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Flat knot 6.1567

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,1,2,1,1,0,1,0,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1567']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.330', '6.531', '6.1076', '6.1079', '6.1567']
Outer characteristic polynomial of the knot is: t^7+23t^5+40t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1567']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1376*K1**4*K2 - 1488*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1504*K1**3*K2*K3 - 384*K1**3*K3 - 832*K1**2*K2**4 + 3456*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 10576*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 8600*K1**2*K2 - 720*K1**2*K3**2 - 5116*K1**2 + 2272*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 2624*K1*K2**2*K3 - 192*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 8568*K1*K2*K3 + 640*K1*K3*K4 + 16*K1*K4*K5 - 96*K2**6 + 192*K2**4*K4 - 3168*K2**4 - 32*K2**3*K6 - 1280*K2**2*K3**2 - 96*K2**2*K4**2 + 2016*K2**2*K4 - 2518*K2**2 - 32*K2*K3**2*K4 + 336*K2*K3*K5 + 24*K2*K4*K6 - 1960*K3**2 - 280*K4**2 - 28*K5**2 - 2*K6**2 + 3958
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1567']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16915', 'vk6.17157', 'vk6.20514', 'vk6.21898', 'vk6.23303', 'vk6.23602', 'vk6.27957', 'vk6.29434', 'vk6.35329', 'vk6.35761', 'vk6.39363', 'vk6.41541', 'vk6.42825', 'vk6.43107', 'vk6.45930', 'vk6.47617', 'vk6.55074', 'vk6.55323', 'vk6.57379', 'vk6.58538', 'vk6.59463', 'vk6.59754', 'vk6.62034', 'vk6.63028', 'vk6.64911', 'vk6.65124', 'vk6.66924', 'vk6.67773', 'vk6.68214', 'vk6.68358', 'vk6.69528', 'vk6.70234']
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 O1O2O3U4O5U1U2O6O4U6U5U3
R3 orbit {'O1O2O3U4O5U1U2O6O4U6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O6O5U2U3O4U6
Gauss code of K* O1O2O3U4U5U3O6U2O4O5U1U6
Gauss code of -K* O1O2O3U4U3O5O6U2O4U1U5U6
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 2 0 1 -1],[ 2 0 1 2 1 1 -1],[ 0 -1 0 1 0 0 -1],[-2 -2 -1 0 0 -1 -1],[ 0 -1 0 0 0 0 -1],[-1 -1 0 1 0 0 0],[ 1 1 1 1 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -1 -2],[-1 1 0 0 0 0 -1],[ 0 0 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -1],[ 1 1 0 1 1 0 1],[ 2 2 1 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,0,1,1,2,0,0,0,1,0,1,1,1,1,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,1,2,1,1,0,1,0,0,0,0,1,1]
Phi of -K [-2,-1,0,0,1,2,2,1,1,2,2,0,0,2,2,0,1,1,1,2,0]
Phi of K* [-2,-1,0,0,1,2,0,1,2,2,2,1,1,2,2,0,0,1,0,1,2]
Phi of -K* [-2,-1,0,0,1,2,-1,1,1,1,2,1,1,0,1,0,0,0,0,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+13t^4+12t^2+1
Outer characteristic polynomial t^7+23t^5+40t^3+4t
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1376*K1**4*K2 - 1488*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1504*K1**3*K2*K3 - 384*K1**3*K3 - 832*K1**2*K2**4 + 3456*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 10576*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 8600*K1**2*K2 - 720*K1**2*K3**2 - 5116*K1**2 + 2272*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 2624*K1*K2**2*K3 - 192*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 8568*K1*K2*K3 + 640*K1*K3*K4 + 16*K1*K4*K5 - 96*K2**6 + 192*K2**4*K4 - 3168*K2**4 - 32*K2**3*K6 - 1280*K2**2*K3**2 - 96*K2**2*K4**2 + 2016*K2**2*K4 - 2518*K2**2 - 32*K2*K3**2*K4 + 336*K2*K3*K5 + 24*K2*K4*K6 - 1960*K3**2 - 280*K4**2 - 28*K5**2 - 2*K6**2 + 3958
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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