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Flat knot 6.958

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,0,2,3,4,-1,1,0,1,0,1,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.958']
Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182']
Outer characteristic polynomial of the knot is: t^7+55t^5+163t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.958']
2-strand cable arrow polynomial of the knot is: -48*K1**4 - 1168*K1**2*K2**2 + 904*K1**2*K2 - 96*K1**2*K3**2 - 1712*K1**2 + 448*K1*K2**3*K3 + 3328*K1*K2*K3 + 232*K1*K3*K4 + 72*K1*K4*K5 - 592*K2**4 - 1040*K2**2*K3**2 - 24*K2**2*K4**2 + 232*K2**2*K4 - 1198*K2**2 + 712*K2*K3*K5 + 40*K2*K4*K6 - 48*K3**4 + 32*K3**2*K6 - 1428*K3**2 - 160*K4**2 - 188*K5**2 - 18*K6**2 + 1718
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.958']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72628', 'vk6.72629', 'vk6.72781', 'vk6.72782', 'vk6.73095', 'vk6.73096', 'vk6.73172', 'vk6.73173', 'vk6.73810', 'vk6.73812', 'vk6.73943', 'vk6.73957', 'vk6.73964', 'vk6.73966', 'vk6.75752', 'vk6.75772', 'vk6.75783', 'vk6.75785', 'vk6.77878', 'vk6.77936', 'vk6.77995', 'vk6.78020', 'vk6.78773', 'vk6.78775', 'vk6.80363', 'vk6.80371', 'vk6.80374', 'vk6.80376', 'vk6.81782', 'vk6.87800', 'vk6.89150', 'vk6.89334']
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 O1O2O3O4U5U2O6O5U1U6U4U3
R3 orbit {'O1O2O3O4U5U2O6O5U1U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U1U5U4O6O5U3U6
Gauss code of K* O1O2O3O4U1U5U4U3O6O5U2U6
Gauss code of -K* O1O2O3O4U5U3O6O5U2U1U6U4
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -1 2 2 0 0],[ 3 0 1 4 3 2 0],[ 1 -1 0 1 0 1 -1],[-2 -4 -1 0 0 -1 -1],[-2 -3 0 0 0 -1 -1],[ 0 -2 -1 1 1 0 0],[ 0 0 1 1 1 0 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 0 -1 -1 0 -3],[-2 0 0 -1 -1 -1 -4],[ 0 1 1 0 0 1 0],[ 0 1 1 0 0 -1 -2],[ 1 0 1 -1 1 0 -1],[ 3 3 4 0 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,0,1,1,0,3,1,1,1,4,0,-1,0,1,2,1]
Phi over symmetry [-3,-1,0,0,2,2,1,0,2,3,4,-1,1,0,1,0,1,1,1,1,0]
Phi of -K [-3,-1,0,0,2,2,1,1,3,1,2,0,2,2,3,0,1,1,1,1,0]
Phi of K* [-2,-2,0,0,1,3,0,1,1,2,1,1,1,3,2,0,0,1,2,3,1]
Phi of -K* [-3,-1,0,0,2,2,1,0,2,3,4,-1,1,0,1,0,1,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial 4w^4z-12w^3z+4w^3+13w^2z+7w
Inner characteristic polynomial t^6+37t^4+71t^2
Outer characteristic polynomial t^7+55t^5+163t^3
Flat arrow polynomial -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial -48*K1**4 - 1168*K1**2*K2**2 + 904*K1**2*K2 - 96*K1**2*K3**2 - 1712*K1**2 + 448*K1*K2**3*K3 + 3328*K1*K2*K3 + 232*K1*K3*K4 + 72*K1*K4*K5 - 592*K2**4 - 1040*K2**2*K3**2 - 24*K2**2*K4**2 + 232*K2**2*K4 - 1198*K2**2 + 712*K2*K3*K5 + 40*K2*K4*K6 - 48*K3**4 + 32*K3**2*K6 - 1428*K3**2 - 160*K4**2 - 188*K5**2 - 18*K6**2 + 1718
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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