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Flat knot 6.1809

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,0,1,1,-1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1809']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1030', '6.1062', '6.1226', '6.1508', '6.1525', '6.1596', '6.1724', '6.1729', '6.1735', '6.1738', '6.1789', '6.1809', '6.1921']
Outer characteristic polynomial of the knot is: t^7+22t^5+54t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1809']
2-strand cable arrow polynomial of the knot is: 2368*K1**4*K2 - 5264*K1**4 - 256*K1**3*K2**2*K3 + 1920*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1408*K1**3*K3 - 128*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 832*K1**2*K2**3 + 736*K1**2*K2**2*K4 - 8400*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 288*K1**2*K2*K4**2 - 1216*K1**2*K2*K4 + 10408*K1**2*K2 - 2128*K1**2*K3**2 - 496*K1**2*K4**2 - 4492*K1**2 + 736*K1*K2**3*K3 - 2016*K1*K2**2*K3 - 288*K1*K2**2*K5 - 1024*K1*K2*K3*K4 + 9120*K1*K2*K3 - 160*K1*K2*K4*K5 + 2856*K1*K3*K4 + 536*K1*K4*K5 - 32*K2**6 + 288*K2**4*K4 - 1080*K2**4 - 64*K2**3*K6 - 480*K2**2*K3**2 - 320*K2**2*K4**2 + 2032*K2**2*K4 - 4586*K2**2 + 424*K2*K3*K5 + 160*K2*K4*K6 - 2536*K3**2 - 1174*K4**2 - 124*K5**2 - 6*K6**2 + 4780
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1809']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13914', 'vk6.14009', 'vk6.14183', 'vk6.14422', 'vk6.14985', 'vk6.15106', 'vk6.15651', 'vk6.16105', 'vk6.16708', 'vk6.16735', 'vk6.16848', 'vk6.18800', 'vk6.19278', 'vk6.19570', 'vk6.23142', 'vk6.23229', 'vk6.25398', 'vk6.26467', 'vk6.33733', 'vk6.33808', 'vk6.34285', 'vk6.35142', 'vk6.37519', 'vk6.42732', 'vk6.44687', 'vk6.54130', 'vk6.54913', 'vk6.54938', 'vk6.56394', 'vk6.56601', 'vk6.59337', 'vk6.64611']
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 O1O2O3U4U3O5O6U5U2O4U1U6
R3 orbit {'O1O2O3U4U3O5O6U5U2O4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U2U6O4O6U1U5
Gauss code of K* O1O2U3O4O5U4U2U6O3O6U1U5
Gauss code of -K* O1O2U3O4O5U1U5O6O3U6U4U2
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 0 1 -1 -1 2],[ 1 0 1 1 -1 0 2],[ 0 -1 0 0 -1 0 1],[-1 -1 0 0 -1 0 -1],[ 1 1 1 1 0 -1 1],[ 1 0 0 0 1 0 1],[-2 -2 -1 1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 1 -1 -1 -1 -2],[-1 -1 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -1],[ 1 1 0 0 0 1 0],[ 1 1 1 1 -1 0 1],[ 1 2 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,0,1,1,-1,0,-1]
Phi over symmetry [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,0,1,1,-1,0,-1]
Phi of -K [-1,-1,-1,0,1,2,-1,0,1,2,2,-1,0,1,2,0,1,1,1,1,2]
Phi of K* [-2,-1,0,1,1,1,2,1,1,2,2,1,1,1,2,0,0,1,-1,0,-1]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,1,1,2,-1,1,1,1,0,0,1,0,1,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+14t^4+29t^2+4
Outer characteristic polynomial t^7+22t^5+54t^3+11t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 2368*K1**4*K2 - 5264*K1**4 - 256*K1**3*K2**2*K3 + 1920*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1408*K1**3*K3 - 128*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 832*K1**2*K2**3 + 736*K1**2*K2**2*K4 - 8400*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 288*K1**2*K2*K4**2 - 1216*K1**2*K2*K4 + 10408*K1**2*K2 - 2128*K1**2*K3**2 - 496*K1**2*K4**2 - 4492*K1**2 + 736*K1*K2**3*K3 - 2016*K1*K2**2*K3 - 288*K1*K2**2*K5 - 1024*K1*K2*K3*K4 + 9120*K1*K2*K3 - 160*K1*K2*K4*K5 + 2856*K1*K3*K4 + 536*K1*K4*K5 - 32*K2**6 + 288*K2**4*K4 - 1080*K2**4 - 64*K2**3*K6 - 480*K2**2*K3**2 - 320*K2**2*K4**2 + 2032*K2**2*K4 - 4586*K2**2 + 424*K2*K3*K5 + 160*K2*K4*K6 - 2536*K3**2 - 1174*K4**2 - 124*K5**2 - 6*K6**2 + 4780
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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