{K=Knot[3,1], d=2}

{Cs,φ}=Rot[K]

n=Length[Cs]

A=IdentityMatrix[2n+1]

Cases[Cs,{s_,i_,j_}:>(A[[{i,j},{i+1,j+1}]]+=(({{-(T^s), (T^s)-1}, {0, -1}})))]

G=Inverse[A]

ρd1=(Subscript[PP, Mold])@Exp[Total[Cases[Cs,{s_,i_,j_}:>Sum[(ϵ^d1)(Subscript[r, d1,s])[i,j],{d1,d}]]]+ Sum[(ϵ^d1)(Subscript[γ, d1,φ[[k]]])[k], {k,2n},{d1,d}]]

ρd2=(Subscript[PP, ExpandedMold])[
Expand[F[{},{}]×Normal@Series[ρd1,{ϵ,0,d}]]//.F[fs_,{es___}]×(((f:(Subscript[(r|γ), ps__])[is__]))^(p_.)):>F[Join[fs,Table[f,p]],DeleteDuplicates@{es,is}]
]

gPair[fs_,w_]:=gPair[fs,w]=(
Print["Running gPair on",{fs,w}];
(Subscript[Zip, Join@@Table[{(Subscript[p, α]),(Subscript[(OverBar[p]), α]),(Subscript[x, α]),(Subscript[(OverBar[x]), α])},{α,w}]])[(Times@@(V/@fs))Exp[Sum[(Subscript[g, α,β])((Subscript[π, α])+(Subscript[(OverBar[π]), α]))((Subscript[ξ, β])+(Subscript[(OverBar[ξ]), β])),{α,w},{β,w}]-Sum[(Subscript[(OverBar[ξ]), α])(Subscript[π, α]),{α,w}]]]
)

ρd3=(Subscript[PP, Pairing])[
ρd2/.F[fs_,es_]:>(gPair[
Replace[fs,Thread[es->Range@Length@es],{2}],
Length@es
]/.(Subscript[g, α_,β_]):>(Subscript[g, es[[α]],es[[β]]]))
]

?gPair

{K=Knot[3,1], d=1}

(Subscript[PP, "Green"])[
{Cs,φ}=Rot[K]; n=Length[Cs];
A=IdentityMatrix[2n+1];
Cases[Cs,{s_,i_,j_}:>(A[[{i,j},{i+1,j+1}]]+=(({{-(T^s), (T^s)-1}, {0, -1}})))];
G=Inverse[A];
]

ρd1=(Subscript[PP, Mold])@Exp[Total[Cases[Cs,{s_,i_,j_}:>Sum[(ϵ^d1)(Subscript[r, d1,s])[i,j],{d1,d}]]]+ Sum[(ϵ^d1)(Subscript[γ, d1,φ[[k]]])[k], {k,2n},{d1,d}]]

ρd2=(Subscript[PP, ExpandedMold])[
Expand[F[{},{}]×Normal@Series[ρd1,{ϵ,0,d}]]//.F[fs_,{es___}]×(((f:(Subscript[(r|γ), ps__])[is__]))^(p_.)):>F[Join[fs,Table[f,p]],DeleteDuplicates@{es,is}]
]

ρd3=Echo@(Subscript[PP, Pairing])[
ρd2/.F[fs_,es_]:>(gPair[
Replace[fs,Thread[es->Range@Length@es],{2}],
Length@es
]/.(Subscript[g, α_,β_]):>(Subscript[g, es[[α]],es[[β]]]))
]

{Δ,ρd3/.ϵ->ϵ (Δ^2)}