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Flat knot 6.134

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,1,2,1,1,1,1,0,1,0,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.134', '7.10424']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1*K2 - 4*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.134', '6.409', '6.424', '6.534', '6.942', '6.969', '6.1192', '6.1280', '6.1310', '6.1325', '6.1858', '6.1925']
Outer characteristic polynomial of the knot is: t^7+41t^5+50t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.134']
2-strand cable arrow polynomial of the knot is: -384*K1**2*K2**4 + 256*K1**2*K2**3 - 992*K1**2*K2**2 + 480*K1**2*K2 - 160*K1**2 + 384*K1*K2**3*K3 + 608*K1*K2*K3 - 192*K2**6 + 192*K2**4*K4 - 448*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 256*K2**2*K4 + 112*K2**2 - 96*K3**2 - 48*K4**2 + 174
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.134']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.327', 'vk6.366', 'vk6.723', 'vk6.772', 'vk6.1458', 'vk6.1516', 'vk6.1823', 'vk6.1958', 'vk6.1996', 'vk6.2270', 'vk6.2459', 'vk6.2512', 'vk6.2661', 'vk6.2776', 'vk6.2997', 'vk6.3121', 'vk6.17651', 'vk6.17659', 'vk6.18737', 'vk6.24209', 'vk6.24860', 'vk6.25321', 'vk6.36467', 'vk6.37056', 'vk6.39748', 'vk6.42008', 'vk6.43566', 'vk6.44209', 'vk6.46310', 'vk6.47887', 'vk6.60255', 'vk6.68540']
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 O1O2O3O4O5O6U4U5U6U3U1U2
R3 orbit {'O1O2O3O4O5O6U4U5U6U3U1U2', 'O1O2O3O4O5U3O6U5U4U6U1U2', 'O1O2O3O4O5U3U4U5U6U1O6U2'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5O6U5U6U4U1U2U3
Gauss code of K* O1O2O3O4O5O6U5U6U4U1U2U3
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 1 0 -2 0 2],[ 1 0 1 0 -2 0 2],[-1 -1 0 0 -2 0 2],[ 0 0 0 0 -2 0 2],[ 2 2 2 2 0 1 2],[ 0 0 0 0 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -2 -2 -2],[-1 2 0 0 0 -1 -2],[ 0 1 0 0 0 0 -1],[ 0 2 0 0 0 0 -2],[ 1 2 1 0 0 0 -2],[ 2 2 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,1,2,2,2,0,0,1,2,0,0,1,0,2,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,1,2,1,1,1,1,0,1,0,1,1,-1]
Phi of -K [-2,-1,0,0,1,2,-1,0,1,1,2,1,1,1,1,0,1,0,1,1,-1]
Phi of K* [-2,-1,0,0,1,2,-1,0,1,1,2,1,1,1,1,0,1,0,1,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,2,2,2,0,0,1,2,0,0,1,0,2,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial -z-1
Enhanced Jones-Krushkal polynomial -10w^3z+9w^2z-w
Inner characteristic polynomial t^6+31t^4+14t^2
Outer characteristic polynomial t^7+41t^5+50t^3
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -384*K1**2*K2**4 + 256*K1**2*K2**3 - 992*K1**2*K2**2 + 480*K1**2*K2 - 160*K1**2 + 384*K1*K2**3*K3 + 608*K1*K2*K3 - 192*K2**6 + 192*K2**4*K4 - 448*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 256*K2**2*K4 + 112*K2**2 - 96*K3**2 - 48*K4**2 + 174
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}]]
If K is slice True
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