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Flat knot 6.820

Min(phi) over symmetries of the knot is: [-3,-3,1,1,1,3,-1,1,3,4,4,0,1,3,2,0,-1,0,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.820']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 6*K1*K2 - 3*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.205', '6.660', '6.775', '6.820']
Outer characteristic polynomial of the knot is: t^7+93t^5+146t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.820']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 768*K1**4*K2 - 1536*K1**4 + 768*K1**3*K2*K3 - 512*K1**3*K3 - 1408*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3648*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9600*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 512*K1**2*K2*K4 + 8928*K1**2*K2 - 320*K1**2*K3**2 - 5032*K1**2 + 2176*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 128*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7088*K1*K2*K3 + 240*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 2512*K2**4 - 1024*K2**2*K3**2 - 56*K2**2*K4**2 + 1400*K2**2*K4 - 2246*K2**2 + 304*K2*K3*K5 + 8*K2*K4*K6 - 1344*K3**2 - 72*K4**2 - 8*K5**2 - 2*K6**2 + 3478
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.820']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11608', 'vk6.11961', 'vk6.12950', 'vk6.13259', 'vk6.20423', 'vk6.21788', 'vk6.27779', 'vk6.29299', 'vk6.31411', 'vk6.32585', 'vk6.32962', 'vk6.39203', 'vk6.41425', 'vk6.47552', 'vk6.53199', 'vk6.53510', 'vk6.57292', 'vk6.61966', 'vk6.64288', 'vk6.64498']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4U1O5U6U5U3U4O6U2
R3 orbit {'O1O2O3O4U1O5U6U5U3U4O6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3O5U1U2U6U5O6U4
Gauss code of K* Same
Gauss code of -K* O1O2O3O4U3O5U1U2U6U5O6U4
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 -3 1 1 3 1 -3],[ 3 0 3 1 2 0 1],[-1 -3 0 0 2 1 -4],[-1 -1 0 0 1 0 -3],[-3 -2 -2 -1 0 0 -4],[-1 0 -1 0 0 0 -1],[ 3 -1 4 3 4 1 0]]
Primitive based matrix [[ 0 3 1 1 1 -3 -3],[-3 0 0 -1 -2 -2 -4],[-1 0 0 0 -1 0 -1],[-1 1 0 0 0 -1 -3],[-1 2 1 0 0 -3 -4],[ 3 2 0 1 3 0 1],[ 3 4 1 3 4 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,3,3,0,1,2,2,4,0,1,0,1,0,1,3,3,4,-1]
Phi over symmetry [-3,-3,1,1,1,3,-1,1,3,4,4,0,1,3,2,0,-1,0,0,1,2]
Phi of -K [-3,-3,1,1,1,3,-1,1,3,4,4,0,1,3,2,0,-1,0,0,1,2]
Phi of K* [-3,-1,-1,-1,3,3,0,1,2,2,4,0,1,0,1,0,1,3,3,4,-1]
Phi of -K* [-3,-3,1,1,1,3,-1,1,3,4,4,0,1,3,2,0,-1,0,0,1,2]
Symmetry type of based matrix +
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+63t^4+72t^2+4
Outer characteristic polynomial t^7+93t^5+146t^3+16t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 6*K1*K2 - 3*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -512*K1**4*K2**2 + 768*K1**4*K2 - 1536*K1**4 + 768*K1**3*K2*K3 - 512*K1**3*K3 - 1408*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3648*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9600*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 512*K1**2*K2*K4 + 8928*K1**2*K2 - 320*K1**2*K3**2 - 5032*K1**2 + 2176*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 128*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7088*K1*K2*K3 + 240*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 2512*K2**4 - 1024*K2**2*K3**2 - 56*K2**2*K4**2 + 1400*K2**2*K4 - 2246*K2**2 + 304*K2*K3*K5 + 8*K2*K4*K6 - 1344*K3**2 - 72*K4**2 - 8*K5**2 - 2*K6**2 + 3478
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}]]
If K is slice False
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