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Flat knot 6.964

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,2,3,3,1,0,1,1,0,0,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.964']
Arrow polynomial of the knot is: 20*K1**3 - 10*K1**2 - 10*K1*K2 - 10*K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.964']
Outer characteristic polynomial of the knot is: t^7+43t^5+71t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.964']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 768*K1**4*K2 - 1792*K1**4 + 224*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**4 + 1696*K1**2*K2**3 - 8560*K1**2*K2**2 - 544*K1**2*K2*K4 + 9992*K1**2*K2 - 96*K1**2*K3**2 - 6724*K1**2 + 1376*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 384*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 8880*K1*K2*K3 + 744*K1*K3*K4 + 96*K1*K4*K5 - 416*K2**6 + 544*K2**4*K4 - 3768*K2**4 - 64*K2**3*K6 - 1184*K2**2*K3**2 - 264*K2**2*K4**2 + 3704*K2**2*K4 - 4416*K2**2 + 680*K2*K3*K5 + 128*K2*K4*K6 - 2332*K3**2 - 938*K4**2 - 120*K5**2 - 24*K6**2 + 5512
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.964']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11286', 'vk6.11364', 'vk6.12549', 'vk6.12660', 'vk6.18366', 'vk6.18704', 'vk6.24816', 'vk6.25273', 'vk6.30961', 'vk6.31085', 'vk6.32142', 'vk6.32261', 'vk6.36994', 'vk6.37444', 'vk6.44176', 'vk6.44495', 'vk6.52038', 'vk6.52125', 'vk6.52883', 'vk6.52948', 'vk6.56138', 'vk6.56364', 'vk6.60661', 'vk6.61006', 'vk6.63653', 'vk6.63698', 'vk6.64083', 'vk6.64128', 'vk6.65792', 'vk6.66048', 'vk6.68794', 'vk6.69002']
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 O1O2O3O4U5U3O6O5U1U6U4U2
R3 orbit {'O1O2O3O4U5U3O6O5U1U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U5U4O6O5U2U6
Gauss code of K* O1O2O3O4U1U4U5U3O6O5U2U6
Gauss code of -K* O1O2O3O4U5U3O6O5U2U6U1U4
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 -3 1 0 2 0 0],[ 3 0 3 1 3 2 0],[-1 -3 0 0 1 -1 -1],[ 0 -1 0 0 0 0 -1],[-2 -3 -1 0 0 -1 -1],[ 0 -2 1 0 1 0 0],[ 0 0 1 1 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 0 -3],[-2 0 -1 0 -1 -1 -3],[-1 1 0 0 -1 -1 -3],[ 0 0 0 0 0 -1 -1],[ 0 1 1 0 0 0 -2],[ 0 1 1 1 0 0 0],[ 3 3 3 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,0,3,1,0,1,1,3,0,1,1,3,0,1,1,0,2,0]
Phi over symmetry [-3,0,0,0,1,2,0,1,2,3,3,1,0,1,1,0,0,0,1,1,1]
Phi of -K [-3,0,0,0,1,2,1,2,3,1,2,0,0,0,1,1,1,2,0,1,0]
Phi of K* [-2,-1,0,0,0,3,0,1,1,2,2,0,0,1,1,0,0,1,1,3,2]
Phi of -K* [-3,0,0,0,1,2,0,1,2,3,3,1,0,1,1,0,0,0,1,1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 4z^2+23z+31
Enhanced Jones-Krushkal polynomial 4w^3z^2-2w^3z+25w^2z+31w
Inner characteristic polynomial t^6+29t^4+30t^2+4
Outer characteristic polynomial t^7+43t^5+71t^3+13t
Flat arrow polynomial 20*K1**3 - 10*K1**2 - 10*K1*K2 - 10*K1 + 5*K2 + 6
2-strand cable arrow polynomial -192*K1**4*K2**2 + 768*K1**4*K2 - 1792*K1**4 + 224*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**4 + 1696*K1**2*K2**3 - 8560*K1**2*K2**2 - 544*K1**2*K2*K4 + 9992*K1**2*K2 - 96*K1**2*K3**2 - 6724*K1**2 + 1376*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 384*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 8880*K1*K2*K3 + 744*K1*K3*K4 + 96*K1*K4*K5 - 416*K2**6 + 544*K2**4*K4 - 3768*K2**4 - 64*K2**3*K6 - 1184*K2**2*K3**2 - 264*K2**2*K4**2 + 3704*K2**2*K4 - 4416*K2**2 + 680*K2*K3*K5 + 128*K2*K4*K6 - 2332*K3**2 - 938*K4**2 - 120*K5**2 - 24*K6**2 + 5512
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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