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Flat knot 6.988

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,1,2,2,0,1,1,2,0,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.988']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 6*K1*K2 - 6*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.360', '6.988', '6.1003']
Outer characteristic polynomial of the knot is: t^7+61t^5+74t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.988']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 1376*K1**4*K2 - 2432*K1**4 - 384*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 576*K1**3*K3 - 640*K1**2*K2**4 + 4160*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 12112*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 11504*K1**2*K2 - 192*K1**2*K3**2 - 6436*K1**2 + 2176*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 - 256*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 8320*K1*K2*K3 + 480*K1*K3*K4 - 96*K2**6 + 160*K2**4*K4 - 3344*K2**4 - 800*K2**2*K3**2 - 72*K2**2*K4**2 + 1976*K2**2*K4 - 2888*K2**2 + 112*K2*K3*K5 - 1548*K3**2 - 256*K4**2 + 4518
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.988']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73276', 'vk6.73417', 'vk6.74016', 'vk6.74558', 'vk6.75188', 'vk6.75418', 'vk6.76036', 'vk6.76766', 'vk6.78149', 'vk6.78384', 'vk6.78993', 'vk6.79550', 'vk6.79978', 'vk6.80131', 'vk6.80517', 'vk6.80985', 'vk6.81869', 'vk6.82163', 'vk6.82187', 'vk6.82585', 'vk6.83580', 'vk6.83768', 'vk6.84042', 'vk6.84597', 'vk6.84925', 'vk6.85585', 'vk6.85712', 'vk6.85931', 'vk6.86738', 'vk6.87674', 'vk6.88930', 'vk6.89968']
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 O1O2O3O4U1U5O6U2O5U3U4U6
R3 orbit {'O1O2O3O4U1U5O6U2O5U3U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U2O6U3O5U6U4
Gauss code of K* O1O2O3U4U5U1U2O4O6U3O5U6
Gauss code of -K* O1O2O3U4O5U1O4O6U2U3U5U6
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 -3 -1 0 2 0 2],[ 3 0 1 2 3 2 3],[ 1 -1 0 0 1 1 2],[ 0 -2 0 0 1 0 1],[-2 -3 -1 -1 0 -2 0],[ 0 -2 -1 0 2 0 2],[-2 -3 -2 -1 0 -2 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 0 -1 -2 -1 -3],[-2 0 0 -1 -2 -2 -3],[ 0 1 1 0 0 0 -2],[ 0 2 2 0 0 -1 -2],[ 1 1 2 0 1 0 -1],[ 3 3 3 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,0,1,2,1,3,1,2,2,3,0,0,2,1,2,1]
Phi over symmetry [-3,-1,0,0,2,2,1,1,1,2,2,0,1,1,2,0,0,0,1,1,0]
Phi of -K [-3,-1,0,0,2,2,1,1,1,2,2,0,1,1,2,0,0,0,1,1,0]
Phi of K* [-2,-2,0,0,1,3,0,0,1,1,2,0,1,2,2,0,0,1,1,1,1]
Phi of -K* [-3,-1,0,0,2,2,1,2,2,3,3,0,1,1,2,0,1,1,2,2,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+43t^4+48t^2+4
Outer characteristic polynomial t^7+61t^5+74t^3+11t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 6*K1*K2 - 6*K1 + 4*K2 + 5
2-strand cable arrow polynomial -512*K1**4*K2**2 + 1376*K1**4*K2 - 2432*K1**4 - 384*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 576*K1**3*K3 - 640*K1**2*K2**4 + 4160*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 12112*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 11504*K1**2*K2 - 192*K1**2*K3**2 - 6436*K1**2 + 2176*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 - 256*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 8320*K1*K2*K3 + 480*K1*K3*K4 - 96*K2**6 + 160*K2**4*K4 - 3344*K2**4 - 800*K2**2*K3**2 - 72*K2**2*K4**2 + 1976*K2**2*K4 - 2888*K2**2 + 112*K2*K3*K5 - 1548*K3**2 - 256*K4**2 + 4518
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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