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Flat knot 6.1007

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-2,0,2,3,3,1,2,2,1,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1007']
Arrow polynomial of the knot is: -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.372', '6.930', '6.1007', '6.1701', '6.1714', '6.1760', '6.1788']
Outer characteristic polynomial of the knot is: t^7+37t^5+52t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1007']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 128*K1**4*K2**2 + 1024*K1**4*K2 - 3120*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1568*K1**3*K3 + 64*K1**2*K2**3 - 2384*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 8760*K1**2*K2 - 1136*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 6728*K1**2 - 800*K1*K2**2*K3 + 32*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 6968*K1*K2*K3 - 32*K1*K3**2*K5 + 2008*K1*K3*K4 + 248*K1*K4*K5 + 8*K1*K5*K6 - 104*K2**4 - 272*K2**2*K3**2 - 64*K2**2*K4**2 + 840*K2**2*K4 - 5256*K2**2 + 440*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 2904*K3**2 - 934*K4**2 - 192*K5**2 - 32*K6**2 + 5484
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1007']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4779', 'vk6.4791', 'vk6.5116', 'vk6.5128', 'vk6.6348', 'vk6.6778', 'vk6.6790', 'vk6.8298', 'vk6.8318', 'vk6.8750', 'vk6.9672', 'vk6.9692', 'vk6.9983', 'vk6.10003', 'vk6.21005', 'vk6.21022', 'vk6.22429', 'vk6.22444', 'vk6.28461', 'vk6.40225', 'vk6.40242', 'vk6.42156', 'vk6.46727', 'vk6.46744', 'vk6.48803', 'vk6.49020', 'vk6.49032', 'vk6.49840', 'vk6.49852', 'vk6.51499', 'vk6.58968', 'vk6.69802']
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 O1O2O3O4U2U4O5U3O6U1U6U5
R3 orbit {'O1O2O3O4U2U4O5U3O6U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U4O6U2O5U1U3
Gauss code of K* O1O2O3U1U4U5U6O4O6U3O5U2
Gauss code of -K* O1O2O3U2O4U1O5O6U5U4U6U3
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 1 2 1],[ 2 0 -2 1 1 3 1],[ 2 2 0 2 1 1 0],[ 0 -1 -2 0 0 1 0],[-1 -1 -1 0 0 0 0],[-2 -3 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 0 0 -1 -1 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 0 -2 -1],[ 2 1 0 1 2 0 2],[ 2 3 1 1 1 -2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,0,0,1,1,3,0,0,0,1,0,1,1,2,1,-2]
Phi over symmetry [-2,-2,0,1,1,2,-2,0,2,3,3,1,2,2,1,1,1,1,0,1,1]
Phi of -K [-2,-2,0,1,1,2,-2,0,2,3,3,1,2,2,1,1,1,1,0,1,1]
Phi of K* [-2,-1,-1,0,2,2,1,1,1,1,3,0,1,2,2,1,2,3,1,0,-2]
Phi of -K* [-2,-2,0,1,1,2,-2,1,1,1,3,2,0,1,1,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 19z+39
Enhanced Jones-Krushkal polynomial 19w^2z+39w
Inner characteristic polynomial t^6+23t^4+23t^2+1
Outer characteristic polynomial t^7+37t^5+52t^3+6t
Flat arrow polynomial -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
2-strand cable arrow polynomial -256*K1**6 - 128*K1**4*K2**2 + 1024*K1**4*K2 - 3120*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1568*K1**3*K3 + 64*K1**2*K2**3 - 2384*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 8760*K1**2*K2 - 1136*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 6728*K1**2 - 800*K1*K2**2*K3 + 32*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 6968*K1*K2*K3 - 32*K1*K3**2*K5 + 2008*K1*K3*K4 + 248*K1*K4*K5 + 8*K1*K5*K6 - 104*K2**4 - 272*K2**2*K3**2 - 64*K2**2*K4**2 + 840*K2**2*K4 - 5256*K2**2 + 440*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 2904*K3**2 - 934*K4**2 - 192*K5**2 - 32*K6**2 + 5484
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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