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Flat knot 6.1661

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,0,1,3,0,0,0,1,1,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1661', '7.33446']
Arrow polynomial of the knot is: 8*K1**3 - 16*K1**2 - 8*K1*K2 - 2*K1 + 8*K2 + 2*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.249', '6.968', '6.1661', '6.1665', '6.2068']
Outer characteristic polynomial of the knot is: t^7+24t^5+48t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1661']
2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 2048*K1**4*K2**2 + 5120*K1**4*K2 - 7168*K1**4 + 2176*K1**3*K2*K3 - 960*K1**3*K3 - 1152*K1**2*K2**4 + 4864*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 15680*K1**2*K2**2 - 1536*K1**2*K2*K4 + 12528*K1**2*K2 - 768*K1**2*K3**2 - 192*K1**2*K4**2 - 1960*K1**2 + 2048*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 3008*K1*K2**2*K3 - 640*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9904*K1*K2*K3 - 64*K1*K2*K4*K5 + 1264*K1*K3*K4 + 192*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 4064*K2**4 - 64*K2**3*K6 - 1184*K2**2*K3**2 - 160*K2**2*K4**2 + 3120*K2**2*K4 - 2028*K2**2 + 688*K2*K3*K5 + 128*K2*K4*K6 - 1480*K3**2 - 560*K4**2 - 112*K5**2 - 12*K6**2 + 3574
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1661']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.396', 'vk6.428', 'vk6.836', 'vk6.877', 'vk6.1585', 'vk6.2032', 'vk6.2062', 'vk6.2705', 'vk6.2739', 'vk6.3147', 'vk6.13531', 'vk6.13722', 'vk6.19458', 'vk6.19753', 'vk6.25813', 'vk6.26631', 'vk6.37925', 'vk6.44915', 'vk6.53675', 'vk6.66250']
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 O1O2O3U2O4U1U4U3O5O6U5U6
R3 orbit {'O1O2O3U2O4U1U4U3O5O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4O5U1U6U3O6U2
Gauss code of K* O1O2O3U4U5O4O5U1U6U3O6U2
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 -2 -1 2 1 -1 1],[ 2 0 0 3 1 0 0],[ 1 0 0 1 0 0 0],[-2 -3 -1 0 0 0 0],[-1 -1 0 0 0 0 0],[ 1 0 0 0 0 0 1],[-1 0 0 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 0 -1 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 0 0],[ 1 0 0 1 0 0 0],[ 1 1 0 0 0 0 0],[ 2 3 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,0,0,1,3,0,0,0,1,1,0,0,0,0,0]
Phi over symmetry [-2,-1,-1,1,1,2,0,0,0,1,3,0,0,0,1,1,0,0,0,0,0]
Phi of -K [-2,-1,-1,1,1,2,1,1,2,3,1,0,2,1,3,2,2,2,0,1,1]
Phi of K* [-2,-1,-1,1,1,2,1,1,2,3,1,0,2,1,3,2,2,2,0,1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,0,0,1,3,0,0,0,1,1,0,0,0,0,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+24z+33
Enhanced Jones-Krushkal polynomial 4w^3z^2+24w^2z+33w
Inner characteristic polynomial t^6+12t^4+12t^2+1
Outer characteristic polynomial t^7+24t^5+48t^3+7t
Flat arrow polynomial 8*K1**3 - 16*K1**2 - 8*K1*K2 - 2*K1 + 8*K2 + 2*K3 + 9
2-strand cable arrow polynomial -1024*K1**6 - 2048*K1**4*K2**2 + 5120*K1**4*K2 - 7168*K1**4 + 2176*K1**3*K2*K3 - 960*K1**3*K3 - 1152*K1**2*K2**4 + 4864*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 15680*K1**2*K2**2 - 1536*K1**2*K2*K4 + 12528*K1**2*K2 - 768*K1**2*K3**2 - 192*K1**2*K4**2 - 1960*K1**2 + 2048*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 3008*K1*K2**2*K3 - 640*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9904*K1*K2*K3 - 64*K1*K2*K4*K5 + 1264*K1*K3*K4 + 192*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 4064*K2**4 - 64*K2**3*K6 - 1184*K2**2*K3**2 - 160*K2**2*K4**2 + 3120*K2**2*K4 - 2028*K2**2 + 688*K2*K3*K5 + 128*K2*K4*K6 - 1480*K3**2 - 560*K4**2 - 112*K5**2 - 12*K6**2 + 3574
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice True
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