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Flat knot 6.67

Min(phi) over symmetries of the knot is: [-4,-4,1,2,2,3,0,2,1,4,3,3,2,5,4,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.67']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1*K2 - 3*K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.67', '6.535', '6.1347', '6.1348', '6.1368']
Outer characteristic polynomial of the knot is: t^7+137t^5+270t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.67']
2-strand cable arrow polynomial of the knot is: -256*K1**2*K2**4 + 320*K1**2*K2**3 - 3008*K1**2*K2**2 - 128*K1**2*K2*K4 + 4048*K1**2*K2 - 3432*K1**2 + 896*K1*K2**3*K3 - 1408*K1*K2**2*K3 + 64*K1*K2**2*K5*K6 - 320*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4528*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K5*K6 + 672*K1*K3*K4 + 192*K1*K4*K5 + 32*K1*K5*K6 - 192*K2**6 + 256*K2**4*K4 - 1696*K2**4 + 192*K2**3*K3*K5 - 96*K2**3*K6 - 1024*K2**2*K3**2 - 64*K2**2*K3*K7 - 24*K2**2*K4**2 + 2104*K2**2*K4 - 192*K2**2*K5**2 - 32*K2**2*K6**2 - 2630*K2**2 - 64*K2*K3**2*K4 + 1008*K2*K3*K5 + 136*K2*K4*K6 + 96*K2*K5*K7 + 48*K3**2*K6 - 1576*K3**2 - 668*K4**2 - 304*K5**2 - 50*K6**2 + 2890
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.67']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74179', 'vk6.74183', 'vk6.74785', 'vk6.74788', 'vk6.76330', 'vk6.76850', 'vk6.76856', 'vk6.79209', 'vk6.79677', 'vk6.79684', 'vk6.81055', 'vk6.81059', 'vk6.82913', 'vk6.83429', 'vk6.85249', 'vk6.85298', 'vk6.85302', 'vk6.86567', 'vk6.87538', 'vk6.89241']
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 O1O2O3O4O5O6U2U1U6U4U5U3
R3 orbit {'O1O2O3O4O5O6U2U1U6U4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U2U3U1U6U5
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U4U2U3U1U6U5
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 -4 -4 2 1 3 2],[ 4 0 0 5 3 4 2],[ 4 0 0 4 2 3 1],[-2 -5 -4 0 -1 1 0],[-1 -3 -2 1 0 1 0],[-3 -4 -3 -1 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 2 2 1 -4 -4],[-3 0 0 -1 -1 -3 -4],[-2 0 0 0 0 -1 -2],[-2 1 0 0 -1 -4 -5],[-1 1 0 1 0 -2 -3],[ 4 3 1 4 2 0 0],[ 4 4 2 5 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,-1,4,4,0,1,1,3,4,0,0,1,2,1,4,5,2,3,0]
Phi over symmetry [-4,-4,1,2,2,3,0,2,1,4,3,3,2,5,4,0,1,1,0,0,1]
Phi of -K [-4,-4,1,2,2,3,0,2,1,4,3,3,2,5,4,0,1,1,0,0,1]
Phi of K* [-3,-2,-2,-1,4,4,0,1,1,3,4,0,0,1,2,1,4,5,2,3,0]
Phi of -K* [-4,-4,1,2,2,3,0,2,1,4,3,3,2,5,4,0,1,1,0,0,1]
Symmetry type of based matrix +
u-polynomial 2t^4-t^3-2t^2-t
Normalized Jones-Krushkal polynomial 8z^2+25z+19
Enhanced Jones-Krushkal polynomial 8w^3z^2+25w^2z+19w
Inner characteristic polynomial t^6+87t^4+40t^2
Outer characteristic polynomial t^7+137t^5+270t^3+4t
Flat arrow polynomial 8*K1**3 - 6*K1*K2 - 3*K1 + K3 + 1
2-strand cable arrow polynomial -256*K1**2*K2**4 + 320*K1**2*K2**3 - 3008*K1**2*K2**2 - 128*K1**2*K2*K4 + 4048*K1**2*K2 - 3432*K1**2 + 896*K1*K2**3*K3 - 1408*K1*K2**2*K3 + 64*K1*K2**2*K5*K6 - 320*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4528*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K5*K6 + 672*K1*K3*K4 + 192*K1*K4*K5 + 32*K1*K5*K6 - 192*K2**6 + 256*K2**4*K4 - 1696*K2**4 + 192*K2**3*K3*K5 - 96*K2**3*K6 - 1024*K2**2*K3**2 - 64*K2**2*K3*K7 - 24*K2**2*K4**2 + 2104*K2**2*K4 - 192*K2**2*K5**2 - 32*K2**2*K6**2 - 2630*K2**2 - 64*K2*K3**2*K4 + 1008*K2*K3*K5 + 136*K2*K4*K6 + 96*K2*K5*K7 + 48*K3**2*K6 - 1576*K3**2 - 668*K4**2 - 304*K5**2 - 50*K6**2 + 2890
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice False
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