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Flat knot 6.968

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,0,0,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.968', '7.42716', '7.45612']
Arrow polynomial of the knot is: 8*K1**3 - 16*K1**2 - 8*K1*K2 - 2*K1 + 8*K2 + 2*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.249', '6.968', '6.1661', '6.1665', '6.2068']
Outer characteristic polynomial of the knot is: t^7+14t^5+25t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.968', '6.2073', '7.42716', '7.45612']
2-strand cable arrow polynomial of the knot is: -1536*K1**4*K2**2 + 4768*K1**4*K2 - 9024*K1**4 + 2016*K1**3*K2*K3 - 1408*K1**3*K3 - 1472*K1**2*K2**4 + 5728*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 18176*K1**2*K2**2 - 1760*K1**2*K2*K4 + 15080*K1**2*K2 - 448*K1**2*K3**2 - 2168*K1**2 + 2272*K1*K2**3*K3 - 3040*K1*K2**2*K3 - 672*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 10720*K1*K2*K3 + 560*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 4288*K2**4 - 64*K2**3*K6 - 768*K2**2*K3**2 - 128*K2**2*K4**2 + 3016*K2**2*K4 - 2156*K2**2 + 392*K2*K3*K5 + 48*K2*K4*K6 - 1252*K3**2 - 272*K4**2 - 36*K5**2 - 4*K6**2 + 3718
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.968']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.338', 'vk6.377', 'vk6.451', 'vk6.735', 'vk6.788', 'vk6.900', 'vk6.1470', 'vk6.1527', 'vk6.1599', 'vk6.1968', 'vk6.2007', 'vk6.2080', 'vk6.2491', 'vk6.2748', 'vk6.3005', 'vk6.3128', 'vk6.3789', 'vk6.3980', 'vk6.7173', 'vk6.7348', 'vk6.18792', 'vk6.19862', 'vk6.24924', 'vk6.25385', 'vk6.25919', 'vk6.26305', 'vk6.26748', 'vk6.37994', 'vk6.38051', 'vk6.45042', 'vk6.50103', 'vk6.60757']
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 O1O2O3O4U5U3O6O5U6U4U1U2
R3 orbit {'O1O2O3O4U5U3O6O5U6U4U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U4U1U5O6O5U2U6
Gauss code of K* O1O2O3O4U3U4U5U2O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U3U6U1U2
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 -1 1 0 1 0 -1],[ 1 0 1 0 1 1 -1],[-1 -1 0 0 1 -1 -1],[ 0 0 0 0 0 0 -1],[-1 -1 -1 0 0 0 -1],[ 0 -1 1 0 0 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 -1 -1 -1],[-1 -1 0 0 0 -1 -1],[ 0 0 0 0 0 0 -1],[ 0 1 0 0 0 -1 -1],[ 1 1 1 0 1 0 -1],[ 1 1 1 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,1,1,1,0,0,1,1,0,0,1,1,1,1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,0,0,1,1,1,-1]
Phi of -K [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,0,0,1,1,1,-1]
Phi of K* [-1,-1,0,0,1,1,-1,1,1,1,1,0,1,1,1,0,0,0,0,1,1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,1,1,0,0,0,0,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+10t^4+15t^2+4
Outer characteristic polynomial t^7+14t^5+25t^3+8t
Flat arrow polynomial 8*K1**3 - 16*K1**2 - 8*K1*K2 - 2*K1 + 8*K2 + 2*K3 + 9
2-strand cable arrow polynomial -1536*K1**4*K2**2 + 4768*K1**4*K2 - 9024*K1**4 + 2016*K1**3*K2*K3 - 1408*K1**3*K3 - 1472*K1**2*K2**4 + 5728*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 18176*K1**2*K2**2 - 1760*K1**2*K2*K4 + 15080*K1**2*K2 - 448*K1**2*K3**2 - 2168*K1**2 + 2272*K1*K2**3*K3 - 3040*K1*K2**2*K3 - 672*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 10720*K1*K2*K3 + 560*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 4288*K2**4 - 64*K2**3*K6 - 768*K2**2*K3**2 - 128*K2**2*K4**2 + 3016*K2**2*K4 - 2156*K2**2 + 392*K2*K3*K5 + 48*K2*K4*K6 - 1252*K3**2 - 272*K4**2 - 36*K5**2 - 4*K6**2 + 3718
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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