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Flat knot 6.1646

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,2,0,1,1,0,0,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1646']
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+21t^5+27t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1646']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 1056*K1**4*K2 - 3968*K1**4 + 288*K1**3*K2*K3 - 864*K1**3*K3 + 128*K1**2*K2**3 - 3152*K1**2*K2**2 - 224*K1**2*K2*K4 + 8216*K1**2*K2 - 352*K1**2*K3**2 - 4440*K1**2 - 384*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 4968*K1*K2*K3 + 632*K1*K3*K4 + 16*K1*K4*K5 - 312*K2**4 - 208*K2**2*K3**2 - 48*K2**2*K4**2 + 616*K2**2*K4 - 3876*K2**2 + 208*K2*K3*K5 + 32*K2*K4*K6 - 1620*K3**2 - 330*K4**2 - 44*K5**2 - 4*K6**2 + 3920
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1646']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3560', 'vk6.3579', 'vk6.3597', 'vk6.3806', 'vk6.3820', 'vk6.3837', 'vk6.3851', 'vk6.6968', 'vk6.6986', 'vk6.6999', 'vk6.7017', 'vk6.7190', 'vk6.7204', 'vk6.7221', 'vk6.15340', 'vk6.15352', 'vk6.15467', 'vk6.15479', 'vk6.33977', 'vk6.34023', 'vk6.34043', 'vk6.34434', 'vk6.48219', 'vk6.48237', 'vk6.48380', 'vk6.49953', 'vk6.49973', 'vk6.49991', 'vk6.53989', 'vk6.54001', 'vk6.54045', 'vk6.54489']
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 O1O2O3U1O4U3U4U5O6O5U2U6
R3 orbit {'O1O2O3U1O4U3U4U5O6O5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O5O4U5U6U1O6U3
Gauss code of K* O1O2O3U4U3O5O4U6U5U1O6U2
Gauss code of -K* O1O2O3U2O4U3U5U4O6O5U1U6
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 0 1 1 0],[ 2 0 2 1 1 2 1],[ 0 -2 0 -1 1 0 0],[ 0 -1 1 0 1 0 0],[-1 -1 -1 -1 0 -1 0],[-1 -2 0 0 1 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 1 0 0 0 -2],[-1 -1 0 0 -1 -1 -1],[ 0 0 0 0 0 0 -1],[ 0 0 1 0 0 1 -1],[ 0 0 1 0 -1 0 -2],[ 2 2 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,-1,0,0,0,2,0,1,1,1,0,0,1,-1,1,2]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,1,2,0,1,1,0,0,1,1,1,0,-1]
Phi of -K [-2,0,0,0,1,1,0,1,1,1,2,0,1,1,0,0,1,1,1,0,-1]
Phi of K* [-1,-1,0,0,0,2,-1,0,0,1,2,1,1,1,1,-1,0,0,0,1,1]
Phi of -K* [-2,0,0,0,1,1,1,1,2,1,2,0,0,0,0,1,1,0,1,0,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+15t^4+10t^2+1
Outer characteristic polynomial t^7+21t^5+27t^3+5t
Flat arrow polynomial -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -192*K1**4*K2**2 + 1056*K1**4*K2 - 3968*K1**4 + 288*K1**3*K2*K3 - 864*K1**3*K3 + 128*K1**2*K2**3 - 3152*K1**2*K2**2 - 224*K1**2*K2*K4 + 8216*K1**2*K2 - 352*K1**2*K3**2 - 4440*K1**2 - 384*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 4968*K1*K2*K3 + 632*K1*K3*K4 + 16*K1*K4*K5 - 312*K2**4 - 208*K2**2*K3**2 - 48*K2**2*K4**2 + 616*K2**2*K4 - 3876*K2**2 + 208*K2*K3*K5 + 32*K2*K4*K6 - 1620*K3**2 - 330*K4**2 - 44*K5**2 - 4*K6**2 + 3920
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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