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Flat knot 6.150

Min(phi) over symmetries of the knot is: [-4,-3,1,2,2,2,0,3,2,3,4,3,1,2,3,0,0,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.150']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 + K1 - 2*K2**2 - K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.150', '6.343', '6.489']
Outer characteristic polynomial of the knot is: t^7+107t^5+131t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.150']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 96*K1**3*K3*K4 + 96*K1**2*K2**2*K4 - 1472*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 704*K1**2*K2*K4 + 4056*K1**2*K2 - 672*K1**2*K3**2 - 64*K1**2*K3*K5 - 384*K1**2*K4**2 - 5120*K1**2 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3*K4**2 - 736*K1*K2*K3*K4 + 4808*K1*K2*K3 - 64*K1*K2*K4*K5 + 2720*K1*K3*K4 + 744*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 - 32*K2**2*K3**2 + 32*K2**2*K4**3 - 440*K2**2*K4**2 + 2264*K2**2*K4 - 4322*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 536*K2*K3*K5 + 304*K2*K4*K6 - 32*K3**2*K4**2 + 8*K3**2*K6 - 2440*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1774*K4**2 - 336*K5**2 - 38*K6**2 + 4372
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.150']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71576', 'vk6.71692', 'vk6.72109', 'vk6.72313', 'vk6.73487', 'vk6.74113', 'vk6.74137', 'vk6.74682', 'vk6.74708', 'vk6.75244', 'vk6.75493', 'vk6.76147', 'vk6.76184', 'vk6.77196', 'vk6.77298', 'vk6.77500', 'vk6.77654', 'vk6.78450', 'vk6.79115', 'vk6.79140', 'vk6.80036', 'vk6.80184', 'vk6.80619', 'vk6.80638', 'vk6.83730', 'vk6.83853', 'vk6.85066', 'vk6.85325', 'vk6.86666', 'vk6.86970', 'vk6.87422', 'vk6.89528']
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 O1O2O3O4O5U1O6U2U5U6U4U3
R3 orbit {'O1O2O3O4O5U1O6U2U5U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U2U6U1U4O6U5
Gauss code of K* O1O2O3O4O5U6U1U5U4U2O6U3
Gauss code of -K* O1O2O3O4O5U3O6U4U2U1U5U6
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 -3 2 2 1 2],[ 4 0 1 4 3 2 2],[ 3 -1 0 4 3 1 2],[-2 -4 -4 0 0 -1 1],[-2 -3 -3 0 0 -1 1],[-1 -2 -1 1 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 2 1 -3 -4],[-2 0 1 0 -1 -3 -3],[-2 -1 0 -1 -1 -2 -2],[-2 0 1 0 -1 -4 -4],[-1 1 1 1 0 -1 -2],[ 3 3 2 4 1 0 -1],[ 4 3 2 4 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,-1,3,4,-1,0,1,3,3,1,1,2,2,1,4,4,1,2,1]
Phi over symmetry [-4,-3,1,2,2,2,0,3,2,3,4,3,1,2,3,0,0,0,0,-1,-1]
Phi of -K [-4,-3,1,2,2,2,0,3,2,3,4,3,1,2,3,0,0,0,0,-1,-1]
Phi of K* [-2,-2,-2,-1,3,4,-1,-1,0,3,4,0,0,1,2,0,2,3,3,3,0]
Phi of -K* [-4,-3,1,2,2,2,1,2,2,3,4,1,2,3,4,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^4+t^3-3t^2-t
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2-2w^3z+28w^2z+25w
Inner characteristic polynomial t^6+69t^4+22t^2+1
Outer characteristic polynomial t^7+107t^5+131t^3+10t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 + K1 - 2*K2**2 - K2 + K3 + 2
2-strand cable arrow polynomial -64*K1**4 + 96*K1**3*K3*K4 + 96*K1**2*K2**2*K4 - 1472*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 704*K1**2*K2*K4 + 4056*K1**2*K2 - 672*K1**2*K3**2 - 64*K1**2*K3*K5 - 384*K1**2*K4**2 - 5120*K1**2 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3*K4**2 - 736*K1*K2*K3*K4 + 4808*K1*K2*K3 - 64*K1*K2*K4*K5 + 2720*K1*K3*K4 + 744*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 - 32*K2**2*K3**2 + 32*K2**2*K4**3 - 440*K2**2*K4**2 + 2264*K2**2*K4 - 4322*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 536*K2*K3*K5 + 304*K2*K4*K6 - 32*K3**2*K4**2 + 8*K3**2*K6 - 2440*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1774*K4**2 - 336*K5**2 - 38*K6**2 + 4372
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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