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Flat knot 6.1710

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,1,2,0,0,1,1,1,0,0,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1710']
Arrow polynomial of the knot is: -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.665', '6.1301', '6.1514', '6.1646', '6.1669', '6.1709', '6.1710', '6.1744', '6.1776']
Outer characteristic polynomial of the knot is: t^7+34t^5+41t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1710']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1408*K1**4*K2 - 3392*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 736*K1**2*K2**3 - 5632*K1**2*K2**2 - 192*K1**2*K2*K4 + 7688*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 3460*K1**2 - 896*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 5224*K1*K2*K3 + 592*K1*K3*K4 + 88*K1*K4*K5 - 904*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 1056*K2**2*K4 - 3052*K2**2 + 272*K2*K3*K5 + 32*K2*K4*K6 - 1328*K3**2 - 354*K4**2 - 68*K5**2 - 4*K6**2 + 3272
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1710']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13944', 'vk6.13945', 'vk6.14038', 'vk6.14041', 'vk6.15012', 'vk6.15015', 'vk6.15134', 'vk6.15135', 'vk6.17444', 'vk6.17465', 'vk6.17471', 'vk6.23952', 'vk6.23962', 'vk6.23985', 'vk6.23995', 'vk6.33757', 'vk6.33830', 'vk6.33833', 'vk6.34298', 'vk6.36255', 'vk6.36259', 'vk6.43415', 'vk6.53892', 'vk6.53893', 'vk6.53924', 'vk6.54437', 'vk6.55588', 'vk6.55596', 'vk6.60080', 'vk6.60093', 'vk6.60099', 'vk6.65307']
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 O1O2O3U1U4O5O4U6U3O6U2U5
R3 orbit {'O1O2O3U1U4O5O4U6U3O6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O5U1U5O6O4U6U3
Gauss code of K* O1O2U1O3O4U5U3U2O5O6U4U6
Gauss code of -K* O1O2U3O4O3U5U1O5O6U4U2U6
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 -2 0 1 1 1 -1],[ 2 0 2 1 2 2 1],[ 0 -2 0 1 0 1 -1],[-1 -1 -1 0 -1 -1 -1],[-1 -2 0 1 0 1 -2],[-1 -2 -1 1 -1 0 -1],[ 1 -1 1 1 2 1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -2 -2],[-1 -1 0 1 -1 -1 -2],[-1 -1 -1 0 -1 -1 -1],[ 0 0 1 1 0 -1 -2],[ 1 2 1 1 1 0 -1],[ 2 2 2 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,-1,0,2,2,-1,1,1,2,1,1,1,1,2,1]
Phi over symmetry [-2,-1,0,1,1,1,0,0,1,1,2,0,0,1,1,1,0,0,-1,-1,-1]
Phi of -K [-2,-1,0,1,1,1,0,0,1,1,2,0,0,1,1,1,0,0,-1,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,0,1,2,-1,0,1,1,1,0,1,0,0,0]
Phi of -K* [-2,-1,0,1,1,1,1,2,1,2,2,1,1,1,2,1,1,0,-1,-1,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+26t^4+24t^2+1
Outer characteristic polynomial t^7+34t^5+41t^3+5t
Flat arrow polynomial -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1408*K1**4*K2 - 3392*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 736*K1**2*K2**3 - 5632*K1**2*K2**2 - 192*K1**2*K2*K4 + 7688*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 3460*K1**2 - 896*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 5224*K1*K2*K3 + 592*K1*K3*K4 + 88*K1*K4*K5 - 904*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 1056*K2**2*K4 - 3052*K2**2 + 272*K2*K3*K5 + 32*K2*K4*K6 - 1328*K3**2 - 354*K4**2 - 68*K5**2 - 4*K6**2 + 3272
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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