Pensieve header: To be run after executing JExperiments-Program.nb.

Testing tm

{{u=⟨"u"⟩, v=⟨"v"⟩, w=⟨"w"⟩};
t1= M[{
x->MakeLieSeries[u+v+w],
y->MakeLieSeries[b[u,v]+b[v,w]]
}, MakeCWSeries[CW["uvw"]]
],
t1 // tm[u,v,u],
t2=t1 // tm[u,v,u]// tm[u,w,u],
t3=t1 // tm[v,w,v] // tm[u,v,u],
t2≡t3
} // ColumnForm

Testing hm

{t1=(SuperPlus[R])[u,x]⋃(SuperPlus[R])[v,y]⋃(SuperPlus[R])[w,z],
t1 // hm[x,y,x],
t2=t1 // hm[x,y,x] // hm[x,z,x],
t1 // hm[y,z,y],
t3=t1 // hm[y,z,y] // hm[x,y,x],
t2≡t3
} // ColumnForm

Testing C and RC

$SeriesShowDegree=6;
{{u=⟨"u"⟩, w=⟨"w"⟩},
λ0= MakeLieSeries[u+⟨"uw"⟩],
t1= MakeLieSeries[⟨"uv"⟩],
t2 = t1 // LieMorphism[u -> Ad[-λ0][u]],
t3 = t2 // RC[u, λ0],
t1 ≡ t3
} // ColumnForm

Testing the RC “composition property”

$SeriesShowDegree=4;
{{u=⟨"u"⟩, v=⟨"v"⟩, w=⟨"w"⟩},
λ=MakeLieSeries[u],
λx=MakeLieSeries[v],
λy=MakeLieSeries[w],
BCH[λx, λy],
t1 = λ // RC[u, BCH[λx, λy]],
t2 =  λ // RC[u, λx] // RC[u, λy // RC[u, λx]],
t1≡t2
} // ColumnForm

$SeriesShowDegree=4;
$SeriesCompareDegree = 6;
{{u=⟨"u"⟩, v=⟨"v"⟩, w=⟨"w"⟩},
λ=MakeLieSeries[u],
λx=MakeLieSeries[v+b[u,v]+b[u,b[u,v]]],
λy=MakeLieSeries[w+b[u,b[v,w]]],
BCH[λx, λy],
t1 = λ // RC[u, BCH[λx, λy]],
t2 =  λ // RC[u, λx] // RC[u, λy // RC[u, λx]],
t1≡t2
} // ColumnForm

Testing the Conjugation Relation

$SeriesShowDegree=3;
{
t1=(SuperPlus[R])[u,x]⋃(SuperPlus[R])[v,y]⋃(SuperPlus[R])[w,z],
t2=t1//tm[v,w,v],
t3=t2//hm[x,y,x],
t4=t3//tha[u,z],
t5=(SuperPlus[R])[v,x]⋃(SuperPlus[R])[w,z]⋃(SuperPlus[R])[u,y],
t6=t5 // tm[v,w,v],
t7=t6 // hm[x,y,x],
t7≡t4
} // ColumnForm

Testing JA

$SeriesShowDegree=3;
$SeriesCompareDegree = 6;
{
{u=⟨"u"⟩, v=⟨"v"⟩, λ0= MakeLieSeries[u+v+b[u,v]]},
t1 = JA[u, (s+ϵ)λ0],
t2 = JA[u, s λ0] // RC[u, ϵ λ0 // RC[u, s λ0]],
t3 = JA[u, ϵ λ0 // RC[u, s λ0]],
t1 ≡ t2+t3
} // ColumnForm

Testing J

$SeriesShowDegree=6;
{{u=⟨"u"⟩, v=⟨"v"⟩, w=⟨"w"⟩},
λx=MakeLieSeries[v+b[u,v]+b[u,b[u,v]]],
div[u,λ // RC[u, s λx]] // LieMorphism[u -> Ad[-s λx][u]]
}

$SeriesShowDegree=6;
$SeriesCompareDegree = 6;
{{t=⟨"t"⟩, u=⟨"u"⟩, v=⟨"v"⟩, w=⟨"w"⟩, uvw=LieMorphism[u->w, v->w]},
λx=MakeLieSeries[v+b[u,v]+b[u,b[t,v]]],
t1 = J[w, λx // uvw] // RC[w, λx // uvw],
t2 = J[u, λx] // RC[u, λx] // RC[v, λx // RC[u, λx]] // uvw,
t3 = J[v, λx // RC[u, λx]] //  RC[v, λx // RC[u, λx]] // uvw,
t1 ≡ t2+t3
} // ColumnForm

$SeriesShowDegree=3;
$SeriesCompareDegree = 6;
{{u=⟨"u"⟩, v=⟨"v"⟩, w=⟨"w"⟩},
λx=MakeLieSeries[v+b[u,v]+b[u,b[u,v]]],
λy=MakeLieSeries[w+b[u,b[v,w]]],
t1 = J[u, BCH[λx, λy]] // RC[u, λx],
t2 = J[u, λx] // RC[u, λx],
t3 = J[u, λy // RC[u, λx]],
t1 ≡ t2+t3
} // ColumnForm

Testing the relationship between J and JA

$SeriesShowDegree=3;
$SeriesCompareDegree = 7;
{{u=⟨"u"⟩, v=⟨"v"⟩},
λ0=MakeLieSeries[v+b[u,v]+b[u,b[u,v]]],
t1 = J[u, λ0],
t2 = JA[u, λ0] // LieMorphism[u -> Ad[-λ0][u]],
t1≡t2
} // ColumnForm

Computing (Subscript[8, 17]).

μ0=(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]

μ0 // dm[1,2,1]

{
μ0=(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[μ0 = μ0 // dm[1,k,1], {k,2,16}]; μ0
} // ColumnForm

μ0[[2]][{6}]

Print /@ {β=-((1-4 X+8 (X^2)-11 (X^3)+8 (X^4)-4 (X^5)+(X^6))/(X^3))/. X->(E^x),
Series[β , {x,0,8}],
Series[Log[β] , {x,0,9}]
};

The Borromean Link

(* {r=⟨r⟩, g=⟨g⟩, b=⟨b⟩}; *)
μ0= (SuperMinus[R])[r,6]⋃(SuperPlus[R])[2,4]⋃(SuperMinus[R])[g,9]⋃(SuperPlus[R])[5,7]⋃(SuperMinus[R])[b,3]⋃(SuperPlus[R])[8,1]

(1/. μ0[[1]])[1] // FullForm

μ0 // dm[r,1,r]

Do[μ0 = μ0 // dm[r,k,r], {k,1,3}];
Do[μ0 = μ0 // dm[g,k,g], {k,4,6}];
Do[μ0 = μ0 // dm[b,k,b], {k,7,9}];
μ0

μ1 = Sort /@ ((LieMorphism[⟨r⟩->⟨g⟩, ⟨g⟩->⟨b⟩, ⟨b⟩->⟨r⟩] /@ μ0) /. Thread[{r,g,b}->{g,b,r}])

μ2 = Sort /@ ((LieMorphism[⟨r⟩->⟨g⟩, ⟨g⟩->⟨r⟩] /@ μ0) /. Thread[{r,g}->{g,r}])

$SeriesShowDegree=4; μ0

$SeriesShowDegree=4; μ1

$SeriesShowDegree=4; μ2

$SeriesShowDegree=5; μ0

(r/.First[μ0])[{5}]

$SeriesShowDegree=6;  Last[μ0]

$SeriesShowDegree=7; r /. First[μ0]

$SeriesShowDegree=7;Last[μ0]