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Flat knot 6.72

Min(phi) over symmetries of the knot is: [-4,-3,-1,2,2,4,0,2,2,4,5,1,1,2,3,1,2,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.72']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 2*K2 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.72']
Outer characteristic polynomial of the knot is: t^7+137t^5+134t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.72']
2-strand cable arrow polynomial of the knot is: -192*K1**4 + 96*K1**3*K2*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 960*K1**2*K2**2 + 824*K1**2*K2 - 272*K1**2*K3**2 - 780*K1**2 + 128*K1*K2**3*K3**3 + 672*K1*K2**3*K3 + 256*K1*K2*K3**3 + 1336*K1*K2*K3 + 168*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 560*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**2*K3**4 - 720*K2**2*K3**2 - 72*K2**2*K4**2 + 224*K2**2*K4 - 32*K2**2*K5**2 - 16*K2**2*K6**2 - 328*K2**2 + 32*K2*K3**3*K5 + 320*K2*K3*K5 + 40*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 112*K3**4 + 32*K3**2*K6 - 392*K3**2 - 112*K4**2 - 52*K5**2 - 24*K6**2 - 2*K8**2 + 776
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.72']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73281', 'vk6.73424', 'vk6.73701', 'vk6.73818', 'vk6.74539', 'vk6.74818', 'vk6.75197', 'vk6.75616', 'vk6.76015', 'vk6.76374', 'vk6.76756', 'vk6.76877', 'vk6.78154', 'vk6.78595', 'vk6.78988', 'vk6.79232', 'vk6.79539', 'vk6.79708', 'vk6.79979', 'vk6.80241', 'vk6.80505', 'vk6.80714', 'vk6.80979', 'vk6.81073', 'vk6.81611', 'vk6.81789', 'vk6.82160', 'vk6.82167', 'vk6.82652', 'vk6.84043', 'vk6.84052', 'vk6.84211', 'vk6.84596', 'vk6.84928', 'vk6.85948', 'vk6.86728', 'vk6.87668', 'vk6.87752', 'vk6.88199', 'vk6.89971']
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 O1O2O3O4O5O6U2U3U5U1U6U4
R3 orbit {'O1O2O3O4O5U1U2U6U3U5O6U4', 'O1O2O3O4O5O6U2U3U5U1U6U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U3U1U6U2U4U5
Gauss code of K* O1O2O3O4O5O6U4U1U2U6U3U5
Gauss code of -K* O1O2O3O4O5O6U2U4U1U5U6U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -4 -2 3 1 4],[ 2 0 -2 0 4 2 4],[ 4 2 0 1 4 2 3],[ 2 0 -1 0 3 1 2],[-3 -4 -4 -3 0 -1 1],[-1 -2 -2 -1 1 0 1],[-4 -4 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 4 3 1 -2 -2 -4],[-4 0 -1 -1 -2 -4 -3],[-3 1 0 -1 -3 -4 -4],[-1 1 1 0 -1 -2 -2],[ 2 2 3 1 0 0 -1],[ 2 4 4 2 0 0 -2],[ 4 3 4 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-1,2,2,4,1,1,2,4,3,1,3,4,4,1,2,2,0,1,2]
Phi over symmetry [-4,-3,-1,2,2,4,0,2,2,4,5,1,1,2,3,1,2,3,0,0,1]
Phi of -K [-4,-2,-2,1,3,4,0,1,3,3,5,0,1,1,2,2,2,4,1,2,0]
Phi of K* [-4,-3,-1,2,2,4,0,2,2,4,5,1,1,2,3,1,2,3,0,0,1]
Phi of -K* [-4,-2,-2,1,3,4,1,2,2,4,3,0,1,3,2,2,4,4,1,1,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -4w^3z+11w^2z+15w
Inner characteristic polynomial t^6+87t^4+42t^2
Outer characteristic polynomial t^7+137t^5+134t^3
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 2*K2 + K4 + 2
2-strand cable arrow polynomial -192*K1**4 + 96*K1**3*K2*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 960*K1**2*K2**2 + 824*K1**2*K2 - 272*K1**2*K3**2 - 780*K1**2 + 128*K1*K2**3*K3**3 + 672*K1*K2**3*K3 + 256*K1*K2*K3**3 + 1336*K1*K2*K3 + 168*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 560*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**2*K3**4 - 720*K2**2*K3**2 - 72*K2**2*K4**2 + 224*K2**2*K4 - 32*K2**2*K5**2 - 16*K2**2*K6**2 - 328*K2**2 + 32*K2*K3**3*K5 + 320*K2*K3*K5 + 40*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 112*K3**4 + 32*K3**2*K6 - 392*K3**2 - 112*K4**2 - 52*K5**2 - 24*K6**2 - 2*K8**2 + 776
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}]]
If K is slice False
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