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Flat knot 6.581

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,0,3,2,1,1,2,2,1,0,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.581']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.576', '6.581', '6.622', '6.627', '6.983', '6.1017']
Outer characteristic polynomial of the knot is: t^7+46t^5+52t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.581']
2-strand cable arrow polynomial of the knot is: -192*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 3328*K1**4*K2 - 4464*K1**4 + 320*K1**3*K2*K3 - 1216*K1**3*K3 - 448*K1**2*K2**4 + 2752*K1**2*K2**3 - 9136*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 12312*K1**2*K2 - 304*K1**2*K3**2 - 6812*K1**2 + 384*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2752*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 9024*K1*K2*K3 - 32*K1*K2*K4*K5 + 872*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1960*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 40*K2**2*K4**2 + 2288*K2**2*K4 - 5322*K2**2 + 288*K2*K3*K5 + 32*K2*K4*K6 - 2244*K3**2 - 522*K4**2 - 40*K5**2 - 6*K6**2 + 5528
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.581']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4201', 'vk6.4280', 'vk6.5456', 'vk6.5567', 'vk6.7568', 'vk6.7656', 'vk6.9070', 'vk6.9149', 'vk6.11191', 'vk6.12275', 'vk6.12382', 'vk6.19384', 'vk6.19679', 'vk6.19772', 'vk6.26168', 'vk6.26207', 'vk6.26586', 'vk6.26650', 'vk6.30777', 'vk6.31978', 'vk6.38168', 'vk6.38187', 'vk6.44829', 'vk6.44928', 'vk6.48523', 'vk6.49218', 'vk6.49325', 'vk6.50313', 'vk6.52757', 'vk6.63597', 'vk6.66320', 'vk6.66335']
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 O1O2O3O4U1O5U3O6U5U6U4U2
R3 orbit {'O1O2O3O4U1O5U3O6U5U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U5U6O5U2O6U4
Gauss code of K* O1O2O3O4U5U4U6U3O5U1O6U2
Gauss code of -K* O1O2O3O4U3O5U4O6U2U5U1U6
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 -1 2 0 1],[ 3 0 3 1 2 1 0],[-1 -3 0 -2 1 0 1],[ 1 -1 2 0 2 1 1],[-2 -2 -1 -2 0 -1 1],[ 0 -1 0 -1 1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 -1 -1 -2 -2],[-1 -1 0 -1 -1 -1 0],[-1 1 1 0 0 -2 -3],[ 0 1 1 0 0 -1 -1],[ 1 2 1 2 1 0 -1],[ 3 2 0 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,1,1,2,2,1,1,1,0,0,2,3,1,1,1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,0,3,2,1,1,2,2,1,0,1,-1,-1,1]
Phi of -K [-3,-1,0,1,1,2,1,2,1,4,3,0,0,1,1,1,0,1,-1,0,2]
Phi of K* [-2,-1,-1,0,1,3,0,2,1,1,3,1,1,0,1,0,1,4,0,2,1]
Phi of -K* [-3,-1,0,1,1,2,1,1,0,3,2,1,1,2,2,1,0,1,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+30t^4+21t^2+1
Outer characteristic polynomial t^7+46t^5+52t^3+11t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -192*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 3328*K1**4*K2 - 4464*K1**4 + 320*K1**3*K2*K3 - 1216*K1**3*K3 - 448*K1**2*K2**4 + 2752*K1**2*K2**3 - 9136*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 12312*K1**2*K2 - 304*K1**2*K3**2 - 6812*K1**2 + 384*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2752*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 9024*K1*K2*K3 - 32*K1*K2*K4*K5 + 872*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1960*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 40*K2**2*K4**2 + 2288*K2**2*K4 - 5322*K2**2 + 288*K2*K3*K5 + 32*K2*K4*K6 - 2244*K3**2 - 522*K4**2 - 40*K5**2 - 6*K6**2 + 5528
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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