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Flat knot 6.1298

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,2,1,1,2,3,1,0,2,0,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1298']
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+45t^5+117t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1298']
2-strand cable arrow polynomial of the knot is: 1504*K1**4*K2 - 3552*K1**4 + 928*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 1120*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6800*K1**2*K2**2 - 960*K1**2*K2*K4 + 7832*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K4**2 - 3544*K1**2 + 576*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 352*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 7256*K1*K2*K3 + 1080*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1352*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 16*K2**2*K4**2 + 1888*K2**2*K4 - 3622*K2**2 + 496*K2*K3*K5 + 16*K2*K4*K6 - 1828*K3**2 - 558*K4**2 - 76*K5**2 - 2*K6**2 + 3644
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1298']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16803', 'vk6.16810', 'vk6.16858', 'vk6.16867', 'vk6.18172', 'vk6.18184', 'vk6.18507', 'vk6.18519', 'vk6.23239', 'vk6.23246', 'vk6.24628', 'vk6.25048', 'vk6.25068', 'vk6.35235', 'vk6.35260', 'vk6.36768', 'vk6.37200', 'vk6.37222', 'vk6.42750', 'vk6.42761', 'vk6.44348', 'vk6.44360', 'vk6.54996', 'vk6.55029', 'vk6.55977', 'vk6.55981', 'vk6.59394', 'vk6.59408', 'vk6.60512', 'vk6.65645', 'vk6.68184', 'vk6.68191']
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 O1O2O3U2O4O5U6U3O6U1U5U4
R3 orbit {'O1O2O3U2O4O5U6U3O6U1U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U3O6U1U6O5O4U2
Gauss code of K* O1O2O3U1U4U5O4U3U2O6O5U6
Gauss code of -K* O1O2O3U4O5O4U2U1O6U5U6U3
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 0 2 2 -1],[ 2 0 -1 2 3 2 1],[ 1 1 0 1 1 1 0],[ 0 -2 -1 0 1 0 0],[-2 -3 -1 -1 0 0 -2],[-2 -2 -1 0 0 0 -2],[ 1 -1 0 0 2 2 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 0 0 -1 -2 -2],[-2 0 0 -1 -1 -2 -3],[ 0 0 1 0 -1 0 -2],[ 1 1 1 1 0 0 1],[ 1 2 2 0 0 0 -1],[ 2 2 3 2 -1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,0,0,1,2,2,1,1,2,3,1,0,2,0,-1,1]
Phi over symmetry [-2,-2,0,1,1,2,0,0,1,2,2,1,1,2,3,1,0,2,0,-1,1]
Phi of -K [-2,-1,-1,0,2,2,0,2,0,1,2,0,1,1,1,0,2,2,1,2,0]
Phi of K* [-2,-2,0,1,1,2,0,1,1,2,1,2,1,2,2,1,0,0,0,0,2]
Phi of -K* [-2,-1,-1,0,2,2,-1,1,2,2,3,0,1,1,1,0,2,2,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+31t^4+66t^2+1
Outer characteristic polynomial t^7+45t^5+117t^3+8t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 1504*K1**4*K2 - 3552*K1**4 + 928*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 1120*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6800*K1**2*K2**2 - 960*K1**2*K2*K4 + 7832*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K4**2 - 3544*K1**2 + 576*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 352*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 7256*K1*K2*K3 + 1080*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1352*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 16*K2**2*K4**2 + 1888*K2**2*K4 - 3622*K2**2 + 496*K2*K3*K5 + 16*K2*K4*K6 - 1828*K3**2 - 558*K4**2 - 76*K5**2 - 2*K6**2 + 3644
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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