Pensieve header: Demo of the free-Lie meta-group-action structure for http://www.math.toronto.edu/~drorbn/Talks/NhaTrang-1305/.

Get["http://drorbn.net/AcademicPensieve/2013-05/FreeLie.m"];
Get["http://drorbn.net/AcademicPensieve/2013-05/muCalculus.m"];

u=⟨"u"⟩; v=⟨"v"⟩; BCH[u, v]@{6}

w=⟨"w"⟩; Print /@ {BCH[BCH[u,v], w], BCH[u, BCH[v,w]]};

(Subscript[J, v])[BCH[u,v]]@{4}

Testing hm[x,y,z] // tha[u,z] ≡ tha[u,x] // tha[u,y] // hm[x,y,z]

Print /@ {
1-> (t1=M[{x -> MakeLieSeries[u+b[u,v]],y -> MakeLieSeries[v+(2/3)b[u,v]]},
 MakeCWSeries[CW["uu"]+CW["uvv"]]]),
2-> (t2=t1 // hm[x,y,z] // tha[u,z]),
3-> (t3=t1 // tha[u,x] // tha[u,y] // hm[x,y,z]),
4-> (t2≡t3)};

Demo 1 - The Knot (Subscript[8, 17])

μ1=(SuperMinus[R])[12,1](SuperMinus[R])[2,7](SuperMinus[R])[8,3](SuperMinus[R])[4,11](SuperPlus[R])[16,5](SuperPlus[R])[6,13](SuperPlus[R])[14,9](SuperPlus[R])[10,15]

Do[μ1 = μ1 // dm[1,k,1], {k,2,16}]; μ1[W]@{6}

Compare with the Alexander polynomial:

Series[Log[-(1/(X^3))+(4/(X^2))-(8/X)+11-8 X+4 (X^2)-(X^3) /. X->(E^x)], {x,0,6}]

Demo 2 - The Borromean Tangle

μ2= (SuperMinus[R])[r,6](SuperPlus[R])[2,4](SuperMinus[R])[g,9](SuperPlus[R])[5,7](SuperMinus[R])[b,3](SuperPlus[R])[8,1];
(Do[μ2 = μ2 // dm[r,k,r], {k,1,3}]; Do[μ2 = μ2 // dm[g,k,g], {k,4,6}];
Do[μ2 = μ2 // dm[b,k,b], {k,7,9}];  {μ2[r]@{4}, μ2[W]@{4}})