Pensieve Header: Cheat sheat β verification program, continues pensieve://2014-04/, continued pensieve://2014-06/. Seems to be the first occurrence of Γ-calculus.

Program

<< KnotTheory`

βSimplify=Simplify;
SetAttributes[βCollect, Listable];
βCollect[B[ω_, σ_, μ_]] := B[
βSimplify[ω], σ,
Collect[μ, _h, Collect[#, _t, βSimplify]&]
];
hL[b_] := Union[Cases[b, h[s_] :> s, Infinity]];
tL[b_] := Union[Cases[b, t[s_]|(Subscript[T, s_]) :> s, Infinity]];
dL[b_] := Union[hL[b], tL[b]];
σ_ ⊢h_ := ((∂_h)σ /. 0->1);
B[ω_, σ_, μ_]@A := Module[
{tails, heads},
tails = tL[B[ω, σ, μ]]; heads = hL[B[ω, σ, μ]];
Outer[βSimplify[(∂_(t[#1],h[#2]))μ]&, tails, heads]
];
βForm[B[ω_, σ_, μ_]] := Module[
{tails, heads, mat},
tails = tL[B[ω, σ, μ]]; heads = hL[B[ω, σ, μ]];
mat = Outer[βSimplify[(∂_(h[#1],t[#2]))μ]&, heads, tails];
PrependTo[mat, t /@ tails];
mat = Join[
{Prepend[h /@ heads, ω]},
Transpose[mat],
{Prepend[(σ⊢h[#])& /@ heads, "1+Σ/ω"]}
];
MatrixForm[mat]
];
βForm[else_] := else /. b_B :> βForm[b];
Format[b_B, StandardForm] := βForm[b];
B /: B[ω1_, σ1_, μ1_] == B[ω2_, σ2_, μ2_] := Module[
{heads, tails},
tails = tL[{B[ω1, σ1, μ1], B[ω2, σ2, μ2]}];
heads = hL[{B[ω1, σ1, μ1], B[ω2, σ2, μ2]}];
(ω1 == ω2) && (σ1==σ2) && (
And @@ Flatten[Outer[
(Coefficient[μ1, t[#1]h[#2]] == Coefficient[μ2, t[#1]h[#2]])&,
tails, heads
]]
)
];

B /: B[ω1_, σ1_, μ1_]B[ω2_, σ2_, μ2_] := B[ω1*ω2, σ1+σ2, ω2 μ1+ω1 μ2];
tm[x_, y_, z_][b_] := b /. {t[x]->t[z], t[y]->t[z], (Subscript[T, x])->(Subscript[T, z]), (Subscript[T, y])->(Subscript[T, z])};
hm[x_, y_, z_][B[ω_, σ_, μ_]] := B[ω, 
h[z](σ⊢h[x])(σ⊢h[y])+(σ /. h[x]|h[y] -> 0),
h[z](D[μ, h[x]] + (σ⊢h[x])(∂_(h[y]))μ) + (μ /. h[x]|h[y] -> 0)
] // βCollect;
swapth[y_,x_][B[ω_, σ_, μ_]] := Module[
{α, β, γ, δ},
(({{α, β}, {γ, δ}}))=(({{Coefficient[μ, t[y]h[x]], D[μ, t[y]] /. h[x] -> 0}, {D[μ, h[x]] /. t[y] -> 0, μ /. h[x]|t[y] -> 0}}));
B[ω+α, σ, {(σ⊢h[x])t[y], 1}.(({{α, β}, {γ, ((ω+α)δ-γ*β)/ω}})).{h[x], 1}] // βCollect
];
dm0[x_,y_,z_][b_] := b // swapth[x,y] // hm[x,y,z] // tm[x,y,z];
dm[a_,b_,c_][B[ω0_, σ_, μ_]] := Module[
{ω,α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ,σa,σb},
ω=ω0 /. {(Subscript[T, a])->(Subscript[T, c]), (Subscript[T, b])->(Subscript[T, c])};
{σa,σb} = {σ⊢h[a], σ⊢h[b]} /. {(Subscript[T, a])->(Subscript[T, c]), (Subscript[T, b])->(Subscript[T, c])};
(({{α, β, θ}, {γ, δ, ϵ}, {ϕ, ψ, Ξ}}))=(({{(∂_(t[a],h[a]))μ, (∂_(t[a],h[b]))μ, (∂_(t[a]))μ /. h[a]|h[b] -> 0}, {(∂_(t[b],h[a]))μ, (∂_(t[b],h[b]))μ, (∂_(t[b]))μ /. h[a]|h[b] -> 0}, {(∂_(h[a]))μ /. t[a]|t[b] -> 0, (∂_(h[b]))μ /. t[a]|t[b] -> 0, μ /. t[a]|t[b]|h[a]|h[b] -> 0}})) /. {(Subscript[T, a])->(Subscript[T, c]), (Subscript[T, b])->(Subscript[T, c])};
B[ω+β,
h[c]σa σb+(σ /. h[a]|h[b] -> 0 /. {(Subscript[T, a])->(Subscript[T, c]), (Subscript[T, b])->(Subscript[T, c])}),
{t[c],1}.(({{γ+σa δ+σb(α+σa β)+((β γ - α δ)/ω), ϵ+σb θ+((β ϵ-δ θ)/ω)}, {ϕ+σa ψ+((β ϕ-α ψ)/ω), Ξ+((β Ξ-ψ θ)/ω)}})).{h[c],1}
] // βCollect
]

Unprotect[NonCommutativeMultiply];
b1_B ** b2_B := Module[
{ρ, σ, labels},
ρ = b1*(b2 /. {h[s_] :> h[σ[s]], t[s_] :> t[σ[s]], (Subscript[T, s_]) :> (Subscript[T, σ[s]])});
labels = dL[{b1,b2}];
Do[ρ = ρ // dm[s, σ[s], s], {s, labels}];
ρ
];

βbRp[x_, y_] := B[1, (Subscript[T, x])h[y], ((Subscript[T, x])-1)*t[x]h[y]];
βbRm[x_, y_] := B[1, h[y]/(Subscript[T, x]), (1/(Subscript[T, x])-1)*t[x]h[y]];

βbZ[L_] := Module[{s, Z, c, k},
s=Skeleton[L];
Z=Times@@PD[L] /. X[i_,j_,k_,l_] :> If[PositiveQ[X[i,j,k,l]], βbRp[l,i], βbRm[j,i]];
Do[Z = Z // dm[s[[c,1]], s[[c,k]], s[[c,1]]], {c, Length[s]},{k, 2, Length[s[[c]]]}];
Z]

R3 for β-better

{βbRp[1,2], βbRm[1,2]}

βbRp[1,2]**βbRp[1,3]

{βbRp[1,2]**βbRp[1,3]**βbRp[2,3], βbRp[2,3]**βbRp[1,3]**βbRp[1,2]}

dm for β-better

{B0=B[ω, {(Subscript[σ, a]),(Subscript[σ, b]),σ}.{h@a, h@b, h@S},{t@a,t@b,t@S}.(({{α, β, θ}, {γ, δ, ϵ}, {ϕ, ψ, Ξ}})).{h@a, h@b, h@S}],
B0 // dm[a,b,c]}

(B0 // swapth[a,b] // hm[a,b,c] // tm[a,b,c])==(B0 // dm[a,b,c]) // Simplify

(({{β+ω, h[c], h[S]}, {t[c], γ+((β γ-α δ)/ω)+δ (Subscript[σ, a])+(α+β (Subscript[σ, a])) (Subscript[σ, b]), ((β ϵ-δ θ+ϵ ω+θ ω (Subscript[σ, b]))/ω)}, {t[S], ((β ϕ-α ψ+ϕ ω+ψ ω (Subscript[σ, a]))/ω), ((β Ξ-θ ψ+Ξ ω)/ω)}, {"1+Σ/ω", (Subscript[σ, a]) (Subscript[σ, b]), σ}})) // FullSimplify // MatrixForm

Back to β (and β-Burau)

B0=B[ω, {(Subscript[σ, a]),(Subscript[σ, b]),σ}.{h@a, h@b, h@S},{t@a,t@b,t@S}.(({{α, β, θ}, {γ, δ, ϵ}, {ϕ, ψ, Ξ}})).{h@a, h@b, h@S}]

hm[a,b,c][B0]@A // MatrixForm

hm and swapth

((ω^(-1))hm[a,b,c][B0]@A /. Thread[{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}->ω{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}]) // FullSimplify // MatrixForm

((((ω+α))^(-1))swapth[a,a][B0]@A /. Thread[{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}->ω{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}]) // FullSimplify // MatrixForm

(1+α)((((ω+α))^(-1))swapth[a,a][B0]@A /. Thread[{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}->ω{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}]) // FullSimplify // MatrixForm

dm

((((ω+β))^(-1))(({{γ+((β γ-α δ)/ω)+δ (Subscript[σ, a])+(α+β (Subscript[σ, a])) (Subscript[σ, b]), ((β ϵ-δ θ+ϵ ω+θ ω (Subscript[σ, b]))/ω)}, {((β ϕ-α ψ+ϕ ω+ψ ω (Subscript[σ, a]))/ω), ((β Ξ-θ ψ+Ξ ω)/ω)}})) /. Thread[{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}->ω{α,β,γ,δ,θ,ϵ,ϕ,ψ,Ξ}]) // FullSimplify // MatrixForm

(({{((γ+β γ-α δ+α (Subscript[σ, b])+(Subscript[σ, a]) (δ+β (Subscript[σ, b])))/(1+β)), ((ϵ+β ϵ-δ θ+θ (Subscript[σ, b]))/(1+β))}, {((ϕ+β ϕ-α ψ+ψ (Subscript[σ, a]))/(1+β)), Ξ-((θ ψ)/(1+β))}}))-(({{γ, ϵ}, {ϕ, Ξ}})) // FullSimplify // MatrixForm

Plus[(({{((γ+β γ-α δ+α (Subscript[σ, b])+(Subscript[σ, a]) (δ+β (Subscript[σ, b])))/(1+β)), ((ϵ+β ϵ-δ θ+θ (Subscript[σ, b]))/(1+β))}, {((ϕ+β ϕ-α ψ+ψ (Subscript[σ, a]))/(1+β)), Ξ-((θ ψ)/(1+β))}})) /. {α->α+(Subscript[σ, a]), δ->δ+(Subscript[σ, b]), Ξ->Ξ+σ},(({{-(Subscript[σ, a]) (Subscript[σ, b]), 0}, {0, -σ}}))] // FullSimplify // MatrixForm

(1+β)(({{γ-((α δ)/(1+β)), ϵ-((δ θ)/(1+β))}, {ϕ-((α ψ)/(1+β)), Ξ-((θ ψ)/(1+β))}})) // FullSimplify // MatrixForm

Plus[(({{((γ+β γ-α δ+α (Subscript[σ, b])+(Subscript[σ, a]) (δ+β (Subscript[σ, b])))/(1+β)), ((ϵ+β ϵ-δ θ+θ (Subscript[σ, b]))/(1+β))}, {((ϕ+β ϕ-α ψ+ψ (Subscript[σ, a]))/(1+β)), Ξ-((θ ψ)/(1+β))}}))/. {α->α+(Subscript[σ, a])-1, δ->δ+(Subscript[σ, b])-1, Ξ->Ξ+σ-1},(({{1-(Subscript[σ, a]) (Subscript[σ, b]), 0}, {0, 1-σ}}))] // FullSimplify // MatrixForm

Plus[(({{((γ+β γ-α δ+α (Subscript[σ, b])+(Subscript[σ, a]) (δ+β (Subscript[σ, b])))/(1+β)), ((ϵ+β ϵ-δ θ+θ (Subscript[σ, b]))/(1+β))}, {((ϕ+β ϕ-α ψ+ψ (Subscript[σ, a]))/(1+β)), Ξ-((θ ψ)/(1+β))}})) /. {α->α-1, δ->δ-1, Ξ->Ξ-1},(({{1, 0}, {0, 1}}))] // FullSimplify // MatrixForm

Alexander with β-better

{Knot[8,17] // βbZ, Alexander[Knot[8,17]][(Subscript[T, 1])] // βSimplify}

The MVA with β-better

βbMVA[L_Link] := Module[{Hs, ω, σ, μ, A, M},
{ω,σ,μ} = List @@ βbZ[L];
Hs = Rest[h /@ (First /@ Skeleton[L])];
A=Outer[Coefficient[μ, #1*#2]&,Hs, Hs /. h[a_] :> t[a]];
M=A-ω DiagonalMatrix[((σ⊢#)-1)& /@ Hs];
Factor[(((ω^(2-Length@Skeleton@L)) Det[M])/(1-(Subscript[T, Skeleton[L][[1,1]]])))]
]

Link["L6a4"] // βbZ

βbMVA[Link["L6a4"]]

Factor[(((MultivariableAlexander[#][T] /. T[i_] :> (Subscript[T, Skeleton[#][[i,1]]])))/(βbMVA[#]))]&  /@ AllLinks[{2,8}]

Burau Calculus

Plus[
(({{β+ω, h[c], h[S]}, {t[c], γ+((β γ-α δ)/ω)+δ (Subscript[σ, a])+(α+β (Subscript[σ, a])) (Subscript[σ, b]), ((β ϵ-δ θ+ϵ ω+θ ω (Subscript[σ, b]))/ω)}, {t[S], ((β ϕ-α ψ+ϕ ω+ψ ω (Subscript[σ, a]))/ω), ((β Ξ-θ ψ+Ξ ω)/ω)}, {"1+Σ/ω", (Subscript[σ, a]) (Subscript[σ, b]), σ}})) /. {α->α+ω (Subscript[σ, a]), δ->δ+ω (Subscript[σ, b]), Ξ->Ξ+ω σ},
-(({{0, 0, 0}, {0, ω (Subscript[σ, a]) (Subscript[σ, b]), 0}, {0, 0, ω σ}, {0, 0, 0}}))
] // FullSimplify // MatrixForm

Plus[
(({{β+ω, h[c], h[S]}, {t[c], γ+((β γ-α δ)/ω)+δ (Subscript[σ, a])+(α+β (Subscript[σ, a])) (Subscript[σ, b]), ((β ϵ-δ θ+ϵ ω+θ ω (Subscript[σ, b]))/ω)}, {t[S], ((β ϕ-α ψ+ϕ ω+ψ ω (Subscript[σ, a]))/ω), ((β Ξ-θ ψ+Ξ ω)/ω)}, {"1+Σ/ω", (Subscript[σ, a]) (Subscript[σ, b]), σ}})) /. {α->α+ω((Subscript[σ, a])-1), δ->δ+ω((Subscript[σ, b])-1), Ξ->Ξ+ω(σ-1)},
-(({{0, 0, 0}, {0, ω((Subscript[σ, a]) (Subscript[σ, b])-1), 0}, {0, 0, ω(σ-1)}, {0, 0, 0}}))
] // FullSimplify // MatrixForm

Plus[
(({{β+ω, h[c], h[S]}, {t[c], γ+((β γ-α δ)/ω)+δ (Subscript[σ, a])+(α+β (Subscript[σ, a])) (Subscript[σ, b]), ((β ϵ-δ θ+ϵ ω+θ ω (Subscript[σ, b]))/ω)}, {t[S], ((β ϕ-α ψ+ϕ ω+ψ ω (Subscript[σ, a]))/ω), ((β Ξ-θ ψ+Ξ ω)/ω)}, {"1+Σ/ω", (Subscript[σ, a]) (Subscript[σ, b]), σ}})) /. {α->α-ω, δ->δ-ω, Ξ->Ξ-ω},
+(({{0, 0, 0}, {0, ω, 0}, {0, 0, ω}, {0, 0, 0}}))
] // FullSimplify // MatrixForm