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Flat knot 6.166

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,2,3,4,2,1,2,2,2,0,0,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.166']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.166', '6.225', '6.296', '6.498']
Outer characteristic polynomial of the knot is: t^7+80t^5+92t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.166']
2-strand cable arrow polynomial of the knot is: -1056*K1**2*K2**2 - 160*K1**2*K2*K4 + 1768*K1**2*K2 - 1584*K1**2 + 320*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 1960*K1*K2*K3 + 376*K1*K3*K4 - 32*K2**4*K4**2 + 128*K2**4*K4 - 752*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 608*K2**2*K3**2 - 176*K2**2*K4**2 + 904*K2**2*K4 - 8*K2**2*K6**2 - 1072*K2**2 - 32*K2*K3**2*K4 + 328*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 640*K3**2 - 270*K4**2 - 40*K5**2 - 8*K6**2 + 1212
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.166']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16372', 'vk6.16413', 'vk6.18115', 'vk6.18451', 'vk6.22706', 'vk6.22807', 'vk6.24568', 'vk6.24985', 'vk6.34675', 'vk6.34754', 'vk6.35533', 'vk6.35981', 'vk6.36701', 'vk6.37123', 'vk6.39428', 'vk6.41622', 'vk6.42328', 'vk6.42372', 'vk6.42945', 'vk6.43238', 'vk6.43977', 'vk6.44291', 'vk6.46008', 'vk6.47680', 'vk6.54656', 'vk6.55212', 'vk6.56219', 'vk6.57436', 'vk6.59177', 'vk6.59610', 'vk6.60818', 'vk6.62106', 'vk6.64653', 'vk6.64701', 'vk6.65571', 'vk6.65881', 'vk6.68003', 'vk6.68029', 'vk6.68651', 'vk6.68864']
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 O1O2O3O4O5U1O6U4U5U6U3U2
R3 orbit {'O1O2O3O4O5U1U3O6U5U4U6U2', 'O1O2O3O4O5U1O6U4U5U6U3U2'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U4U3U6U1U2O6U5
Gauss code of K* O1O2O3O4O5U6U5U4U1U2O6U3
Gauss code of -K* O1O2O3O4O5U3O6U4U5U2U1U6
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 -4 1 1 -1 1 2],[ 4 0 4 3 1 2 2],[-1 -4 0 0 -2 0 2],[-1 -3 0 0 -2 0 2],[ 1 -1 2 2 0 1 2],[-1 -2 0 0 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 1 1 1 -1 -4],[-2 0 -1 -2 -2 -2 -2],[-1 1 0 0 0 -1 -2],[-1 2 0 0 0 -2 -3],[-1 2 0 0 0 -2 -4],[ 1 2 1 2 2 0 -1],[ 4 2 2 3 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,1,4,1,2,2,2,2,0,0,1,2,0,2,3,2,4,1]
Phi over symmetry [-4,-1,1,1,1,2,1,2,3,4,2,1,2,2,2,0,0,1,0,2,2]
Phi of -K [-4,-1,1,1,1,2,2,1,2,3,4,0,0,1,1,0,0,-1,0,-1,0]
Phi of K* [-2,-1,-1,-1,1,4,-1,-1,0,1,4,0,0,0,1,0,0,2,1,3,2]
Phi of -K* [-4,-1,1,1,1,2,1,2,3,4,2,1,2,2,2,0,0,1,0,2,2]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+56t^4+15t^2
Outer characteristic polynomial t^7+80t^5+92t^3+3t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 + K2 + 2
2-strand cable arrow polynomial -1056*K1**2*K2**2 - 160*K1**2*K2*K4 + 1768*K1**2*K2 - 1584*K1**2 + 320*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 1960*K1*K2*K3 + 376*K1*K3*K4 - 32*K2**4*K4**2 + 128*K2**4*K4 - 752*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 608*K2**2*K3**2 - 176*K2**2*K4**2 + 904*K2**2*K4 - 8*K2**2*K6**2 - 1072*K2**2 - 32*K2*K3**2*K4 + 328*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 640*K3**2 - 270*K4**2 - 40*K5**2 - 8*K6**2 + 1212
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}]]
If K is slice False
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