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Flat knot 6.561

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,0,0,2,1,1,1,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.561']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 8*K1*K2 - 2*K1 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.561', '6.1296']
Outer characteristic polynomial of the knot is: t^7+51t^5+53t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.561']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 256*K1**4*K2**3 - 960*K1**4*K2**2 + 2304*K1**4*K2 - 3728*K1**4 - 512*K1**3*K2**2*K3 + 1184*K1**3*K2*K3 - 1088*K1**3*K3 - 640*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3200*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 11104*K1**2*K2**2 - 1472*K1**2*K2*K4 + 11488*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 5856*K1**2 + 2208*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 - 512*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9128*K1*K2*K3 + 1032*K1*K3*K4 + 104*K1*K4*K5 - 64*K2**6 + 480*K2**4*K4 - 2824*K2**4 - 64*K2**3*K6 - 992*K2**2*K3**2 - 480*K2**2*K4**2 + 2392*K2**2*K4 - 3716*K2**2 + 344*K2*K3*K5 + 96*K2*K4*K6 - 2004*K3**2 - 626*K4**2 - 44*K5**2 - 4*K6**2 + 4808
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.561']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4702', 'vk6.5005', 'vk6.6196', 'vk6.6667', 'vk6.8189', 'vk6.8607', 'vk6.9567', 'vk6.9906', 'vk6.17386', 'vk6.20923', 'vk6.20980', 'vk6.22335', 'vk6.22402', 'vk6.23557', 'vk6.23894', 'vk6.28399', 'vk6.36146', 'vk6.40061', 'vk6.40173', 'vk6.42114', 'vk6.43061', 'vk6.43365', 'vk6.46589', 'vk6.46678', 'vk6.48734', 'vk6.49530', 'vk6.49733', 'vk6.51430', 'vk6.55536', 'vk6.58923', 'vk6.65282', 'vk6.69767']
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 O1O2O3O4O5U4U2U3O6U5U1U6
R3 orbit {'O1O2O3O4O5U4U2U3O6U5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U1O6U3U4U2
Gauss code of K* O1O2O3U4O5O6O4U6U2U3U1U5
Gauss code of -K* O1O2O3U1O4O5O6U3U6U4U5U2
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 -1 -2 0 -1 2 2],[ 1 0 -2 0 -1 3 2],[ 2 2 0 1 0 3 1],[ 0 0 -1 0 0 2 1],[ 1 1 0 0 0 1 1],[-2 -3 -3 -2 -1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 1 -2 -1 -3 -3],[-2 -1 0 -1 -1 -2 -1],[ 0 2 1 0 0 0 -1],[ 1 1 1 0 0 1 0],[ 1 3 2 0 -1 0 -2],[ 2 3 1 1 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,-1,2,1,3,3,1,1,2,1,0,0,1,-1,0,2]
Phi over symmetry [-2,-2,0,1,1,2,-1,1,1,2,3,0,0,2,1,1,1,1,-1,-1,1]
Phi of -K [-2,-1,-1,0,2,2,-1,1,1,1,3,1,1,0,1,1,2,2,0,1,-1]
Phi of K* [-2,-2,0,1,1,2,-1,1,1,2,3,0,0,2,1,1,1,1,-1,-1,1]
Phi of -K* [-2,-1,-1,0,2,2,0,2,1,1,3,1,0,1,1,0,2,3,1,2,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+37t^4+22t^2+1
Outer characteristic polynomial t^7+51t^5+53t^3+11t
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 8*K1*K2 - 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -256*K1**6 + 256*K1**4*K2**3 - 960*K1**4*K2**2 + 2304*K1**4*K2 - 3728*K1**4 - 512*K1**3*K2**2*K3 + 1184*K1**3*K2*K3 - 1088*K1**3*K3 - 640*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3200*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 11104*K1**2*K2**2 - 1472*K1**2*K2*K4 + 11488*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 5856*K1**2 + 2208*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 - 512*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9128*K1*K2*K3 + 1032*K1*K3*K4 + 104*K1*K4*K5 - 64*K2**6 + 480*K2**4*K4 - 2824*K2**4 - 64*K2**3*K6 - 992*K2**2*K3**2 - 480*K2**2*K4**2 + 2392*K2**2*K4 - 3716*K2**2 + 344*K2*K3*K5 + 96*K2*K4*K6 - 2004*K3**2 - 626*K4**2 - 44*K5**2 - 4*K6**2 + 4808
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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