Pensieve header: Formula testing for the future double-tree paper; continues pensieve://2013-10/.

Double-Tree Formula Testing

SetDirectory["C:\\drorbn\\AcademicPensieve\\2013-11\\DoubleTree"];
<< "../../2013-05/FreeLie.m"
<< "muCalculus.m"
<< "../../2013-10/WKOSolution8-1.m"
α=MakeLieSeries[{"1","2"},αs]; β=MakeLieSeries[{"1","2"},βs];
γ=MakeCWSeries[{"1","2"},γs];
V=M[{1->α, 2->β}, γ];
κs[d_, 1] := If[OddQ[d], 0, κs[d]]; κ=MakeCWSeries[{"1"},κs];
Unprotect[C]; C=M[{1->MakeLieSeries[0]}, κ];
Ci=M[{1->MakeLieSeries[0]}, -κ];
$SeriesShowDegree=5; $SeriesCompareDegree = 6;
{V,C // dc[1]}

Vb=(C*(C//dP[2]))**V**(Ci//dP[12]);

General Equations

The Hard R4 Equation

(SuperPlus[R])[2,3]**(SuperPlus[R])[1,3]**V ≡ V**((SuperPlus[R])[1,2] // dP[12,3])

The Twist Equation

V**Θ[1,2] ≡ (SuperPlus[R])[1,2]**(V // dP[2,1])

The Non-Degeneracy Equations

{(V // dη[1])≡dϵ[2], (V // dη[2])≡dϵ[1]}

Cup / Cap

C≡(C//dS[1])

The Asosciator Φ

Φ=Module[{Vi=V // dA[1,2]},
(Vi//dP[12,3])**Vi**(V//dP[2,3])**(V//dP[1,23])]

The Pentagon

Φ**(Φ//dP[1,23,4])**(Φ//dP[2,3,4])≡(Φ//dP[12,3,4])**(Φ//dP[1,2,34])

TBD

Horizontal Flip for Φ

Φ**(Φ//dP[3,2,1]) ≡ dϵ[1,2,3]

Vertical Flip for Φ

Φ**(Φ//dS[1,2,3]) ≡ dϵ[1,2,3]

The Overhand Equation

(Φ // dP[{{0,1},{2},{3}}] // dS[2,3] // dm[0,3,0] // dm[1,2,1]) ≡ dϵ[0,1]

The definition of ν

ν=Φ // dS[2] // dm[3,2,1,1]

C and ν

C**C**C**C**ν ≡ dϵ[1]

V with left cup and right puncture

((V // hη[1] // tη[2])dϵ[1] // dm[2,1,1])≡R[1,1, 1/2]

V with right cup and left puncture

((V//hη[2])hϵ[2]//tη[1] // dm[1,2,1])≡ dϵ[1]

((Vb//hη[2])hϵ[2]//tη[1] // dm[1,2,1])≡ dϵ[1]

V//tη[1]

V//tη[2]

TBD

V with top cap and left puncture

(V // tη[1] // dS[2] // hm[1,2,1])≡ dϵ[1]

V // tη[1]

{(V // tη[1])[1]@{8}, (V // tη[1])[2]@{8}, (V // tη[1])[W]@{8}}

The Buckle

ZB=(Φ//dP[4,2,13])**(Φ//dP[1,3,2])**Θ[3,2]**(Φ//dP[3,2,1])**(Φ//dP[12,3,4]);

Buckle 2 V

VB = (ZB // tη[1] // hη[2])hϵ[2] // dm[1,2,1];
VB = (VB// tη[3] // hη[4])hϵ[4] // dm[3,4,2]

Equations for Unitary V

TBD

The Unitarity Equation

V**(V // dA[1] // dA[2]) ≡ dϵ[1]⋃dϵ[2]

The Vertical Flip Equation

V**(V // dS[1] // dS[2]) ≡ (SuperPlus[R])[1,2]

The Cap Equation

(V**(C // dP[12])  // dc[1] // dc[2]) ≡ (C⋃(C//dP[2])// dc[1] // dc[2])

TBD

TBD

TBD

The Bucke Equation V~VB

C**(C//dP[2])**V ≡ VB**(C//dP[12])

V**(C**(C//dP[2])) ≡ VB**(C//dP[12])

$SeriesShowDegree=6;
{VB // dS[1] // dm[1,2,1] // dc[1], C*C*C*C//dc[1]}

$SeriesShowDegree=8;
{V // dS[1] // dm[1,2,1] // dc[1], C*C//dc[1]}

C//dP[12] // dS[1] // dm[1,2,1] // dc[1]

Equations with Simple Buckle

TBD

The Unitarity Equation

Vb**((C*C)//dP[12])**(Vb // dA[1] // dA[2]) ≡ C*C*(C//dP[2])*(C//dP[2])

The Vertical Flip Equation

Vb**((C*C)//dP[12])**(Vb // dS[1] // dS[2]) ≡ (SuperPlus[R])[1,2]**(C*C*(C//dP[2])*(C//dP[2]))

The Cap Equation

(Vb**(C *C// dP[12])  // dc[1] // dc[2]) ≡ ((C*C)⋃((C*C)//dP[2])// dc[1] // dc[2])

TBD

TBD

TBD

The Buckle Equation V~VB

Vb ≡ VB

Equations with Trivial Caps