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Flat knot 6.202

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,2,4,3,3,1,2,2,2,1,1,2,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.202']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.91', '6.202', '6.210']
Outer characteristic polynomial of the knot is: t^7+97t^5+41t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.202']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1536*K1**4*K2**2 + 2752*K1**4*K2 - 2592*K1**4 - 128*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 928*K1**3*K3 - 512*K1**2*K2**4 + 3552*K1**2*K2**3 - 8400*K1**2*K2**2 - 1536*K1**2*K2*K4 + 7656*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K3*K5 - 160*K1**2*K4**2 - 4500*K1**2 + 1440*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 480*K1*K2**2*K5 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7608*K1*K2*K3 - 160*K1*K2*K4*K5 + 1552*K1*K3*K4 + 392*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 2568*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 1344*K2**2*K3**2 - 32*K2**2*K3*K7 - 808*K2**2*K4**2 + 2952*K2**2*K4 - 160*K2**2*K5**2 - 8*K2**2*K6**2 - 2956*K2**2 - 32*K2*K3**2*K4 + 944*K2*K3*K5 + 296*K2*K4*K6 + 8*K2*K5*K7 - 1844*K3**2 - 980*K4**2 - 224*K5**2 - 20*K6**2 + 3866
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.202']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16334', 'vk6.16375', 'vk6.18075', 'vk6.18411', 'vk6.22665', 'vk6.22746', 'vk6.24522', 'vk6.24941', 'vk6.34609', 'vk6.34686', 'vk6.36659', 'vk6.37082', 'vk6.42304', 'vk6.42333', 'vk6.43937', 'vk6.44254', 'vk6.54597', 'vk6.54636', 'vk6.55895', 'vk6.56182', 'vk6.59076', 'vk6.59119', 'vk6.60419', 'vk6.60777', 'vk6.64628', 'vk6.64666', 'vk6.65529', 'vk6.65844', 'vk6.67987', 'vk6.68009', 'vk6.68611', 'vk6.68828']
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 O1O2O3O4O5U2O6U3U4U6U1U5
R3 orbit {'O1O2O3O4O5U2O6U3U4U6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U5U6U2U3O6U4
Gauss code of K* O1O2O3O4O5U4U6U1U2U5O6U3
Gauss code of -K* O1O2O3O4O5U3O6U1U4U5U6U2
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 -3 -2 0 4 2],[ 1 0 -2 -1 1 4 2],[ 3 2 0 1 2 3 2],[ 2 1 -1 0 1 3 2],[ 0 -1 -2 -1 0 2 1],[-4 -4 -3 -3 -2 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 4 2 0 -1 -2 -3],[-4 0 0 -2 -4 -3 -3],[-2 0 0 -1 -2 -2 -2],[ 0 2 1 0 -1 -1 -2],[ 1 4 2 1 0 -1 -2],[ 2 3 2 1 1 0 -1],[ 3 3 2 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,1,2,3,0,2,4,3,3,1,2,2,2,1,1,2,1,2,1]
Phi over symmetry [-4,-2,0,1,2,3,0,2,4,3,3,1,2,2,2,1,1,2,1,2,1]
Phi of -K [-3,-2,-1,0,2,4,0,0,1,3,4,0,1,2,3,0,1,1,1,2,2]
Phi of K* [-4,-2,0,1,2,3,2,2,1,3,4,1,1,2,3,0,1,1,0,0,0]
Phi of -K* [-3,-2,-1,0,2,4,1,2,2,2,3,1,1,2,3,1,2,4,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+63t^4+17t^2+1
Outer characteristic polynomial t^7+97t^5+41t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1536*K1**4*K2**2 + 2752*K1**4*K2 - 2592*K1**4 - 128*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 928*K1**3*K3 - 512*K1**2*K2**4 + 3552*K1**2*K2**3 - 8400*K1**2*K2**2 - 1536*K1**2*K2*K4 + 7656*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K3*K5 - 160*K1**2*K4**2 - 4500*K1**2 + 1440*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 480*K1*K2**2*K5 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7608*K1*K2*K3 - 160*K1*K2*K4*K5 + 1552*K1*K3*K4 + 392*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 2568*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 1344*K2**2*K3**2 - 32*K2**2*K3*K7 - 808*K2**2*K4**2 + 2952*K2**2*K4 - 160*K2**2*K5**2 - 8*K2**2*K6**2 - 2956*K2**2 - 32*K2*K3**2*K4 + 944*K2*K3*K5 + 296*K2*K4*K6 + 8*K2*K5*K7 - 1844*K3**2 - 980*K4**2 - 224*K5**2 - 20*K6**2 + 3866
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
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