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Flat knot 6.61

Min(phi) over symmetries of the knot is: [-4,-4,0,2,2,4,0,1,2,4,3,2,3,5,4,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.61']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**2*K2 + 4*K1**2 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.61', '6.177', '6.254', '6.357', '6.477']
Outer characteristic polynomial of the knot is: t^7+154t^5+242t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.61']
2-strand cable arrow polynomial of the knot is: -1152*K2**8 + 1024*K2**6*K4 - 1472*K2**6 - 288*K2**4*K4**2 + 1056*K2**4*K4 - 384*K2**4 + 32*K2**3*K4*K6 - 208*K2**2*K4**2 + 1152*K2**2*K4 + 176*K2**2 + 80*K2*K4*K6 - 272*K4**2 - 16*K6**2 + 270
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.61']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74151', 'vk6.74153', 'vk6.74741', 'vk6.74744', 'vk6.76268', 'vk6.76276', 'vk6.76815', 'vk6.76818', 'vk6.79171', 'vk6.79175', 'vk6.79633', 'vk6.79639', 'vk6.80653', 'vk6.80657', 'vk6.81037', 'vk6.81038', 'vk6.82858', 'vk6.82898', 'vk6.83040', 'vk6.83331', 'vk6.83336', 'vk6.85549', 'vk6.85551', 'vk6.85722', 'vk6.85783', 'vk6.85853', 'vk6.85856', 'vk6.87843', 'vk6.89374', 'vk6.89376', 'vk6.89589', 'vk6.90206']
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 O1O2O3O4O5O6U2U1U4U5U6U3
R3 orbit {'O1O2O3O4O5O6U2U1U4U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U1U2U3U6U5
Gauss code of K* O1O2O3O4O5O6U2U1U6U3U4U5
Gauss code of -K* O1O2O3O4O5O6U2U3U4U1U6U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 -4 2 0 2 4],[ 4 0 0 5 2 3 4],[ 4 0 0 4 1 2 3],[-2 -5 -4 0 -2 0 2],[ 0 -2 -1 2 0 1 2],[-2 -3 -2 0 -1 0 1],[-4 -4 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 4 2 2 0 -4 -4],[-4 0 -1 -2 -2 -3 -4],[-2 1 0 0 -1 -2 -3],[-2 2 0 0 -2 -4 -5],[ 0 2 1 2 0 -1 -2],[ 4 3 2 4 1 0 0],[ 4 4 3 5 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-2,0,4,4,1,2,2,3,4,0,1,2,3,2,4,5,1,2,0]
Phi over symmetry [-4,-4,0,2,2,4,0,1,2,4,3,2,3,5,4,1,2,2,0,1,2]
Phi of -K [-4,-4,0,2,2,4,0,2,1,3,4,3,2,4,5,0,1,2,0,0,1]
Phi of K* [-4,-2,-2,0,4,4,0,1,2,4,5,0,0,1,2,1,3,4,2,3,0]
Phi of -K* [-4,-4,0,2,2,4,0,1,2,4,3,2,3,5,4,1,2,2,0,1,2]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 2z^2+5z+3
Enhanced Jones-Krushkal polynomial 4w^5z^2-8w^4z^2+4w^4z+6w^3z^2+w^2z+3
Inner characteristic polynomial t^6+98t^4+26t^2
Outer characteristic polynomial t^7+154t^5+242t^3
Flat arrow polynomial -8*K1**4 + 4*K1**2*K2 + 4*K1**2 + 1
2-strand cable arrow polynomial -1152*K2**8 + 1024*K2**6*K4 - 1472*K2**6 - 288*K2**4*K4**2 + 1056*K2**4*K4 - 384*K2**4 + 32*K2**3*K4*K6 - 208*K2**2*K4**2 + 1152*K2**2*K4 + 176*K2**2 + 80*K2*K4*K6 - 272*K4**2 - 16*K6**2 + 270
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}]]
If K is slice False
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