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Flat knot 6.924

Min(phi) over symmetries of the knot is: [-3,0,1,2,0,3,3,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.924']
Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314']
Outer characteristic polynomial of the knot is: t^5+35t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.924']
2-strand cable arrow polynomial of the knot is: -320*K1**6 + 288*K1**4*K2 - 896*K1**4 - 272*K1**2*K2**2 + 968*K1**2*K2 - 256*K1**2*K3**2 - 128*K1**2*K4**2 - 476*K1**2 + 744*K1*K2*K3 + 680*K1*K3*K4 + 256*K1*K4*K5 + 24*K1*K5*K6 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 48*K2**2*K4 - 510*K2**2 + 40*K2*K3*K5 + 16*K2*K4*K6 - 512*K3**2 - 366*K4**2 - 124*K5**2 - 18*K6**2 + 844
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.924']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4136', 'vk6.4167', 'vk6.5378', 'vk6.5409', 'vk6.5474', 'vk6.5585', 'vk6.7504', 'vk6.7670', 'vk6.9009', 'vk6.9040', 'vk6.11185', 'vk6.12273', 'vk6.12380', 'vk6.12433', 'vk6.12464', 'vk6.13362', 'vk6.13587', 'vk6.13618', 'vk6.14273', 'vk6.14720', 'vk6.14733', 'vk6.15189', 'vk6.15880', 'vk6.15891', 'vk6.26201', 'vk6.26644', 'vk6.30834', 'vk6.30865', 'vk6.32022', 'vk6.32053', 'vk6.33080', 'vk6.33111', 'vk6.38144', 'vk6.38177', 'vk6.44805', 'vk6.44922', 'vk6.49232', 'vk6.49343', 'vk6.52755', 'vk6.53534']
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 O1O2O3O4U3U5O6O5U1U6U4U2
R3 orbit {'O1O2O3O4U3U5O6O5U1U6U4U2', 'O1O2O3U2O4U5O6O5U1U6U3U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U3U1U5U4O6O5U6U2
Gauss code of K* O1O2O3O4U1U4U5U3O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U2U6U1U4
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 -3 1 -1 2 1 0],[ 3 0 3 0 3 3 0],[-1 -3 0 -1 1 0 -1],[ 1 0 1 0 1 1 0],[-2 -3 -1 -1 0 -1 -1],[-1 -3 0 -1 1 0 0],[ 0 0 1 0 1 0 0]]
Primitive based matrix [[ 0 2 1 0 -3],[-2 0 -1 -1 -3],[-1 1 0 -1 -3],[ 0 1 1 0 0],[ 3 3 3 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,0,3,1,1,3,1,3,0]
Phi over symmetry [-3,0,1,2,0,3,3,1,1,1]
Phi of -K [-3,0,1,2,3,1,2,0,1,0]
Phi of K* [-2,-1,0,3,0,1,2,0,1,3]
Phi of -K* [-3,0,1,2,0,3,3,1,1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial 11w^2z+23w
Inner characteristic polynomial t^4+21t^2
Outer characteristic polynomial t^5+35t^3+19t
Flat arrow polynomial -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2
2-strand cable arrow polynomial -320*K1**6 + 288*K1**4*K2 - 896*K1**4 - 272*K1**2*K2**2 + 968*K1**2*K2 - 256*K1**2*K3**2 - 128*K1**2*K4**2 - 476*K1**2 + 744*K1*K2*K3 + 680*K1*K3*K4 + 256*K1*K4*K5 + 24*K1*K5*K6 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 48*K2**2*K4 - 510*K2**2 + 40*K2*K3*K5 + 16*K2*K4*K6 - 512*K3**2 - 366*K4**2 - 124*K5**2 - 18*K6**2 + 844
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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