Pensieve Header: An attempt on the ribbon property using Γ-calculus; continues pensieve://2014-05/.

dir=SetDirectory["C:/drorbn/AcademicPensieve/2014-06/"];
<< KnotTheory`
<< MetaCalculi/MetaCalculi-Program.m
Format[(Subscript[α, a_,b_]), StandardForm]:= Interpretation[(Subscript[α, 10a+b]), (Subscript[α, a,b])];
Format[(Subscript[β, a_,b_]), StandardForm]:= Interpretation[(Subscript[β, 10a+b]), (Subscript[β, a,b])];

{n=4; γ0=Γ[ω, (∑_(a=0)^n)(Subscript[h, a])(Subscript[σ, a]), (∑_(a=1)^n)(∑_(b=1)^n)(Subscript[t, a])(Subscript[h, b])(Subscript[α, a,b])], γ1= γ0 // dS[2] // dS[4]}

{Ov = Xp[o1,1]Xp[o2,2]Xp[o3,3]Xp[o4,4] // Γ // dm[o1,o2,o] // dm[o,o3,o] // dm[o,o4,o],
t1=Ov**(γ0*Γ[ϵ[o]]), t2=(γ0*Γ[ϵ[o]])**Ov, ocond=Simplify[t1==t2]}

ocond=FullSimplify[ocond /. {(Subscript[T, o])-> 0}]

{U=Xm[1,u1]Xm[2,u2]Xm[3,u3]Xm[4,u4] // Γ // dm[u1,u2,u]//dm[u,u3,u]//dm[u,u4,u],
t1=U**(γ0*Γ[ϵ[u]]), t2=(γ0*Γ[ϵ[u]])**U, ucond=FullSimplify[t1==t2]}

{cert = γ1 // dm[1,2,1] // dm[3,4,2], eqns=(ϵ[1]ϵ[2] // Γ)==(cert /. (Subscript[σ, _])->1) // Simplify}

γ1 // dm[2,1,1] // dm[4,3,3]

alex= FullSimplify[(γ1 // dm[4,3,3] // dm[3,2,2] // dm[2,1,1])@ω]

{FullSimplify[alex, eqns], FullSimplify[cert@ω], FullSimplify[alex, eqns&&ucond&&ocond]}

Ω[n_] :=  Ω[n] = Table[Which[i<j, 0,i==j, 1,i>j, 1-T], {i,n}, {j,n}];
Ω[3] // MatrixForm

Simplify[
And@@Thread[Flatten[Ω[4]]==Flatten[Transpose[γ0[A]].Ω[4].(γ0[A]/.α ->β)]]
]

Simplify[And[
And@@Thread[Flatten[Ω[4]]==Flatten[Transpose[γ0[A]].Ω[4].(γ0[A]/.α ->β)]],
And@@Thread[Flatten[Ω[4]/.T->1/T]==Flatten[Transpose[(γ0[A]/.α ->β)].(Ω[4]/.T->1/T).γ0[A]]]
]]