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Flat knot 6.2068

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,0,0,0,0,1,0,1,1,0,0,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.2068', '7.44729']
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+10t^5+17t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2068', '7.44729']
2-strand cable arrow polynomial of the knot is: -1536*K1**4*K2**2 + 4768*K1**4*K2 - 8192*K1**4 + 2016*K1**3*K2*K3 - 1280*K1**3*K3 - 1472*K1**2*K2**4 + 5728*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 17536*K1**2*K2**2 - 1600*K1**2*K2*K4 + 14312*K1**2*K2 - 512*K1**2*K3**2 - 2520*K1**2 + 2272*K1*K2**3*K3 - 3232*K1*K2**2*K3 - 640*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 10336*K1*K2*K3 + 592*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 4160*K2**4 - 64*K2**3*K6 - 864*K2**2*K3**2 - 128*K2**2*K4**2 + 2952*K2**2*K4 - 2220*K2**2 + 424*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 :['6.2068']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.329', 'vk6.370', 'vk6.442', 'vk6.724', 'vk6.775', 'vk6.891', 'vk6.1467', 'vk6.1526', 'vk6.1598', 'vk6.1963', 'vk6.2004', 'vk6.2075', 'vk6.2494', 'vk6.2749', 'vk6.3016', 'vk6.3137', 'vk6.3788', 'vk6.3981', 'vk6.7172', 'vk6.7349', 'vk6.18785', 'vk6.19853', 'vk6.24909', 'vk6.25372', 'vk6.25917', 'vk6.26294', 'vk6.26739', 'vk6.37996', 'vk6.38053', 'vk6.45035', 'vk6.50102', 'vk6.60758']
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 O1O2U1U2O3O4U5U3O6O5U6U4
R3 orbit {'O1O2U1U2O3O4U5U3O6O5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2U3U4O3O4U1U5O6O5U2U6
Gauss code of K* O1O2U3U1O3O4U5U6O5O6U2U4
Gauss code of -K* O1O2U1U2O3O4U5U4O6O5U3U6
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 0 1 0],[-1 -1 0 0 0 -1 0],[ 0 0 0 0 0 0 -1],[-1 0 0 0 0 0 -1],[ 0 -1 1 0 0 0 -1],[ 1 0 0 1 1 1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 0],[ 0 0 0 0 0 0 -1],[ 0 0 1 0 0 -1 -1],[ 1 0 1 0 1 0 0],[ 1 1 0 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,0,0,0,0,1,0,1,1,0,0,0,1,1,1,0]
Phi over symmetry [-1,-1,0,0,1,1,0,0,0,0,1,0,1,1,0,0,0,1,1,1,0]
Phi of -K [-1,-1,0,0,1,1,0,0,0,1,2,0,1,2,1,0,1,0,1,1,0]
Phi of K* [-1,-1,0,0,1,1,0,0,1,1,2,1,1,2,1,0,0,0,1,0,0]
Phi of -K* [-1,-1,0,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
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+6t^4+7t^2
Outer characteristic polynomial t^7+10t^5+17t^3+4t
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 - 8192*K1**4 + 2016*K1**3*K2*K3 - 1280*K1**3*K3 - 1472*K1**2*K2**4 + 5728*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 17536*K1**2*K2**2 - 1600*K1**2*K2*K4 + 14312*K1**2*K2 - 512*K1**2*K3**2 - 2520*K1**2 + 2272*K1*K2**3*K3 - 3232*K1*K2**2*K3 - 640*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 10336*K1*K2*K3 + 592*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 4160*K2**4 - 64*K2**3*K6 - 864*K2**2*K3**2 - 128*K2**2*K4**2 + 2952*K2**2*K4 - 2220*K2**2 + 424*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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