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Flat knot 6.663

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,2,4,0,0,0,1,0,1,1,2,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.663']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370']
Outer characteristic polynomial of the knot is: t^7+55t^5+52t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.663']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 992*K1**2*K2**3 - 3536*K1**2*K2**2 - 96*K1**2*K2*K4 + 4136*K1**2*K2 - 64*K1**2*K3**2 - 3320*K1**2 + 384*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3848*K1*K2*K3 + 336*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1152*K2**4 - 336*K2**2*K3**2 - 8*K2**2*K4**2 + 1072*K2**2*K4 - 1992*K2**2 + 208*K2*K3*K5 - 1104*K3**2 - 240*K4**2 - 40*K5**2 + 2318
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.663']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4739', 'vk6.5065', 'vk6.6275', 'vk6.6716', 'vk6.8240', 'vk6.8689', 'vk6.9633', 'vk6.9949', 'vk6.20647', 'vk6.22080', 'vk6.28133', 'vk6.29564', 'vk6.39571', 'vk6.41804', 'vk6.46186', 'vk6.47806', 'vk6.48771', 'vk6.48979', 'vk6.49579', 'vk6.49786', 'vk6.50785', 'vk6.50998', 'vk6.51273', 'vk6.51471', 'vk6.57563', 'vk6.58735', 'vk6.62237', 'vk6.63185', 'vk6.67041', 'vk6.67916', 'vk6.69666', 'vk6.70349']
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 O1O2O3O4U2O5U4U3O6U1U6U5
R3 orbit {'O1O2O3O4U2O5U4U3O6U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U4O6U2U1O5U3
Gauss code of K* O1O2O3U1U4U5U6O4U3O6O5U2
Gauss code of -K* O1O2O3U2O4O5U1O6U5U4U6U3
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 -2 0 0 3 1],[ 2 0 -2 1 1 4 1],[ 2 2 0 2 1 2 0],[ 0 -1 -2 0 0 2 0],[ 0 -1 -1 0 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 0 -1 -2 -2 -4],[-1 0 0 0 0 0 -1],[ 0 1 0 0 0 -1 -1],[ 0 2 0 0 0 -2 -1],[ 2 2 0 1 2 0 2],[ 2 4 1 1 1 -2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,0,1,2,2,4,0,0,0,1,0,1,1,2,1,-2]
Phi over symmetry [-3,-1,0,0,2,2,0,1,2,2,4,0,0,0,1,0,1,1,2,1,-2]
Phi of -K [-2,-2,0,0,1,3,-2,0,1,3,3,1,1,2,1,0,1,1,1,2,2]
Phi of K* [-3,-1,0,0,2,2,2,1,2,1,3,1,1,2,3,0,1,0,1,1,-2]
Phi of -K* [-2,-2,0,0,1,3,-2,1,1,1,4,1,2,0,2,0,0,1,0,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+37t^4+20t^2
Outer characteristic polynomial t^7+55t^5+52t^3+4t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 992*K1**2*K2**3 - 3536*K1**2*K2**2 - 96*K1**2*K2*K4 + 4136*K1**2*K2 - 64*K1**2*K3**2 - 3320*K1**2 + 384*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3848*K1*K2*K3 + 336*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1152*K2**4 - 336*K2**2*K3**2 - 8*K2**2*K4**2 + 1072*K2**2*K4 - 1992*K2**2 + 208*K2*K3*K5 - 1104*K3**2 - 240*K4**2 - 40*K5**2 + 2318
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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