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Flat knot 6.1267

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,2,2,2,3,1,1,0,1,-1,2,1,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1267']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 10*K1*K2 - 4*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.653', '6.1267']
Outer characteristic polynomial of the knot is: t^7+57t^5+70t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1267']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 704*K1**4*K2 - 896*K1**4 + 192*K1**3*K2*K3 - 160*K1**3*K3 - 1984*K1**2*K2**4 + 4128*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 32*K1**2*K2**2*K4 - 11328*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 8424*K1**2*K2 - 256*K1**2*K3**2 - 4900*K1**2 - 256*K1*K2**4*K3 + 3296*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 - 512*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 9008*K1*K2*K3 + 664*K1*K3*K4 + 40*K1*K4*K5 - 352*K2**6 + 608*K2**4*K4 - 3808*K2**4 - 64*K2**3*K6 - 1712*K2**2*K3**2 - 264*K2**2*K4**2 + 2720*K2**2*K4 - 2204*K2**2 + 696*K2*K3*K5 + 48*K2*K4*K6 - 1900*K3**2 - 504*K4**2 - 48*K5**2 - 4*K6**2 + 3822
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1267']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16906', 'vk6.17150', 'vk6.20520', 'vk6.21908', 'vk6.23294', 'vk6.23595', 'vk6.27965', 'vk6.29440', 'vk6.35312', 'vk6.35750', 'vk6.39373', 'vk6.41557', 'vk6.42813', 'vk6.43097', 'vk6.45944', 'vk6.47627', 'vk6.55061', 'vk6.55308', 'vk6.57381', 'vk6.58547', 'vk6.59453', 'vk6.59744', 'vk6.62036', 'vk6.63034', 'vk6.64902', 'vk6.65117', 'vk6.66930', 'vk6.67783', 'vk6.68207', 'vk6.68353', 'vk6.69536', 'vk6.70240']
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 O1O2O3U1O4O5U2U3O6U4U6U5
R3 orbit {'O1O2O3U1O4O5U2U3O6U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U6O5U1U2O4O6U3
Gauss code of K* O1O2O3U4U5U6O4U1U3O5O6U2
Gauss code of -K* O1O2O3U2O4O5U1U3O6U4U5U6
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 1 2 2 2 0],[ 2 -1 0 1 2 3 1],[ 0 -2 -1 0 1 2 1],[ 0 -2 -2 -1 0 2 1],[-3 -2 -3 -2 -2 0 0],[-1 0 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 0 -2 -2 -2 -3],[-1 0 0 -1 -1 0 -1],[ 0 2 1 0 1 -2 -1],[ 0 2 1 -1 0 -2 -2],[ 2 2 0 2 2 0 1],[ 2 3 1 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,0,2,2,2,3,1,1,0,1,-1,2,1,2,2,-1]
Phi over symmetry [-3,-1,0,0,2,2,0,2,2,2,3,1,1,0,1,-1,2,1,2,2,-1]
Phi of -K [-2,-2,0,0,1,3,-1,0,0,3,3,0,1,2,2,1,0,1,0,1,2]
Phi of K* [-3,-1,0,0,2,2,2,1,1,2,3,0,0,2,3,-1,0,0,1,0,-1]
Phi of -K* [-2,-2,0,0,1,3,-1,1,2,1,3,2,2,0,2,1,1,2,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2-2w^3z+28w^2z+25w
Inner characteristic polynomial t^6+39t^4+26t^2
Outer characteristic polynomial t^7+57t^5+70t^3+9t
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 10*K1*K2 - 4*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -384*K1**4*K2**2 + 704*K1**4*K2 - 896*K1**4 + 192*K1**3*K2*K3 - 160*K1**3*K3 - 1984*K1**2*K2**4 + 4128*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 32*K1**2*K2**2*K4 - 11328*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 8424*K1**2*K2 - 256*K1**2*K3**2 - 4900*K1**2 - 256*K1*K2**4*K3 + 3296*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 - 512*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 9008*K1*K2*K3 + 664*K1*K3*K4 + 40*K1*K4*K5 - 352*K2**6 + 608*K2**4*K4 - 3808*K2**4 - 64*K2**3*K6 - 1712*K2**2*K3**2 - 264*K2**2*K4**2 + 2720*K2**2*K4 - 2204*K2**2 + 696*K2*K3*K5 + 48*K2*K4*K6 - 1900*K3**2 - 504*K4**2 - 48*K5**2 - 4*K6**2 + 3822
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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