Pensieve header: Testing the expansion extension property for the 4-strand braid group.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2015-02"];
<< ../Projects/WKO4/FreeLie.m

n=4;
Clear[s,cs];
Do[
Do[cs[i][k,j]=0, {j,1,n-1}, {k, j+1,n}];
cs[i][i+1,i]=1/2;
s[i]=DKS[n,cs[i]],
{i,1,n-1}
]

s[3]

SeriesSolve[{s[1],s[2],s[3]},
s[1]**(s[2]//σ[2,1,3,4])**(s[1]//σ[2,3,1,4]) ≡ s[2]**(s[1]//σ[1,3,2,4])**(s[2]//σ[3,1,2,4])
&& s[2]**(s[3]//σ[1,3,2,4])**(s[2]//σ[1,3,4,2]) ≡ s[3]**(s[2]//σ[1,2,4,3])**(s[3]//σ[1,4,2,3])
&& s[1]**(s[3]//σ[2,1,3,4]) ≡ s[3]**(s[1]//σ[1,2,4,3])
]

{s[1],s[2],s[3]}

s[1]@{6}