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Flat knot 6.221

Min(phi) over symmetries of the knot is: [-4,-2,-2,2,3,3,0,1,2,4,5,0,0,1,2,1,3,4,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.221']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**2*K2 + 6*K1**2 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.221', '6.257']
Outer characteristic polynomial of the knot is: t^7+123t^5+248t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.221']
2-strand cable arrow polynomial of the knot is: -256*K1**2*K2**4 + 896*K1**2*K2**3 - 3456*K1**2*K2**2 - 224*K1**2*K2*K4 + 3440*K1**2*K2 - 256*K1**2*K3**2 - 2728*K1**2 + 1440*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 - 352*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4424*K1*K2*K3 + 696*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1088*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 1120*K2**4*K4 - 3016*K2**4 + 160*K2**3*K3*K5 - 128*K2**3*K6 - 1600*K2**2*K3**2 - 336*K2**2*K4**2 + 2800*K2**2*K4 - 64*K2**2*K5**2 - 1120*K2**2 + 928*K2*K3*K5 + 72*K2*K4*K6 + 8*K2*K5*K7 - 1224*K3**2 - 586*K4**2 - 128*K5**2 + 2312
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.221']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73192', 'vk6.73208', 'vk6.73632', 'vk6.74308', 'vk6.74402', 'vk6.74950', 'vk6.75011', 'vk6.75106', 'vk6.75125', 'vk6.75568', 'vk6.75594', 'vk6.76518', 'vk6.76580', 'vk6.76929', 'vk6.78044', 'vk6.78064', 'vk6.78532', 'vk6.78561', 'vk6.79360', 'vk6.79782', 'vk6.79853', 'vk6.79964', 'vk6.80818', 'vk6.80881', 'vk6.83693', 'vk6.84699', 'vk6.84809', 'vk6.85264', 'vk6.85638', 'vk6.87709', 'vk6.88386', 'vk6.89496']
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 O1O2O3O4O5U3O6U1U2U6U5U4
R3 orbit {'O1O2O3O4O5U3O6U1U2U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U1U6U4U5O6U3
Gauss code of K* O1O2O3O4O5U1U2U6U5U4O6U3
Gauss code of -K* O1O2O3O4O5U3O6U2U1U6U4U5
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 -4 -2 -2 3 3 2],[ 4 0 1 0 5 4 2],[ 2 -1 0 0 4 3 1],[ 2 0 0 0 2 1 0],[-3 -5 -4 -2 0 0 0],[-3 -4 -3 -1 0 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 3 2 -2 -2 -4],[-3 0 0 0 -1 -3 -4],[-3 0 0 0 -2 -4 -5],[-2 0 0 0 0 -1 -2],[ 2 1 2 0 0 0 0],[ 2 3 4 1 0 0 -1],[ 4 4 5 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-2,2,2,4,0,0,1,3,4,0,2,4,5,0,1,2,0,0,1]
Phi over symmetry [-4,-2,-2,2,3,3,0,1,2,4,5,0,0,1,2,1,3,4,0,0,0]
Phi of -K [-4,-2,-2,2,3,3,1,2,4,2,3,0,3,1,2,4,3,4,1,1,0]
Phi of K* [-3,-3,-2,2,2,4,0,1,1,3,2,1,2,4,3,3,4,4,0,1,2]
Phi of -K* [-4,-2,-2,2,3,3,0,1,2,4,5,0,0,1,2,1,3,4,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial -4w^4z^2+10w^3z^2-4w^3z+23w^2z+15w
Inner characteristic polynomial t^6+77t^4+57t^2+1
Outer characteristic polynomial t^7+123t^5+248t^3+12t
Flat arrow polynomial -8*K1**4 + 4*K1**2*K2 + 6*K1**2 - K2
2-strand cable arrow polynomial -256*K1**2*K2**4 + 896*K1**2*K2**3 - 3456*K1**2*K2**2 - 224*K1**2*K2*K4 + 3440*K1**2*K2 - 256*K1**2*K3**2 - 2728*K1**2 + 1440*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 - 352*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4424*K1*K2*K3 + 696*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1088*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 1120*K2**4*K4 - 3016*K2**4 + 160*K2**3*K3*K5 - 128*K2**3*K6 - 1600*K2**2*K3**2 - 336*K2**2*K4**2 + 2800*K2**2*K4 - 64*K2**2*K5**2 - 1120*K2**2 + 928*K2*K3*K5 + 72*K2*K4*K6 + 8*K2*K5*K7 - 1224*K3**2 - 586*K4**2 - 128*K5**2 + 2312
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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