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Flat knot 6.1789

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,2,2,0,1,-1,0,0,0,1,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1789']
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+23t^5+47t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1789']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1760*K1**4*K2 - 2976*K1**4 + 288*K1**3*K2*K3 - 1152*K1**3*K3 + 2400*K1**2*K2**3 - 6768*K1**2*K2**2 - 1248*K1**2*K2*K4 + 6848*K1**2*K2 - 160*K1**2*K3**2 - 3188*K1**2 + 256*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 256*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6384*K1*K2*K3 + 1024*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1528*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 128*K2**2*K4**2 + 1904*K2**2*K4 - 2698*K2**2 + 416*K2*K3*K5 + 104*K2*K4*K6 - 1532*K3**2 - 698*K4**2 - 144*K5**2 - 22*K6**2 + 3032
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1789']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81842', 'vk6.81892', 'vk6.82064', 'vk6.82082', 'vk6.82562', 'vk6.82612', 'vk6.82777', 'vk6.82786', 'vk6.82834', 'vk6.82846', 'vk6.82948', 'vk6.83057', 'vk6.83068', 'vk6.83283', 'vk6.83326', 'vk6.83368', 'vk6.83530', 'vk6.84538', 'vk6.84639', 'vk6.84910', 'vk6.84954', 'vk6.85830', 'vk6.86102', 'vk6.86128', 'vk6.86163', 'vk6.86841', 'vk6.88451', 'vk6.88886', 'vk6.89033', 'vk6.89704', 'vk6.89934', 'vk6.90013']
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 O1O2O3U4U1O5O6U3U2O4U5U6
R3 orbit {'O1O2O3U4U1O5O6U3U2O4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U2U1O4O5U3U6
Gauss code of K* O1O2U3O4O5U6U2U1O3O6U4U5
Gauss code of -K* O1O2U3O4O5U1U2O6O3U5U4U6
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 0 -1 0 2],[ 1 0 2 1 -1 1 1],[ 0 -2 0 0 -1 1 2],[ 0 -1 0 0 0 0 1],[ 1 1 1 0 0 0 1],[ 0 -1 -1 0 0 0 1],[-2 -1 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 0 0 0 -1 -1],[-2 0 -1 -1 -2 -1 -1],[ 0 1 0 0 0 0 -1],[ 0 1 0 0 -1 0 -1],[ 0 2 0 1 0 -1 -2],[ 1 1 0 0 1 0 1],[ 1 1 1 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,0,0,0,1,1,1,1,2,1,1,0,0,0,1,1,0,1,1,2,-1]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,2,2,0,1,-1,0,0,0,1,0,1,-1]
Phi of -K [-1,-1,0,0,0,2,-1,0,1,1,2,-1,0,0,2,-1,0,0,0,1,1]
Phi of K* [-2,0,0,0,1,1,0,1,1,2,2,0,1,-1,0,0,0,1,0,1,-1]
Phi of -K* [-1,-1,0,0,0,2,-1,1,1,2,1,0,0,1,1,0,-1,1,0,1,2]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2+26w^2z+25w
Inner characteristic polynomial t^6+17t^4+26t^2+4
Outer characteristic polynomial t^7+23t^5+47t^3+9t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1760*K1**4*K2 - 2976*K1**4 + 288*K1**3*K2*K3 - 1152*K1**3*K3 + 2400*K1**2*K2**3 - 6768*K1**2*K2**2 - 1248*K1**2*K2*K4 + 6848*K1**2*K2 - 160*K1**2*K3**2 - 3188*K1**2 + 256*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 256*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6384*K1*K2*K3 + 1024*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1528*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 128*K2**2*K4**2 + 1904*K2**2*K4 - 2698*K2**2 + 416*K2*K3*K5 + 104*K2*K4*K6 - 1532*K3**2 - 698*K4**2 - 144*K5**2 - 22*K6**2 + 3032
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice False
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