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Flat knot 6.1020

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,2,4,1,0,0,1,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1020']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.240', '6.577', '6.625', '6.1020']
Outer characteristic polynomial of the knot is: t^7+42t^5+40t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.611', '6.1017', '6.1020']
2-strand cable arrow polynomial of the knot is: -1360*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 576*K1**3*K3 + 96*K1**2*K2**2*K4 - 1616*K1**2*K2**2 - 608*K1**2*K2*K4 + 4656*K1**2*K2 - 544*K1**2*K3**2 - 176*K1**2*K4**2 - 3748*K1**2 - 640*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4272*K1*K2*K3 + 1704*K1*K3*K4 + 192*K1*K4*K5 + 8*K1*K5*K6 - 40*K2**4 - 128*K2**2*K3**2 - 56*K2**2*K4**2 + 920*K2**2*K4 - 3198*K2**2 + 360*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1872*K3**2 - 870*K4**2 - 148*K5**2 - 10*K6**2 + 3212
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1020']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4864', 'vk6.5207', 'vk6.6442', 'vk6.6861', 'vk6.8407', 'vk6.8826', 'vk6.9763', 'vk6.10054', 'vk6.11686', 'vk6.12039', 'vk6.13032', 'vk6.20500', 'vk6.20757', 'vk6.21867', 'vk6.27912', 'vk6.29408', 'vk6.29722', 'vk6.32675', 'vk6.33018', 'vk6.39341', 'vk6.39797', 'vk6.46357', 'vk6.47611', 'vk6.47934', 'vk6.48830', 'vk6.49099', 'vk6.51353', 'vk6.51564', 'vk6.53285', 'vk6.57361', 'vk6.64346', 'vk6.66918']
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 O1O2O3O4U3U2O5U1O6U5U6U4
R3 orbit {'O1O2O3O4U3U2O5U1O6U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U6O5U4O6U3U2
Gauss code of K* O1O2O3U4U5U6U3O6O5U1O4U2
Gauss code of -K* O1O2O3U2O4U3O5O6U1U6U5U4
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 3 0 1],[ 2 0 0 0 4 1 1],[ 1 0 0 0 2 0 0],[ 1 0 0 0 1 0 0],[-3 -4 -2 -1 0 -1 1],[ 0 -1 0 0 1 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -1 -1 -2 -4],[-1 -1 0 -1 0 0 -1],[ 0 1 1 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 2 0 0 0 0 0],[ 2 4 1 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,-1,1,1,2,4,1,0,0,1,0,0,1,0,0,0]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,2,4,1,0,0,1,0,0,1,0,0,0]
Phi of -K [-2,-1,-1,0,1,3,1,1,1,2,1,0,1,2,2,1,2,3,0,2,3]
Phi of K* [-3,-1,0,1,1,2,3,2,2,3,1,0,2,2,2,1,1,1,0,1,1]
Phi of -K* [-2,-1,-1,0,1,3,0,0,1,1,4,0,0,0,1,0,0,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+26t^4+19t^2
Outer characteristic polynomial t^7+42t^5+40t^3+4t
Flat arrow polynomial -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial -1360*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 576*K1**3*K3 + 96*K1**2*K2**2*K4 - 1616*K1**2*K2**2 - 608*K1**2*K2*K4 + 4656*K1**2*K2 - 544*K1**2*K3**2 - 176*K1**2*K4**2 - 3748*K1**2 - 640*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4272*K1*K2*K3 + 1704*K1*K3*K4 + 192*K1*K4*K5 + 8*K1*K5*K6 - 40*K2**4 - 128*K2**2*K3**2 - 56*K2**2*K4**2 + 920*K2**2*K4 - 3198*K2**2 + 360*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1872*K3**2 - 870*K4**2 - 148*K5**2 - 10*K6**2 + 3212
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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