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Flat knot 6.1196

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,2,3,0,2,-1,1,1,0,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1196']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.324', '6.672', '6.953', '6.1196', '6.1215', '6.1216', '6.1699']
Outer characteristic polynomial of the knot is: t^7+39t^5+83t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1196']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 512*K1**4*K2 - 1376*K1**4 + 256*K1**3*K2*K3 - 128*K1**3*K3 + 512*K1**2*K2**5 - 2560*K1**2*K2**4 + 5568*K1**2*K2**3 - 11264*K1**2*K2**2 - 512*K1**2*K2*K4 + 8400*K1**2*K2 - 96*K1**2*K3**2 - 4080*K1**2 - 512*K1*K2**4*K3 + 3104*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2336*K1*K2**2*K3 - 320*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6512*K1*K2*K3 + 312*K1*K3*K4 + 24*K1*K4*K5 - 864*K2**6 + 1056*K2**4*K4 - 5136*K2**4 - 32*K2**3*K6 - 992*K2**2*K3**2 - 384*K2**2*K4**2 + 3280*K2**2*K4 - 710*K2**2 + 256*K2*K3*K5 + 56*K2*K4*K6 - 1024*K3**2 - 480*K4**2 - 32*K5**2 - 2*K6**2 + 3214
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1196']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70511', 'vk6.70513', 'vk6.70515', 'vk6.70517', 'vk6.70576', 'vk6.70580', 'vk6.70584', 'vk6.70588', 'vk6.70731', 'vk6.70735', 'vk6.70739', 'vk6.70743', 'vk6.70830', 'vk6.70832', 'vk6.71084', 'vk6.71088', 'vk6.71092', 'vk6.71096', 'vk6.71217', 'vk6.71221', 'vk6.71225', 'vk6.71229', 'vk6.71288', 'vk6.71290', 'vk6.74727', 'vk6.74730', 'vk6.76242', 'vk6.76246', 'vk6.76258', 'vk6.76262', 'vk6.89176', 'vk6.89179']
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 O1O2O3O4U2U3U1O5O6U4U5U6
R3 orbit {'O1O2O3O4U2U3U1O5O6U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1O5O6U4U2U3
Gauss code of K* O1O2O3U4U5O6O4O5U3U1U2U6
Gauss code of -K* O1O2O3U1U2O4O5O6U3U5U6U4
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 0 2],[ 1 0 -1 1 3 1 1],[ 2 1 0 1 2 1 1],[ 0 -1 -1 0 1 1 1],[-1 -3 -2 -1 0 1 2],[ 0 -1 -1 -1 -1 0 1],[-2 -1 -1 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -1 -1 -1],[-1 2 0 1 -1 -3 -2],[ 0 1 -1 0 -1 -1 -1],[ 0 1 1 1 0 -1 -1],[ 1 1 3 1 1 0 -1],[ 2 1 2 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,1,1,1,1,-1,1,3,2,1,1,1,1,1,1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,2,3,0,2,-1,1,1,0,1,0,1,0]
Phi of -K [-2,-1,0,0,1,2,0,1,1,1,3,0,0,-1,2,-1,0,1,2,1,-1]
Phi of K* [-2,-1,0,0,1,2,-1,1,1,2,3,0,2,-1,1,1,0,1,0,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,1,2,1,1,1,3,1,-1,-1,1,1,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial t^6+29t^4+25t^2+1
Outer characteristic polynomial t^7+39t^5+83t^3+9t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -256*K1**4*K2**2 + 512*K1**4*K2 - 1376*K1**4 + 256*K1**3*K2*K3 - 128*K1**3*K3 + 512*K1**2*K2**5 - 2560*K1**2*K2**4 + 5568*K1**2*K2**3 - 11264*K1**2*K2**2 - 512*K1**2*K2*K4 + 8400*K1**2*K2 - 96*K1**2*K3**2 - 4080*K1**2 - 512*K1*K2**4*K3 + 3104*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2336*K1*K2**2*K3 - 320*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6512*K1*K2*K3 + 312*K1*K3*K4 + 24*K1*K4*K5 - 864*K2**6 + 1056*K2**4*K4 - 5136*K2**4 - 32*K2**3*K6 - 992*K2**2*K3**2 - 384*K2**2*K4**2 + 3280*K2**2*K4 - 710*K2**2 + 256*K2*K3*K5 + 56*K2*K4*K6 - 1024*K3**2 - 480*K4**2 - 32*K5**2 - 2*K6**2 + 3214
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {1, 4}, {3}]]
If K is slice False
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