Pensieve header: Searching for the EK-type equation for V.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2012-05\\beta5.1"];
<< betaCalculus.m
Clear[ℏ]; Unprotect[C];
$PerturbativeDegree=8;
βSimplify[expr_]:= Replace[
Series[Normal[expr], {ℏ, 0, $PerturbativeDegree}],sd_SeriesData :> MapAt[Expand, sd, 3]
];
βCollect[B[ω_, μ_]] := B[
βSimplify[ω],
βSimplify[μ]
];
{V, C, sol} = Get["SolutionToDegree8-120524.m"];
{V,C} = {
βCollect[
B[ω[ℏ (Subscript[c, 1]),ℏ (Subscript[c, 2])],α[ℏ (Subscript[c, 1]),ℏ (Subscript[c, 2])]t[1]h[1]+β[ℏ (Subscript[c, 1]),ℏ (Subscript[c, 2])]t[1]h[2]+γ[ℏ (Subscript[c, 1]),ℏ (Subscript[c, 2])]t[2]h[1]+δ[ℏ (Subscript[c, 1]),ℏ (Subscript[c, 2])]t[2]h[2]]
] /. {
(ϵ:(α|β|γ|δ|ω|κ))[___] :> (Subscript[ϵ, 0]), Derivative[k___][(ϵ:(α|β|γ|δ|ω|κ))][___] :> (Subscript[ϵ, FromDigits[{k}]])
},
βCollect[B[κ[ℏ (Subscript[c, 1])],0]] /. {
(ϵ:(α|β|γ|δ|ω|κ))[___] :> (Subscript[ϵ, 0]), Derivative[k___][(ϵ:(α|β|γ|δ|ω|κ))][___] :> (Subscript[ϵ, FromDigits[{k}]])
}
} /. sol

{
"R4" -> R[2,3]**R[1,3]**V==V**(R[1,3] // dΔ[1,1,2]),
"TwistEq" -> V**Θ[1,2]==R[1,2]**(V // dP[2,1]),
"Unitarity" -> V**(V // dA[1] // dA[2]) == B[1,0],
"VerticalFlipEquation" -> V**(V // dS[1] // dS[2]) == R[1,2],
"CapEquation" -> (V**(C // dP[12])  // dcap[1] // dcap[2]) == (C*(C // dP[2]) // dcap[1] // dcap[2]),
"SidesNonDegeneracy" -> (V // dη[1])==B[1,0] && (V // dη[2])==B[1,0],
"CapsAndCups" -> Simplify[C ==(C//dS[1])],
"OverhandEquation" -> (Φ[V] // dΔ[1,0,1] // dS[2] // dS[3] // dm[0,3,0] // dm[1,2,1])==B[1,0]
}

C[[1]]

Log[C[[1]]] // Simplify

{V,C}={V,C}/.(Subscript[κ, 1])->0;

Simplify[C ==(C//dS[1])]

Simplify[C ==(C//dA[1])]

C

Log[C[[1]]] /. (Subscript[c, 1])->2 // Simplify

Cup equation

Simplify[
(((C//dS[1]//dP[12])**(V//Inverse))//hη[1]//hη[2]) == (C//dS[1])(C//dS[1]//dP[2])
]

Top cap left punctured.

ek1 = V*(C//dP[3]) // tη[1] // dm[2,3,2] // dS[2] // hm[1,2,2] // dP[2->0]

Right cup left punctured.

ek2 = V*(C//dP[3]//dS[3]) // dm[3,2,2] // hη[2] // tη[1] // dm[1,2,0]

Right cup top punctured.

ek3 = V*(C//dP[3]//dS[3]) // dm[3,2,2] // hη[2] // dS[1] // dm[2,1,0]

Top cap right punctured.

ek4=V*(C//dP[3]) // tη[2] // dm[1,3,1] // dS[1] // dm[2,1,0]

Left cup right punctured.

ek5 = V*(C//dP[3]//dS[3]) // dm[3,1,1] // hη[1] // tη[2] // dm[2,1,0]

Left cup top punctured.

ek6 = V*(C//dP[3]//dS[3]) // dm[3,1,1] // hη[1] // dS[2] // dm[1,2,0]

Identifications

ek2*ek3

(ek1*ek1 // Inverse)

Series[(Sqrt[((Sinh[ℏ/2])/(ℏ/2))]), {ℏ,0,8}]

Log[ek5[[1]]]

Log[ek6[[1]]]