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Flat knot 6.1847

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,1,1,1,1,0,1,0,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1847']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^7+39t^5+105t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1847']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 1664*K1**4*K2 - 6720*K1**4 + 704*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1408*K1**3*K3 - 4384*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 12264*K1**2*K2 - 1472*K1**2*K3**2 - 32*K1**2*K3*K5 - 192*K1**2*K4**2 - 6524*K1**2 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 8104*K1*K2*K3 + 2272*K1*K3*K4 + 256*K1*K4*K5 - 176*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 968*K2**2*K4 - 6156*K2**2 + 224*K2*K3*K5 + 32*K2*K4*K6 - 3000*K3**2 - 980*K4**2 - 140*K5**2 - 12*K6**2 + 6338
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1847']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3666', 'vk6.3761', 'vk6.3952', 'vk6.4047', 'vk6.4475', 'vk6.4572', 'vk6.5861', 'vk6.5990', 'vk6.7157', 'vk6.7332', 'vk6.7423', 'vk6.7906', 'vk6.8027', 'vk6.9340', 'vk6.17911', 'vk6.18006', 'vk6.18755', 'vk6.24446', 'vk6.24880', 'vk6.25341', 'vk6.37502', 'vk6.43881', 'vk6.44235', 'vk6.44538', 'vk6.48290', 'vk6.48353', 'vk6.50079', 'vk6.50189', 'vk6.50567', 'vk6.50632', 'vk6.55854', 'vk6.60735']
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 O1O2O3U2U4O5U6O4O6U1U5U3
R3 orbit {'O1O2O3U2U4O5U6O4O6U1U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U3O5O6U5O4U6U2
Gauss code of K* O1O2O3U1U4U3O4O5U2O6U5U6
Gauss code of -K* O1O2O3U4U5O4U2O5O6U1U6U3
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 2 0 0 1],[ 2 0 0 3 1 0 3],[ 1 0 0 1 1 0 1],[-2 -3 -1 0 -2 -1 -1],[ 0 -1 -1 2 0 1 0],[ 0 0 0 1 -1 0 0],[-1 -3 -1 1 0 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -1 -3],[-1 1 0 0 0 -1 -3],[ 0 1 0 0 -1 0 0],[ 0 2 0 1 0 -1 -1],[ 1 1 1 0 1 0 0],[ 2 3 3 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,2,1,3,0,0,1,3,1,0,0,1,1,0]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,2,1,1,1,1,0,1,0,1,1,2,1]
Phi of -K [-2,-1,0,0,1,2,1,1,2,0,1,0,1,1,2,-1,1,0,1,1,0]
Phi of K* [-2,-1,0,0,1,2,0,0,1,2,1,1,1,1,0,1,0,1,1,2,1]
Phi of -K* [-2,-1,0,0,1,2,0,0,1,3,3,0,1,1,1,-1,0,1,0,2,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+29t^4+75t^2+4
Outer characteristic polynomial t^7+39t^5+105t^3+8t
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -128*K1**6 + 1664*K1**4*K2 - 6720*K1**4 + 704*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1408*K1**3*K3 - 4384*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 12264*K1**2*K2 - 1472*K1**2*K3**2 - 32*K1**2*K3*K5 - 192*K1**2*K4**2 - 6524*K1**2 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 8104*K1*K2*K3 + 2272*K1*K3*K4 + 256*K1*K4*K5 - 176*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 968*K2**2*K4 - 6156*K2**2 + 224*K2*K3*K5 + 32*K2*K4*K6 - 3000*K3**2 - 980*K4**2 - 140*K5**2 - 12*K6**2 + 6338
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice False
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