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Flat knot 6.986

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,2,3,4,1,-1,1,0,-1,1,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.986']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.237', '6.602', '6.956', '6.986', '6.992', '6.1052', '6.1059']
Outer characteristic polynomial of the knot is: t^7+39t^5+86t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.986']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 320*K1**2*K2**3 - 1520*K1**2*K2**2 + 1688*K1**2*K2 - 48*K1**2*K3**2 - 1192*K1**2 + 96*K1*K2**3*K3 + 1296*K1*K2*K3 + 208*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 792*K2**4 - 272*K2**2*K3**2 - 72*K2**2*K4**2 + 584*K2**2*K4 - 642*K2**2 + 176*K2*K3*K5 + 40*K2*K4*K6 - 428*K3**2 - 246*K4**2 - 44*K5**2 - 14*K6**2 + 1108
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.986']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11285', 'vk6.11365', 'vk6.12548', 'vk6.12661', 'vk6.18360', 'vk6.18698', 'vk6.24808', 'vk6.25265', 'vk6.30960', 'vk6.31086', 'vk6.31209', 'vk6.31552', 'vk6.32141', 'vk6.32262', 'vk6.32381', 'vk6.32790', 'vk6.36986', 'vk6.37438', 'vk6.39632', 'vk6.41871', 'vk6.44170', 'vk6.44489', 'vk6.46236', 'vk6.47841', 'vk6.52037', 'vk6.52478', 'vk6.52882', 'vk6.53364', 'vk6.56370', 'vk6.57606', 'vk6.61014', 'vk6.62266', 'vk6.63652', 'vk6.63699', 'vk6.64082', 'vk6.64129', 'vk6.65800', 'vk6.66054', 'vk6.68800', 'vk6.69008']
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 O1O2O3O4U1U4O5U3O6U5U2U6
R3 orbit {'O1O2O3O4U1U4O5U3O6U5U2U6', 'O1O2O3O4U1U4U2O5O6U3U5U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U5U3U6O5U2O6U1U4
Gauss code of K* O1O2O3U4U2U5U6O4O6U1O5U3
Gauss code of -K* O1O2O3U1O4U3O5O6U5U4U2U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 0 0 1 0 2],[ 3 0 3 2 1 1 1],[ 0 -3 0 -1 0 1 2],[ 0 -2 1 0 0 1 1],[-1 -1 0 0 0 0 0],[ 0 -1 -1 -1 0 0 1],[-2 -1 -2 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 0 -3],[-2 0 0 -1 -1 -2 -1],[-1 0 0 0 0 0 -1],[ 0 1 0 0 1 1 -2],[ 0 1 0 -1 0 -1 -1],[ 0 2 0 -1 1 0 -3],[ 3 1 1 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,0,3,0,1,1,2,1,0,0,0,1,-1,-1,2,1,1,3]
Phi over symmetry [-3,0,0,0,1,2,0,1,2,3,4,1,-1,1,0,-1,1,1,1,1,1]
Phi of -K [-3,0,0,0,1,2,0,1,2,3,4,1,-1,1,0,-1,1,1,1,1,1]
Phi of K* [-2,-1,0,0,0,3,1,0,1,1,4,1,1,1,3,-1,1,0,1,1,2]
Phi of -K* [-3,0,0,0,1,2,1,2,3,1,1,-1,-1,0,1,1,0,1,0,2,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -4w^3z+11w^2z+15w
Inner characteristic polynomial t^6+25t^4+21t^2
Outer characteristic polynomial t^7+39t^5+86t^3
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 320*K1**2*K2**3 - 1520*K1**2*K2**2 + 1688*K1**2*K2 - 48*K1**2*K3**2 - 1192*K1**2 + 96*K1*K2**3*K3 + 1296*K1*K2*K3 + 208*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 792*K2**4 - 272*K2**2*K3**2 - 72*K2**2*K4**2 + 584*K2**2*K4 - 642*K2**2 + 176*K2*K3*K5 + 40*K2*K4*K6 - 428*K3**2 - 246*K4**2 - 44*K5**2 - 14*K6**2 + 1108
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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