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Flat knot 6.1859

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,0,1,2,1,1,0,0,1,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1859', '7.41754']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^7+38t^5+117t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1859', '7.41754']
2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 1536*K1**4 + 768*K1**3*K2*K3 - 128*K1**2*K2**4 + 288*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2832*K1**2*K2**2 - 160*K1**2*K2*K4 + 2248*K1**2*K2 - 768*K1**2*K3**2 - 32*K1**2*K4**2 - 208*K1**2 + 320*K1*K2**3*K3 - 864*K1*K2**2*K3 - 160*K1*K2**2*K5 + 2112*K1*K2*K3 + 632*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 16*K2**2*K4**2 + 496*K2**2*K4 - 558*K2**2 + 264*K2*K3*K5 + 16*K2*K4*K6 - 396*K3**2 - 126*K4**2 - 36*K5**2 - 2*K6**2 + 708
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1859']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.514', 'vk6.607', 'vk6.639', 'vk6.1016', 'vk6.1111', 'vk6.1154', 'vk6.1874', 'vk6.2295', 'vk6.2515', 'vk6.2561', 'vk6.2594', 'vk6.2799', 'vk6.2896', 'vk6.2919', 'vk6.3082', 'vk6.3204', 'vk6.4608', 'vk6.5897', 'vk6.6026', 'vk6.6537', 'vk6.8071', 'vk6.9384', 'vk6.17844', 'vk6.17861', 'vk6.19061', 'vk6.19880', 'vk6.22553', 'vk6.24361', 'vk6.25677', 'vk6.26324', 'vk6.26769', 'vk6.28574', 'vk6.29806', 'vk6.39905', 'vk6.43786', 'vk6.45069', 'vk6.46847', 'vk6.48012', 'vk6.48086', 'vk6.50660']
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 O1O2O3U4U5O4U6O5O6U1U3U2
R3 orbit {'O1O2O3U4U5U6O4O6O5U3U1U2', 'O1O2O3U4U5O4U6O5O6U1U3U2'}
R3 orbit length 2
Gauss code of -K O1O2O3U2U1U3O4O5U4O6U5U6
Gauss code of K* O1O2O3U1U3U2O4O5U4O6U5U6
Gauss code of -K* O1O2O3U4U5O4U6O5O6U2U1U3
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 1 1 -1 0 1],[ 2 0 2 1 1 2 3],[-1 -2 0 0 -2 -1 0],[-1 -1 0 0 -2 -1 0],[ 1 -1 2 2 0 0 1],[ 0 -2 1 1 0 0 0],[-1 -3 0 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -1 -3],[-1 0 0 0 -1 -2 -1],[-1 0 0 0 -1 -2 -2],[ 0 0 1 1 0 0 -2],[ 1 1 2 2 0 0 -1],[ 2 3 1 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,0,0,0,1,3,0,1,2,1,1,2,2,0,2,1]
Phi over symmetry [-2,-1,0,1,1,1,0,0,0,1,2,1,1,0,0,1,0,0,0,0,0]
Phi of -K [-2,-1,0,1,1,1,0,0,0,1,2,1,1,0,0,1,0,0,0,0,0]
Phi of K* [-1,-1,-1,0,1,2,0,0,0,0,1,0,0,0,2,1,1,0,1,0,0]
Phi of -K* [-2,-1,0,1,1,1,1,2,1,2,3,0,2,2,1,1,1,0,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+30t^4+88t^2+1
Outer characteristic polynomial t^7+38t^5+117t^3+4t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 768*K1**4*K2 - 1536*K1**4 + 768*K1**3*K2*K3 - 128*K1**2*K2**4 + 288*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2832*K1**2*K2**2 - 160*K1**2*K2*K4 + 2248*K1**2*K2 - 768*K1**2*K3**2 - 32*K1**2*K4**2 - 208*K1**2 + 320*K1*K2**3*K3 - 864*K1*K2**2*K3 - 160*K1*K2**2*K5 + 2112*K1*K2*K3 + 632*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 16*K2**2*K4**2 + 496*K2**2*K4 - 558*K2**2 + 264*K2*K3*K5 + 16*K2*K4*K6 - 396*K3**2 - 126*K4**2 - 36*K5**2 - 2*K6**2 + 708
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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