Pensieve header: The $k$-Logos in the quantized case; evolved from pensieve://Talks/Sydney-1708/nb/ExtraDetails.pdf.

Some Shortcuts

ME[x_] := MatrixExp[x]; MF[x_] := MatrixForm[x];

Representing ϵyax

(Borrowed from pensieve://2017-08/Multi-beta-yax.nb)

q=(E^(ℏ γ ϵ)); ρI=(({{1, 0}, {0, 1}})); ρy=(({{0, 0}, {-1, 0}})); ρa=(({{0, 0}, {0, -γ}}));ρx=(({{0, ((-1+(E^(-γ ϵ ℏ)))/ℏ)}, {0, 0}})); ρb=(({{ϵ, 0}, {0, 0}}));
ρt=Simplify[ϵ ρa-γ ρb]; ρA=ME[-ℏ ϵ ρa];  ρB=ME[-ℏ γ ρb];  ρT=ME[ℏ ρt];
(#->MF@Simplify@ToExpression@#)& /@ {"{{q}}", "ρy","ρa","ρx","ρb", "ρt", "ρA", "ρB", "ρT", "ρI"}

{ρa.ρx-ρx.ρa==γ ρx, ρx.ρA==q ρA.ρx, ρa.ρy-ρy.ρa==-γ ρy, ρb.ρy-ρy.ρb==-ϵ ρy, ρx.ρy-q ρy.ρx==(ρI-ρT.ρA.ρA)/ℏ} // Simplify

Likely, the PBW property for ϵyax follows from the local injectivity below; meaning, that from the matrix below one may recover (τ,η,α,ξ).

Normal@Series[ME[ℏ τ ρt].ME[ℏ η ρy].ME[ℏ α ρa].ME[ℏ ξ ρx], {ℏ,0,1}] // MF

Some ϵyax lemmas

Lemma 1. 𝕆((e^(α a+ξ x))|x a)=𝕆((e^(α a+(e^(-γ α))ξ x))|a x).
Lemma 2. 𝕆((e^(α a+η y))|a y)=𝕆((e^(α a+(e^(γ α))η y))|y a).

Proofs.

MF /@ {lhs=ME[ξ ρx].ME[α ρa], rhs=ME[α ρa].ME[(E^(-γ α))ξ ρx], lhs==rhs}

MF /@ {lhs=ME[α ρa].ME[η ρy], rhs=ME[(E^(-γ α))η ρy].ME[α ρa], lhs==rhs}

Lemma 3 at λ=0. 𝕆((e^(ξ x+η  y))|x y)=𝕆((Subscript[ω, 0])(e^((Subscript[η, 0])y+(Subscript[α, 0])a+(Subscript[ξ, 0])x))|y a x)=𝕆((e^((Subscript[τ, 0])t+(Subscript[η, 0])y+(Subscript[α, 0])a+(Subscript[ξ, 0])x))|t y a x), with {(Subscript[ω, 0])->((η ξ-(e^(-γ ϵ ℏ)) η ξ+ℏ)/ℏ), (Subscript[η, 0])->(((e^(γ ϵ ℏ)) η ℏ)/(-η ξ+(e^(γ ϵ ℏ)) (η ξ+ℏ))), (Subscript[α, 0])->-((2 (γ ϵ ℏ+Log[ℏ]-Log[-η ξ+(e^(γ ϵ ℏ)) (η ξ+ℏ)]))/γ), (Subscript[ξ, 0])->(((e^(γ ϵ ℏ)) ξ ℏ)/(-η ξ+(e^(γ ϵ ℏ)) (η ξ+ℏ))), (Subscript[τ, 0])->-((Log[((η ξ-(e^(-γ ϵ ℏ)) η ξ+ℏ)/ℏ)])/(γ ϵ))}.

Derivation.

Column[
sol=First@Solve[Thread[Flatten/@(ME[ξ ρx].ME[η ρy]==ω0 ME[η0 ρy].ME[α0 ρa].ME[ξ0 ρx])], {ω0,η0,α0,ξ0}] /. (v_->ℰ_) :> (v->FullSimplify@PowerExpand[ℰ]);
AppendTo[sol, τ0 -> Simplify[((Log[ω0 /. sol])/(-γ ϵ))]]
]

Lemma 3 at ϵ=0. With ν=(((1+δ t))^(-1)), we have 𝕆((e^(ξ x + η y))|x y)=𝕆(ν (e^(ν(δ x y - ξ η t + ξ x + η y)))|t y x).

Derivation.

MapAt[Limit[#, ϵ->0]&, sol, {All, 2}]

And so at ϵ=0, 𝕆((e^(ξ x + η y))|x y)=𝕆((e^(ξ x+η y - ξ η t))|t y x). Hence 𝕆((e^(ξ x + η y+δ x y))|x y)= (e^(δ (∂_ξ)(∂_η)))𝕆((e^(ξ x + η y))|x y)= (e^(δ (∂_ξ)(∂_η)))𝕆((e^(ξ x+η y - ξ η t))|t y x)= 𝕆(ψ|t y x), where ψ =  (e^(δ (∂_ξ)(∂_η)))(e^(ξ x+η y - ξ η t)) satisfies and is determined by (Subscript[ψ, δ=0])= (e^(ξ x+η y - ξ η t)) and (∂_δ)ψ=(∂_(ξ,η))ψ.

With[{ψ=ν (E^(ν(δ x y - ξ η t + ξ x + η y))) /. ν->(((1+δ t))^(-1))}, Simplify@{(∂_δ)ψ-(∂_(ξ,η))ψ, ψ /. δ->0}]

Aside.

MapAt[Series[#,{ϵ,0,2}]&, sol, {All, 2}]

MapAt[Simplify[#/.ℏ->1]&, sol, {All, 2}]//Column

A Lemma 3 for (Subscript[yax, k]):=ϵyax/((ϵ^(k+1))=0).

Lemma (Subscript[3, k]). 𝕆((e^(ξ x+η y+δ x y))|x y)=𝕆(ν (e^(ν(δ x y - ξ η t + ξ x + η y)))(Subscript[Λ, k])(ϵ,y,a,x,η,ξ,δ)|y a x), with ν=(((1+t δ))^(-1)) and where for any fixed k, (Subscript[Λ, k])(ϵ,y,a,x,η,ξ,δ) is a fixed polynomial of degree at most 4k in y,(Sqrt[a]),x,η,ξ, δ, with scalar coefficients.

Comment. And hence the (Subscript[yax, k]) invariant is computable in polynomial time.

Proof of Lemma (Subscript[3, k]). We know that 𝕆((e^(ξ x+η  y))|x y)=𝕆((e^((Subscript[τ, 0])t+(Subscript[η, 0])y+(Subscript[α, 0])a+(Subscript[ξ, 0])x))|y a x), with {(Subscript[η, 0])->(((e^(γ ϵ ℏ)) η ℏ)/(-η ξ+(e^(γ ϵ ℏ)) (η ξ+ℏ))), (Subscript[α, 0])->-((2 (γ ϵ ℏ+Log[ℏ]-Log[-η ξ+(e^(γ ϵ ℏ)) (η ξ+ℏ)]))/γ), (Subscript[ξ, 0])->(((e^(γ ϵ ℏ)) ξ ℏ)/(-η ξ+(e^(γ ϵ ℏ)) (η ξ+ℏ))), (Subscript[τ, 0])->-((Log[((η ξ-(e^(-γ ϵ ℏ)) η ξ+ℏ)/ℏ)])/(γ ϵ))}. Expanding in ϵ we get 𝕆((e^(ξ x+η  y))|x y)= 𝕆((Subscript[λ, ϵ])(η,ξ)𝕆((e^(ξ x+η y - ξ η t))|y a x) = 𝕆((Subscript[λ, ϵ])((∂_y),(∂_x))(e^(ξ x+η y - ξ η t))|y a x) and so 𝕆((e^(ξ x+η y+δ x y))|x y)=  𝕆((Subscript[λ, ϵ])((∂_y),(∂_x))(e^(δ(∂_ξ)(∂_η)))(e^(ξ x+η y - ξ η t))|y a x) = 𝕆(ν (Subscript[λ, ϵ])((∂_y),(∂_x))(e^(ν(δ x y - ξ η t + ξ x + η y)))|y a x).

(Subscript[DP, η_->(Subscript[D, y_]),ξ_->(Subscript[D, x_])])[P_][λ_]:=Total[CoefficientRules[P,{η,ξ}]/.({m_,n_}->c_):>c D[λ,{y,m},{x,n}]]

FullSimplify[(E^(-ξ x-η y+ξ η t+τ0 t+η0 y+α0 a+ξ0 x))/.sol]

Normal@Series[(E^(-ξ x-η y+ξ η t+τ0 t+η0 y+α0 a+ξ0 x))/.sol,{ϵ,0,2}]

Expand[(E^(-ν(δ x y - ξ η t + ξ x + η y)))(Subscript[DP, η->(Subscript[D, y]),ξ->(Subscript[D, x])])[Normal@Series[(E^(-ξ x-η y+ξ η t+τ0 t+η0 y+α0 a+ξ0 x))/.sol,{ϵ,0,1}]][(E^(ν(δ x y - ξ η t + ξ x + η y)))]]

(Subscript[Λ, k_]):=Collect[
(E^(-ν(δ x y - ξ η t + ξ x + η y)))(Subscript[DP, η->(Subscript[D, y]),ξ->(Subscript[D, x])])[Normal@Series[(E^(-ξ x-η y+ξ η t+τ0 t+η0 y+α0 a+ξ0 x))/.sol,{ϵ,0,k}]]
[(E^(ν(δ x y - ξ η t + ξ x + η y)))],
ϵ,Simplify];

(Subscript[Λ, 0])

(Subscript[Λ, 1])

(Subscript[Λ, 2])

Collect[(Subscript[Λ, 2]) /. ν->(((1+δ t))^(-1)) ,ϵ,Simplify]

Comment. Even better, log((Subscript[Λ, k])) is of degree at most 2k+2 in said variables.

The Main (Subscript[𝔤, k]) Theorem

The (Subscript[𝔤, k]) invariant of any S-component tangle T can be written in the form Z(T)=𝕆(ω (e^(L+Q+P))|(∏_(i∈S))(Subscript[e, i])(Subscript[l, i])(Subscript[f, i])), where ω is a scalar (meaning, a rational function in the variables (Subscript[h, i]) and their exponentials (Subscript[t, i])=(e^(Subscript[h, i]))), where L=∑(Subscript[a, ij])(Subscript[h, i])(Subscript[l, j]) is a balanced quadratic in the variables (Subscript[h, i]) and (Subscript[l, j]) with integer coefficients (Subscript[a, ij]), where Q=∑(Subscript[b, ij])(Subscript[e, i])(Subscript[f, j]) is a balanced quadratic in the variables (Subscript[e, i]) and (Subscript[f, j]) with scalar coefficients (Subscript[b, ij]), and where P is a polynomial in {ϵ, (Subscript[e, i]), (Subscript[l, i]), (Subscript[f, i])} (with scalar coefficients) whose (ϵ^d)-term is of degree at most 2d+2 in {(Subscript[e, i]),(Sqrt[(Subscript[l, i])]), (Subscript[f, i])}. Furthermore, after setting (Subscript[h, i])=h and (Subscript[t, i])=t for all i, the invariant Z(T) is poly-time computable.

Partial Proof. Indeed,
0. (R^±)=?, (n^±)=?.
1. 𝕆(𝒫(l,e)(e^(γ l+β e))|l e)=𝕆(𝒫((∂_γ),(∂_β))(e^(γ l+(e^γ)β e))|e l),
2. 𝕆(𝒫(l,f)(e^(γ l+β f))|f l)=𝕆(𝒫((∂_γ),(∂_β))(e^(γ l+(e^γ)β f))|l f),
3. 𝕆(𝒫(e,f)(e^(β e+α f+δ e f))|f e)=𝕆(ν 𝒫((∂_β),(∂_α))(e^(ν(-α β h+β e+α f+δ e f)))(Subscript[Λ, k])(ϵ,e,l,f,α,β,δ)|e f), with ν=(((1+hδ))^(-1)), and (Subscript[Λ, k])(ϵ,e,l,f,α,β,δ) as above.

Pushforwards of distributions, 0-dimensional QFT, Feynman diagrams and what had really happened here.