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Flat knot 6.1776

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,2,1,1,0,1,0,1,0,1,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1776']
Arrow polynomial of the knot is: -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.665', '6.1301', '6.1514', '6.1646', '6.1669', '6.1709', '6.1710', '6.1744', '6.1776']
Outer characteristic polynomial of the knot is: t^7+19t^5+24t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1776']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 320*K1**4*K2**2 + 1664*K1**4*K2 - 7008*K1**4 + 960*K1**3*K2*K3 + 64*K1**3*K3*K4 - 832*K1**3*K3 + 128*K1**2*K2**2*K4 - 5792*K1**2*K2**2 - 672*K1**2*K2*K4 + 12312*K1**2*K2 - 1248*K1**2*K3**2 - 240*K1**2*K4**2 - 5352*K1**2 - 704*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8208*K1*K2*K3 + 1992*K1*K3*K4 + 232*K1*K4*K5 - 216*K2**4 - 160*K2**2*K3**2 - 48*K2**2*K4**2 + 904*K2**2*K4 - 5644*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2732*K3**2 - 826*K4**2 - 76*K5**2 - 4*K6**2 + 5800
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1776']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3660', 'vk6.3755', 'vk6.3950', 'vk6.4045', 'vk6.4489', 'vk6.4586', 'vk6.5871', 'vk6.6000', 'vk6.7143', 'vk6.7322', 'vk6.7413', 'vk6.7928', 'vk6.8049', 'vk6.9358', 'vk6.17912', 'vk6.18009', 'vk6.18742', 'vk6.24451', 'vk6.24861', 'vk6.25324', 'vk6.37489', 'vk6.43878', 'vk6.44216', 'vk6.44521', 'vk6.48300', 'vk6.48363', 'vk6.50081', 'vk6.50195', 'vk6.50589', 'vk6.50654', 'vk6.55857', 'vk6.60716']
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 O1O2O3U4U3O5O4U1U5O6U2U6
R3 orbit {'O1O2O3U4U3O5O4U1U5O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O4U5U3O6O5U1U6
Gauss code of K* O1O2U3O4O3U1U4U5O6O5U2U6
Gauss code of -K* O1O2U1O3O4U5U3O6O5U6U2U4
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 1 0 0 1],[ 2 0 2 1 1 0 1],[ 0 -2 0 1 0 -1 1],[-1 -1 -1 0 -1 -1 0],[ 0 -1 0 1 0 0 1],[ 0 0 1 1 0 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 0 0 -1 -1 -1],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 1 0 0],[ 0 1 1 -1 0 0 -2],[ 0 1 1 0 0 0 -1],[ 2 1 1 0 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,0,0,1,1,1,1,1,1,1,-1,0,0,0,2,1]
Phi over symmetry [-2,0,0,0,1,1,0,1,2,1,1,0,1,0,1,0,1,1,1,1,0]
Phi of -K [-2,0,0,0,1,1,0,1,2,2,2,0,1,0,0,0,0,0,0,1,0]
Phi of K* [-1,-1,0,0,0,2,0,0,0,0,2,0,0,1,2,0,-1,0,0,1,2]
Phi of -K* [-2,0,0,0,1,1,0,1,2,1,1,0,1,0,1,0,1,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+13t^4+11t^2+1
Outer characteristic polynomial t^7+19t^5+24t^3+4t
Flat arrow polynomial -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -384*K1**6 - 320*K1**4*K2**2 + 1664*K1**4*K2 - 7008*K1**4 + 960*K1**3*K2*K3 + 64*K1**3*K3*K4 - 832*K1**3*K3 + 128*K1**2*K2**2*K4 - 5792*K1**2*K2**2 - 672*K1**2*K2*K4 + 12312*K1**2*K2 - 1248*K1**2*K3**2 - 240*K1**2*K4**2 - 5352*K1**2 - 704*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8208*K1*K2*K3 + 1992*K1*K3*K4 + 232*K1*K4*K5 - 216*K2**4 - 160*K2**2*K3**2 - 48*K2**2*K4**2 + 904*K2**2*K4 - 5644*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2732*K3**2 - 826*K4**2 - 76*K5**2 - 4*K6**2 + 5800
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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