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Flat knot 6.1257

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,1,1,2,0,1,0,2,0,1,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1257']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.541', '6.1257']
Outer characteristic polynomial of the knot is: t^7+29t^5+43t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1257']
2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 864*K1**2*K2**3 - 4496*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 1120*K1**2*K2*K4 + 5024*K1**2*K2 - 64*K1**2*K3**2 - 64*K1**2*K5**2 - 4552*K1**2 + 448*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 672*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6248*K1*K2*K3 - 96*K1*K2*K4*K5 + 1272*K1*K3*K4 + 472*K1*K4*K5 + 112*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1432*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 624*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 368*K2**2*K4**2 + 2704*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3962*K2**2 - 32*K2*K3*K4*K5 + 1384*K2*K3*K5 - 32*K2*K4**2*K6 + 296*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 2104*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1172*K4**2 - 568*K5**2 - 102*K6**2 - 8*K7**2 - 2*K8**2 + 3940
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1257']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4740', 'vk6.5066', 'vk6.6274', 'vk6.6715', 'vk6.8241', 'vk6.8690', 'vk6.9632', 'vk6.9948', 'vk6.20651', 'vk6.22084', 'vk6.28137', 'vk6.29568', 'vk6.39575', 'vk6.41808', 'vk6.46190', 'vk6.47810', 'vk6.48772', 'vk6.48980', 'vk6.49578', 'vk6.49785', 'vk6.50786', 'vk6.50999', 'vk6.51272', 'vk6.51470', 'vk6.57559', 'vk6.58731', 'vk6.62233', 'vk6.63181', 'vk6.67037', 'vk6.67912', 'vk6.69662', 'vk6.70345']
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 O1O2O3O4U5U3U2O5O6U4U1U6
R3 orbit {'O1O2O3O4U5U3U2O5O6U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U1O5O6U3U2U6
Gauss code of K* O1O2O3U1U4O5O6O4U6U3U2U5
Gauss code of -K* O1O2O3U1U4O5O6O4U3U6U5U2
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 -1 0 0 1 -2 2],[ 1 0 0 0 2 -1 2],[ 0 0 0 0 1 -1 1],[ 0 0 0 0 0 0 1],[-1 -2 -1 0 0 -1 1],[ 2 1 1 0 1 0 2],[-2 -2 -1 -1 -1 -2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -2 -1],[ 0 1 0 0 0 0 0],[ 0 1 1 0 0 0 -1],[ 1 2 2 0 0 0 -1],[ 2 2 1 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,1,2,2,0,1,2,1,0,0,0,0,1,1]
Phi over symmetry [-2,-1,0,0,1,2,0,1,1,1,2,0,1,0,2,0,1,1,1,2,0]
Phi of -K [-2,-1,0,0,1,2,0,1,2,2,2,1,1,0,1,0,0,1,1,1,0]
Phi of K* [-2,-1,0,0,1,2,0,1,1,1,2,0,1,0,2,0,1,1,1,2,0]
Phi of -K* [-2,-1,0,0,1,2,1,0,1,1,2,0,0,2,2,0,0,1,1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2-4w^3z+28w^2z+21w
Inner characteristic polynomial t^6+19t^4+23t^2+1
Outer characteristic polynomial t^7+29t^5+43t^3+9t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial -192*K1**2*K2**4 + 864*K1**2*K2**3 - 4496*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 1120*K1**2*K2*K4 + 5024*K1**2*K2 - 64*K1**2*K3**2 - 64*K1**2*K5**2 - 4552*K1**2 + 448*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 672*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6248*K1*K2*K3 - 96*K1*K2*K4*K5 + 1272*K1*K3*K4 + 472*K1*K4*K5 + 112*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1432*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 624*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 368*K2**2*K4**2 + 2704*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3962*K2**2 - 32*K2*K3*K4*K5 + 1384*K2*K3*K5 - 32*K2*K4**2*K6 + 296*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 2104*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1172*K4**2 - 568*K5**2 - 102*K6**2 - 8*K7**2 - 2*K8**2 + 3940
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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