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Flat knot 6.859

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,3,4,0,0,2,1,1,1,1,0,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.859']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.362', '6.624', '6.789', '6.859', '6.882', '6.975', '6.989', '6.1048', '6.1057', '6.1158']
Outer characteristic polynomial of the knot is: t^7+46t^5+58t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.859']
2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 160*K1**2*K2**3 - 1344*K1**2*K2**2 + 792*K1**2*K2 - 16*K1**2*K3**2 - 788*K1**2 + 352*K1*K2**3*K3 + 1568*K1*K2*K3 + 184*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 488*K2**4 - 240*K2**2*K3**2 - 56*K2**2*K4**2 + 408*K2**2*K4 - 504*K2**2 + 80*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 572*K3**2 - 238*K4**2 - 32*K5**2 - 16*K6**2 + 828
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.859']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10510', 'vk6.10519', 'vk6.10581', 'vk6.10598', 'vk6.10770', 'vk6.10785', 'vk6.10887', 'vk6.10894', 'vk6.17683', 'vk6.17687', 'vk6.17730', 'vk6.17734', 'vk6.24289', 'vk6.24293', 'vk6.24776', 'vk6.25233', 'vk6.30191', 'vk6.30200', 'vk6.30260', 'vk6.30277', 'vk6.30389', 'vk6.30404', 'vk6.30639', 'vk6.30734', 'vk6.36517', 'vk6.36954', 'vk6.43621', 'vk6.43629', 'vk6.43723', 'vk6.43733', 'vk6.52723', 'vk6.52829', 'vk6.60347', 'vk6.60365', 'vk6.60643', 'vk6.60982', 'vk6.63447', 'vk6.63456', 'vk6.65418', 'vk6.65772']
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 O1O2O3O4U1U4O5O6U3U5U6U2
R3 orbit {'O1O2O3O4U1U4O5O6U3U5U6U2', 'O1O2O3O4U1U4O5U2O6U5U3U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U3U5U6U2O5O6U1U4
Gauss code of K* O1O2O3O4U5U4U1U6O5O6U2U3
Gauss code of -K* O1O2O3O4U2U3O5O6U5U4U1U6
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 1 -1 1 0 2],[ 3 0 3 2 1 1 1],[-1 -3 0 -2 0 0 2],[ 1 -2 2 0 0 1 2],[-1 -1 0 0 0 0 0],[ 0 -1 0 -1 0 0 1],[-2 -1 -2 -2 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -1 -2 -1],[-1 0 0 0 0 0 -1],[-1 2 0 0 0 -2 -3],[ 0 1 0 0 0 -1 -1],[ 1 2 0 2 1 0 -2],[ 3 1 1 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,2,1,2,1,0,0,0,1,0,2,3,1,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,3,4,0,0,2,1,1,1,1,0,-1,1]
Phi of -K [-3,-1,0,1,1,2,0,2,1,3,4,0,0,2,1,1,1,1,0,-1,1]
Phi of K* [-2,-1,-1,0,1,3,-1,1,1,1,4,0,1,0,1,1,2,3,0,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,1,3,1,1,0,2,2,0,0,1,0,0,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z+7
Enhanced Jones-Krushkal polynomial -8w^3z+11w^2z+7w
Inner characteristic polynomial t^6+30t^4+17t^2
Outer characteristic polynomial t^7+46t^5+58t^3
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
2-strand cable arrow polynomial -192*K1**2*K2**4 + 160*K1**2*K2**3 - 1344*K1**2*K2**2 + 792*K1**2*K2 - 16*K1**2*K3**2 - 788*K1**2 + 352*K1*K2**3*K3 + 1568*K1*K2*K3 + 184*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 488*K2**4 - 240*K2**2*K3**2 - 56*K2**2*K4**2 + 408*K2**2*K4 - 504*K2**2 + 80*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 572*K3**2 - 238*K4**2 - 32*K5**2 - 16*K6**2 + 828
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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