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Flat knot 6.79

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,2,4,0,2,3,2,5,1,1,1,2,1,1,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.79']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.79']
Outer characteristic polynomial of the knot is: t^7+119t^5+50t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.79']
2-strand cable arrow polynomial of the knot is: -1568*K1**4 + 1728*K1**3*K2*K3 - 576*K1**3*K3 - 256*K1**2*K2**4 + 1344*K1**2*K2**3 - 1024*K1**2*K2**2*K3**2 - 11584*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1728*K1**2*K2*K4 + 11488*K1**2*K2 - 992*K1**2*K3**2 - 96*K1**2*K4**2 - 7296*K1**2 + 256*K1*K2**3*K3**3 + 5312*K1*K2**3*K3 + 704*K1*K2**2*K3*K4 - 3072*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 1472*K1*K2**2*K5 + 384*K1*K2*K3**3 - 384*K1*K2*K3*K4 + 12240*K1*K2*K3 - 192*K1*K2*K4*K5 + 1648*K1*K3*K4 + 176*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 768*K2**4*K3**2 - 64*K2**4*K4**2 + 384*K2**4*K4 - 5104*K2**4 + 576*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 - 3200*K2**2*K3**2 - 576*K2**2*K4**2 + 4304*K2**2*K4 - 256*K2**2*K5**2 - 16*K2**2*K6**2 - 3860*K2**2 + 1504*K2*K3*K5 + 208*K2*K4*K6 + 48*K2*K5*K7 - 64*K3**4 - 2888*K3**2 - 840*K4**2 - 152*K5**2 - 28*K6**2 + 5654
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.79']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72567', 'vk6.72668', 'vk6.72982', 'vk6.73142', 'vk6.74220', 'vk6.74849', 'vk6.76417', 'vk6.76899', 'vk6.77854', 'vk6.77883', 'vk6.77998', 'vk6.79264', 'vk6.79741', 'vk6.80753', 'vk6.81150', 'vk6.82311', 'vk6.83985', 'vk6.86358', 'vk6.87282', 'vk6.88231']
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 O1O2O3O4O5O6U2U4U5U1U6U3
R3 orbit {'O1O2O3O4O5O6U2U4U5U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U1U6U2U3U5
Gauss code of K* O1O2O3O4O5O6U4U1U6U2U3U5
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 -2 -4 2 -1 1 4],[ 2 0 -2 3 0 2 4],[ 4 2 0 4 1 2 3],[-2 -3 -4 0 -2 0 2],[ 1 0 -1 2 0 1 2],[-1 -2 -2 0 -1 0 1],[-4 -4 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 4 2 1 -1 -2 -4],[-4 0 -2 -1 -2 -4 -3],[-2 2 0 0 -2 -3 -4],[-1 1 0 0 -1 -2 -2],[ 1 2 2 1 0 0 -1],[ 2 4 3 2 0 0 -2],[ 4 3 4 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,2,4,2,1,2,4,3,0,2,3,4,1,2,2,0,1,2]
Phi over symmetry [-4,-2,-1,1,2,4,0,2,3,2,5,1,1,1,2,1,1,3,1,2,0]
Phi of -K [-4,-2,-1,1,2,4,0,2,3,2,5,1,1,1,2,1,1,3,1,2,0]
Phi of K* [-4,-2,-1,1,2,4,0,2,3,2,5,1,1,1,2,1,1,3,1,2,0]
Phi of -K* [-4,-2,-1,1,2,4,2,1,2,4,3,0,2,3,4,1,2,2,0,1,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+77t^4+20t^2
Outer characteristic polynomial t^7+119t^5+50t^3+4t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -1568*K1**4 + 1728*K1**3*K2*K3 - 576*K1**3*K3 - 256*K1**2*K2**4 + 1344*K1**2*K2**3 - 1024*K1**2*K2**2*K3**2 - 11584*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1728*K1**2*K2*K4 + 11488*K1**2*K2 - 992*K1**2*K3**2 - 96*K1**2*K4**2 - 7296*K1**2 + 256*K1*K2**3*K3**3 + 5312*K1*K2**3*K3 + 704*K1*K2**2*K3*K4 - 3072*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 1472*K1*K2**2*K5 + 384*K1*K2*K3**3 - 384*K1*K2*K3*K4 + 12240*K1*K2*K3 - 192*K1*K2*K4*K5 + 1648*K1*K3*K4 + 176*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 768*K2**4*K3**2 - 64*K2**4*K4**2 + 384*K2**4*K4 - 5104*K2**4 + 576*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 - 3200*K2**2*K3**2 - 576*K2**2*K4**2 + 4304*K2**2*K4 - 256*K2**2*K5**2 - 16*K2**2*K6**2 - 3860*K2**2 + 1504*K2*K3*K5 + 208*K2*K4*K6 + 48*K2*K5*K7 - 64*K3**4 - 2888*K3**2 - 840*K4**2 - 152*K5**2 - 28*K6**2 + 5654
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
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