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Flat knot 6.1358

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,2,2,3,0,1,1,1,1,2,2,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1358']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370']
Outer characteristic polynomial of the knot is: t^7+54t^5+62t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1358']
2-strand cable arrow polynomial of the knot is: 2240*K1**4*K2 - 5376*K1**4 + 1024*K1**3*K2*K3 - 1344*K1**3*K3 - 128*K1**2*K2**4 + 896*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 5760*K1**2*K2**2 - 768*K1**2*K2*K4 + 8784*K1**2*K2 - 1280*K1**2*K3**2 - 96*K1**2*K4**2 - 3256*K1**2 + 192*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6704*K1*K2*K3 + 1728*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 720*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 1264*K2**2*K4 - 3592*K2**2 + 128*K2*K3*K5 - 1896*K3**2 - 628*K4**2 - 48*K5**2 + 3682
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1358']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20153', 'vk6.20159', 'vk6.21441', 'vk6.21445', 'vk6.27277', 'vk6.27283', 'vk6.28937', 'vk6.28943', 'vk6.38698', 'vk6.38708', 'vk6.40886', 'vk6.47273', 'vk6.47277', 'vk6.56978', 'vk6.56995', 'vk6.58130', 'vk6.62681', 'vk6.67467', 'vk6.70040', 'vk6.70047']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U2O4O5U1U6U3O6U5U4
R3 orbit {'O1O2O3U2O4O5U1U6U3O6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U1U6U3O5O4U2
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U5O6U1U6U3O5O4U2
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -1 1 2 2 -1],[ 3 0 0 2 3 2 2],[ 1 0 0 1 1 1 0],[-1 -2 -1 0 1 0 -1],[-2 -3 -1 -1 0 0 -2],[-2 -2 -1 0 0 0 -2],[ 1 -2 0 1 2 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 0 -1 -2 -2],[-2 0 0 -1 -1 -2 -3],[-1 0 1 0 -1 -1 -2],[ 1 1 1 1 0 0 0],[ 1 2 2 1 0 0 -2],[ 3 2 3 2 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,0,1,2,2,1,1,2,3,1,1,2,0,0,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,2,2,3,0,1,1,1,1,2,2,0,1,0]
Phi of -K [-3,-1,-1,1,2,2,0,2,2,2,3,0,1,1,1,1,2,2,0,1,0]
Phi of K* [-2,-2,-1,1,1,3,0,0,1,2,2,1,1,2,3,1,1,2,0,0,2]
Phi of -K* [-3,-1,-1,1,2,2,0,2,2,2,3,0,1,1,1,1,2,2,0,1,0]
Symmetry type of based matrix +
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+34t^4+34t^2+1
Outer characteristic polynomial t^7+54t^5+62t^3+5t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial 2240*K1**4*K2 - 5376*K1**4 + 1024*K1**3*K2*K3 - 1344*K1**3*K3 - 128*K1**2*K2**4 + 896*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 5760*K1**2*K2**2 - 768*K1**2*K2*K4 + 8784*K1**2*K2 - 1280*K1**2*K3**2 - 96*K1**2*K4**2 - 3256*K1**2 + 192*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6704*K1*K2*K3 + 1728*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 720*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 1264*K2**2*K4 - 3592*K2**2 + 128*K2*K3*K5 - 1896*K3**2 - 628*K4**2 - 48*K5**2 + 3682
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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