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Flat knot 6.12

Min(phi) over symmetries of the knot is: [-5,-3,2,2,2,2,1,2,3,4,5,1,2,3,4,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.12']
Arrow polynomial of the knot is: -2*K1*K4 + K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.4', '6.12', '6.15', '6.48']
Outer characteristic polynomial of the knot is: t^7+135t^5+200t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.12']
2-strand cable arrow polynomial of the knot is: 160*K1*K2*K3 - 2*K10**2 + 8*K10*K2*K8 - 128*K2**4 + 256*K2**3*K3*K5 - 448*K2**2*K3**2 - 320*K2**2*K5**2 - 8*K2**2*K8**2 - 196*K2**2 + 896*K2*K3*K5 + 96*K2*K5*K7 + 8*K2*K6*K8 - 328*K3**2 + 16*K3*K5*K8 - 328*K5**2 - 2*K6**2 - 12*K8**2 + 330
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.12']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74007', 'vk6.74009', 'vk6.74533', 'vk6.74536', 'vk6.76008', 'vk6.76750', 'vk6.76754', 'vk6.78983', 'vk6.80975', 'vk6.80976', 'vk6.83032', 'vk6.83712', 'vk6.83965', 'vk6.85088', 'vk6.85096', 'vk6.86643', 'vk6.87487', 'vk6.87836', 'vk6.89161', 'vk6.89829']
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 O1O2O3O4O5O6U1U2U6U5U4U3
R3 orbit {'O1O2O3O4O5O6U1U2U6U5U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U3U2U1U5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U4U3U2U1U5U6
Diagrammatic symmetry type +
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 -3 2 2 2 2],[ 5 0 1 5 4 3 2],[ 3 -1 0 4 3 2 1],[-2 -5 -4 0 0 0 0],[-2 -4 -3 0 0 0 0],[-2 -3 -2 0 0 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 2 2 -3 -5],[-2 0 0 0 0 -1 -2],[-2 0 0 0 0 -2 -3],[-2 0 0 0 0 -3 -4],[-2 0 0 0 0 -4 -5],[ 3 1 2 3 4 0 -1],[ 5 2 3 4 5 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,-2,3,5,0,0,0,1,2,0,0,2,3,0,3,4,4,5,1]
Phi over symmetry [-5,-3,2,2,2,2,1,2,3,4,5,1,2,3,4,0,0,0,0,0,0]
Phi of -K [-5,-3,2,2,2,2,1,2,3,4,5,1,2,3,4,0,0,0,0,0,0]
Phi of K* [-2,-2,-2,-2,3,5,0,0,0,1,2,0,0,2,3,0,3,4,4,5,1]
Phi of -K* [-5,-3,2,2,2,2,1,2,3,4,5,1,2,3,4,0,0,0,0,0,0]
Symmetry type of based matrix +
u-polynomial t^5+t^3-4t^2
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial -16w^5z+16w^4z+w^2z+3w
Inner characteristic polynomial t^6+85t^4+20t^2
Outer characteristic polynomial t^7+135t^5+200t^3
Flat arrow polynomial -2*K1*K4 + K3 + K5 + 1
2-strand cable arrow polynomial 160*K1*K2*K3 - 2*K10**2 + 8*K10*K2*K8 - 128*K2**4 + 256*K2**3*K3*K5 - 448*K2**2*K3**2 - 320*K2**2*K5**2 - 8*K2**2*K8**2 - 196*K2**2 + 896*K2*K3*K5 + 96*K2*K5*K7 + 8*K2*K6*K8 - 328*K3**2 + 16*K3*K5*K8 - 328*K5**2 - 2*K6**2 - 12*K8**2 + 330
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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