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Flat knot 6.846

Min(phi) over symmetries of the knot is: [-3,-2,0,0,2,3,-1,2,2,2,4,1,2,1,2,0,2,2,1,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.846']
Arrow polynomial of the knot is: 16*K1**3 - 12*K1**2 - 12*K1*K2 - 6*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.141', '6.846', '6.918', '6.941', '6.2064', '6.2066']
Outer characteristic polynomial of the knot is: t^7+71t^5+44t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.846']
2-strand cable arrow polynomial of the knot is: -2400*K1**4 + 2048*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1024*K1**3*K3 + 704*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 - 10176*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 1600*K1**2*K2*K4 + 14128*K1**2*K2 - 1632*K1**2*K3**2 - 128*K1**2*K4**2 - 9712*K1**2 + 2432*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2560*K1*K2**2*K3 - 896*K1*K2**2*K5 + 512*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 13312*K1*K2*K3 - 64*K1*K2*K4*K5 + 2016*K1*K3*K4 + 144*K1*K4*K5 + 32*K1*K5*K6 - 128*K2**6 + 192*K2**4*K4 - 2608*K2**4 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 176*K2**2*K4**2 + 3264*K2**2*K4 - 6996*K2**2 + 1056*K2*K3*K5 + 160*K2*K4*K6 - 128*K3**4 - 3456*K3**2 - 916*K4**2 - 112*K5**2 - 36*K6**2 + 7306
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.846']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71352', 'vk6.71404', 'vk6.71867', 'vk6.71930', 'vk6.74324', 'vk6.74969', 'vk6.76540', 'vk6.76946', 'vk6.76998', 'vk6.77059', 'vk6.77389', 'vk6.79377', 'vk6.79802', 'vk6.80837', 'vk6.81267', 'vk6.81466', 'vk6.83846', 'vk6.87062', 'vk6.88042', 'vk6.89562']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U1U3O5O6U2U5U4U6
R3 orbit {'O1O2O3O4U1U3O5O6U2U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U6U3O5O6U2U4
Gauss code of K* O1O2O3O4U5U1U6U3O5O6U2U4
Gauss code of -K* Same
Diagrammatic symmetry type -
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 -2 0 2 0 3],[ 3 0 2 1 3 1 2],[ 2 -2 0 0 3 1 3],[ 0 -1 0 0 1 0 1],[-2 -3 -3 -1 0 0 2],[ 0 -1 -1 0 0 0 1],[-3 -2 -3 -1 -2 -1 0]]
Primitive based matrix [[ 0 3 2 0 0 -2 -3],[-3 0 -2 -1 -1 -3 -2],[-2 2 0 0 -1 -3 -3],[ 0 1 0 0 0 -1 -1],[ 0 1 1 0 0 0 -1],[ 2 3 3 1 0 0 -2],[ 3 2 3 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,0,2,3,2,1,1,3,2,0,1,3,3,0,1,1,0,1,2]
Phi over symmetry [-3,-2,0,0,2,3,-1,2,2,2,4,1,2,1,2,0,2,2,1,2,-1]
Phi of -K [-3,-2,0,0,2,3,-1,2,2,2,4,1,2,1,2,0,2,2,1,2,-1]
Phi of K* [-3,-2,0,0,2,3,-1,2,2,2,4,1,2,1,2,0,2,2,1,2,-1]
Phi of -K* [-3,-2,0,0,2,3,2,1,1,3,2,0,1,3,3,0,1,1,0,1,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+45t^4+12t^2
Outer characteristic polynomial t^7+71t^5+44t^3+4t
Flat arrow polynomial 16*K1**3 - 12*K1**2 - 12*K1*K2 - 6*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -2400*K1**4 + 2048*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1024*K1**3*K3 + 704*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 - 10176*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 1600*K1**2*K2*K4 + 14128*K1**2*K2 - 1632*K1**2*K3**2 - 128*K1**2*K4**2 - 9712*K1**2 + 2432*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2560*K1*K2**2*K3 - 896*K1*K2**2*K5 + 512*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 13312*K1*K2*K3 - 64*K1*K2*K4*K5 + 2016*K1*K3*K4 + 144*K1*K4*K5 + 32*K1*K5*K6 - 128*K2**6 + 192*K2**4*K4 - 2608*K2**4 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 176*K2**2*K4**2 + 3264*K2**2*K4 - 6996*K2**2 + 1056*K2*K3*K5 + 160*K2*K4*K6 - 128*K3**4 - 3456*K3**2 - 916*K4**2 - 112*K5**2 - 36*K6**2 + 7306
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice True
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