Pensieve header: Computations related to the weight system of pA.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2010-08"];
<< pA.m

"Points" P

pA[P[i_,j_]] := AHD[
t[i]==t[j], {i}, W[j],
W[j]-W[i]
]

Ptest1=pA[CircuitDiagram[Xm[i,j,k,l], P[l,m]]]

Ptest2=pA[CircuitDiagram[Xm[i,j,k,m]]]

Last[Ptest1]==Last[Ptest2]

The "SemiVirtual" SVXp

diff1=pA[Xp[j,k,l,i]]-pA[CircuitDiagram[P[i,k],P[j,l]]]

diff2= Expand[diff1 /. AHD[_,_,W[k,l], p_] :> p]

diff3 = Expand[diff2 /. t[i_] :> 1+x[i]]

pA[SVXp[i_,j_,k_,l_]] := AHD[
(t[i]==t[k])(t[j]==t[l]), {i,j}, W[k,l],
-W[i,k]+t[j] W[i,k]+W[i,l]-t[i] W[i,l]-W[k,l]+t[i] W[k,l]
]

The "Blob" B

pA[CircuitDiagram[SVXp[i,l,k,j],P[k,l]]]

pA[CircuitDiagram[SVXp[l,i,j,k],P[k,l]]]

pA[B[i_, j_]] := AHD[
(t[i]==t[j]), {i}, W[j], W[i]-t[i] W[i]-W[j]+t[i] W[j]
]

The Archibald Relation

ArchL1=pA[CircuitDiagram[SVXp[i,k,l,p],SVXp[j,p,m,n]]]

ArchL2 = pA[CircuitDiagram[SVXp[j,k,m,p],SVXp[i,p,l,n]]]

ArchL = Expand[
(ArchL1-ArchL2) /. AHD[_,_,W[l,m,n], p_] :> p /. t[i_] :> (1+x[i])
]

ArchR1=pA[CircuitDiagram[SVXp[i,k,l,n],B[j,m]]]

ArchR2=pA[CircuitDiagram[SVXp[j,k,m,n],B[i,l]]]

ArchR = Expand[
(-ArchR1+ArchR2) /. AHD[_,_,W[l,m,n], p_] :> p /. t[i_] :> (1+x[i])
]

ArchL==ArchR

Horizontal Arrow Diagrams

pW[n_, had_HAD] := Module[
{l = Length[had], tail, head, tops=Range[n]},
cd=CircuitDiagram @@ Table[
{tail, head}= IntegerDigits[had[[i]]];
If[tail==head,
B[tops[[head]], tops[[head]]+=10],
SVXp[tops[[tail]], tops[[head]], tops[[tail]]+=10, tops[[head]]+=10]
],
{i,l}
];
Print[cd];
Expand[Signature[tops]*Last[pA[cd]] /. t[i_] :> 1+x[i]] /. Thread[tops -> 10+Range[n]]
];
pW[n_, expr_] := expr /. had_HAD :> pW[n, had]

pW[5, HAD[11,21,34,22]]

pW[2,HAD[11, 22]]

Simplify[pW[2,HAD[12]]]

Simplify[pW[2,HAD[12, 12]]]

Simplify[pW[2,HAD[12, 12, 12]]]

Simplify[pW[2,HAD[12, 12, 12, 12]]]

Simplify[pW[2,HAD[12, 21]]]

Simplify[pW[2,HAD[21,12]]]

4T and OC

Print /@ (FourT = pW[3, {HAD[12,23], HAD[13,23], HAD[23,12], HAD[23,13]}]);

Expand[{1,1,-1,-1}.FourT]

FourT2 = pW[3, {HAD[21,23], HAD[31,23], HAD[23,21], HAD[23,31]}]

{1,1,-1,-1}.FourT2

OC=pW[3, {HAD[12,13], HAD[13, 12]}]

{1,-1}.OC

No-vertex relations?

pW[6, HAD[14,25,36]]

pW[6, Total[
(Signature[#]*HAD@@({10,20,30}+#))&  /@ Permutations[{4,5,6}]
]]

pW[6, Total[
(HAD@@({10,20,30}+#))&  /@ Permutations[{4,5,6}]
]]

(P[1234]-P[2134]-P[1243]+P[2143])-(P[3412]-P[4312]-P[3421]+P[4321]) /. P[p_] :> pW[8, HAD@@({50,60,70,80}+IntegerDigits[p])]