Pensieve Header: The depth-mirror property in Gassner calculus.

dir=SetDirectory["C:/drorbn/AcademicPensieve/2014-06/"];
<< KnotTheory`
<< "MetaCalculi/MetaCalculi-Program.m"
ΓSimp=Factor;

V // A

V1=V //A /. (Subscript[c, a_]):> -(Subscript[c, a])

V//Γ

V1//Γ

V1==(V//A//dA[All]//dS[All]) // FullSimplify  (* result is mess for Simplify, did not finish for FullSimplify *)

The Garside Element

G[1] = ϵ[1] // Γ;
G[n_] /; n>1 := G[n]= (G[n-1] // qΔ[n-1,n-1,n])**(Xp[n-1,n] // Γ);
Gi[n_] := Gi[n] = ((G[n])^(-1));

{G[1], G[2], Xp[1,2]//Γ}

{G[3], Xp[1,2]**Xp[1,3]**Xp[2,3]//Γ, G[3]**Gi[3]}

{G[4], G[5]}

{Gi[4], Gi[5]}

{Xp[1,2]ϵ[3]//Γ,G[3]**(Xp[2,1]//Γ)**(Gi[3]//dσ[2,1])}

{Xp[2,3]ϵ[1]//Γ,G[3]**(Xp[3,2]//Γ)**(Gi[3]//dσ[1,3,2])}

{t1=Xp[1,2]**Xm[3,1]**Xp[2,3]**Xm[1,2]**Xp[3,1]**Xm[2,3] // Γ,
t2=G[3]**(t3=Xp[2,1]**Xm[1,3]**Xp[3,2]**Xm[2,1]**Xp[1,3]**Xm[3,2] // Γ)**Gi[3],
t1==t2 // Simplify} // ColumnForm

(t1//dA[All])**t2

t1

t3

t3 // dS[All] // dA[All]

Thread[((t1[A]//Transpose//Flatten) /. (Subscript[T, _])->T)==Flatten[(t3 // dS[All])[A] /. (Subscript[T, _])->T]] // Simplify

(t3 // dS[All])[A]

{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])], G[4]**γ0**Gi[4]==γ0 // Simplify}

t1=V//Γ//dA[All]//dS[All]

t2=(V//Γ) /. (Subscript[T, a_]):>1/(Subscript[T, a])

t1==t2 // Simplify

(t2^(-1))**(V//Γ)  // ΓCollect[FullSimplify]

t3=V//A//dA[All]//dS[All]

t4=(V//A) /. (Subscript[c, a_]):>-(Subscript[c, a])

(t4//dA[All])**(V//A)  // ACollect

t3==t4 // Simplify