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Flat knot 6.242

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,1,4,0,2,1,3,5,0,0,0,1,0,1,3,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.242']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.77', '6.242', '6.331', '6.862']
Outer characteristic polynomial of the knot is: t^7+90t^5+91t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.242']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 512*K1**4*K2**2 + 960*K1**4*K2 - 2112*K1**4 + 960*K1**3*K2*K3 - 384*K1**3*K3 - 768*K1**2*K2**4 + 960*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 5248*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 6416*K1**2*K2 - 1024*K1**2*K3**2 - 64*K1**2*K3*K5 - 64*K1**2*K4**2 - 3576*K1**2 + 2112*K1*K2**3*K3 + 576*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 - 384*K1*K2**2*K5 + 384*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 5808*K1*K2*K3 + 1200*K1*K3*K4 + 48*K1*K4*K5 - 192*K2**6 - 384*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 1392*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 + 64*K2**2*K3**2*K6 - 1344*K2**2*K3**2 - 272*K2**2*K4**2 + 1392*K2**2*K4 - 32*K2**2*K5**2 - 16*K2**2*K6**2 - 2464*K2**2 - 64*K2*K3**2*K4 + 528*K2*K3*K5 + 64*K2*K4*K6 - 64*K3**4 + 32*K3**2*K6 - 1496*K3**2 - 400*K4**2 - 32*K5**2 - 8*K6**2 + 3094
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.242']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14089', 'vk6.14282', 'vk6.15511', 'vk6.16011', 'vk6.16270', 'vk6.16282', 'vk6.22585', 'vk6.34045', 'vk6.34086', 'vk6.34485', 'vk6.34541', 'vk6.34572', 'vk6.42267', 'vk6.54064', 'vk6.54287', 'vk6.54518', 'vk6.54563', 'vk6.59017', 'vk6.64516', 'vk6.64624']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4O5U4O6U1U6U2U3U5
R3 orbit {'O1O2O3O4O5U4O6U1U6U2U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U3U4U6U5O6U2
Gauss code of K* O1O2O3O4O5U1U3U4U6U5O6U2
Gauss code of -K* Same
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 -1 1 -1 4 1],[ 4 0 2 3 0 5 1],[ 1 -2 0 1 0 3 0],[-1 -3 -1 0 0 2 0],[ 1 0 0 0 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 1 1 -1 -1 -4],[-4 0 0 -2 -1 -3 -5],[-1 0 0 0 0 0 -1],[-1 2 0 0 0 -1 -3],[ 1 1 0 0 0 0 0],[ 1 3 0 1 0 0 -2],[ 4 5 1 3 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,1,1,4,0,2,1,3,5,0,0,0,1,0,1,3,0,0,2]
Phi over symmetry [-4,-1,-1,1,1,4,0,2,1,3,5,0,0,0,1,0,1,3,0,0,2]
Phi of -K [-4,-1,-1,1,1,4,1,3,2,4,3,0,1,2,2,2,2,4,0,1,3]
Phi of K* [-4,-1,-1,1,1,4,1,3,2,4,3,0,1,2,2,2,2,4,0,1,3]
Phi of -K* [-4,-1,-1,1,1,4,0,2,1,3,5,0,0,0,1,0,1,3,0,0,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+54t^4+29t^2+1
Outer characteristic polynomial t^7+90t^5+91t^3+5t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**6 - 512*K1**4*K2**2 + 960*K1**4*K2 - 2112*K1**4 + 960*K1**3*K2*K3 - 384*K1**3*K3 - 768*K1**2*K2**4 + 960*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 5248*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 6416*K1**2*K2 - 1024*K1**2*K3**2 - 64*K1**2*K3*K5 - 64*K1**2*K4**2 - 3576*K1**2 + 2112*K1*K2**3*K3 + 576*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 - 384*K1*K2**2*K5 + 384*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 5808*K1*K2*K3 + 1200*K1*K3*K4 + 48*K1*K4*K5 - 192*K2**6 - 384*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 1392*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 + 64*K2**2*K3**2*K6 - 1344*K2**2*K3**2 - 272*K2**2*K4**2 + 1392*K2**2*K4 - 32*K2**2*K5**2 - 16*K2**2*K6**2 - 2464*K2**2 - 64*K2*K3**2*K4 + 528*K2*K3*K5 + 64*K2*K4*K6 - 64*K3**4 + 32*K3**2*K6 - 1496*K3**2 - 400*K4**2 - 32*K5**2 - 8*K6**2 + 3094
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}]]
If K is slice True
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