**2 + 1024*K1**4*K2 - 2848*K1**4 + 192*K1**3*K2*K3 + 128*K1**3*K3*K4 + 64*K1**3*K4*K5 + 128*K1**2*K2**3 - 2016*K1**2*K2**2 + 4080*K1**2*K2 - 1088*K1**2*K3**2 - 704*K1**2*K4**2 - 128*K1**2*K5**2 - 2800*K1**2 + 3072*K1*K2*K3 + 2176*K1*K3*K4 + 848*K1*K4*K5 + 112*K1*K5*K6 - 128*K2**4 - 96*K2**2*K3**2 - 96*K2**2*K4**2 + 336*K2**2*K4 - 2608*K2**2 + 480*K2*K3*K5 + 192*K2*K4*K6 - 32*K3**4 - 128*K3**2*K4**2 + 64*K3**2*K6 - 1872*K3**2 + 128*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1128*K4**2 - 496*K5**2 - 128*K6**2 - 32*K7**2 - 4*K8**2 + 3578
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice True
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