Initialization

SetDirectory["C:\\drorbn\\AcademicPensieve\\People\\Ens"];
<< "../../Projects/WKO4/FreeLie.m";
$IterationLimit = ∞ ; $SeriesShowDegree=1;
Arb=Arbitrator -> (Replace[#, _:>RandomInteger[{-3,3}], 1]&);
{BDegr,EDegr}={5,5};

Braidors Extend

Clear[B,Bs];
Bs[3,2]=(1/2); Bs[2,1]=0;
B=DKS[3,Bs];
BInfo=SeriesSolve[B,((B^(σ[1,2,3]))**(B^(σ[13,2,4]))**(B^(σ[1,3,4]))≡(B^(σ[12,3,4]))**(B^(σ[1,2,4]))**(B^(σ[14,2,3])))∧((1/2)((DKS[t[1,2]])^(σ[12,3]))≡(1/2)B**DKS[t[1,3]]**(B^(σ[1,3,2]))),Arb];
B@{6}//Timing
ArbAll = Length[Last[#]] & /@ Read[BInfo]

(B^(σ[{},1,2]))@{7}

Table[Length@AllLyndonWords[n,{1,2}], {n, 10}]

Syzygy a)

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0]),Arb];
EE@{7};//Timing

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧((EE^(σ[{},2,3,4]))≡DKS[0]),Arb];
EE@{7};//Timing

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧((EE^(σ[{},2,3,4]))≡DKS[0])∧((EE^(σ[1,{},3,4]))≡DKS[0]),Arb];
EE@{7};//Timing

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧((EE^(σ[{},2,3,4]))≡DKS[0])∧((EE^(σ[1,{},3,4]))≡DKS[0])∧((EE^(σ[1,2,{},4]))≡DKS[0])∧((EE^(σ[1,2,3,{}]))≡DKS[0]),Arb];
EE@{7};//Timing

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧((EE^(σ[1,{},3,4]))≡DKS[0])∧((EE^(σ[1,2,{},4]))≡DKS[0])∧((EE^(σ[1,2,3,{}]))≡DKS[0]),Arb];
EE@{7};//Timing

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0]),Arb];
EE@{8};//Timing

Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧((EE^(σ[{},2,3,4]))≡DKS[0]),Arb];
EE@{8};//Timing

Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[2,1,3,4]))-EE≡DKS[0]),Arb];
EE@{8};//Timing

Length[Last[#]] & /@ Read[SyzInfo]

Clear[EE,EEs]; EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[13,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0]),Arb];
EE@{8};//Timing

Length[Last[#]] & /@ Read[SyzInfo]

Syzygy b)

Clear[EE,EEs];
EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,2,3,4]))≡(EE^(σ[1,2,4,3]))),Arb];

Block[{$IterationLimit = ∞ },
EE@{EDegr};//Timing
]

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Syzygy c)

Clear[EE,EEs];
EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,(ℏ^(-1))(b[DKS[t[2,3]],(EE^(σ[1,2,4,5]))+(EE^(σ[12,3,4,5]))]≡-b[DKS[t[4,5]],(EE^(σ[1,4,2,3]))+(EE^(σ[14,5,2,3]))]),Arb];

Block[{$IterationLimit = ∞ },
EE@{EDegr};//Timing
]

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Syzygy ab)

Clear[EE,EEs]
EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧((EE^(σ[1,2,3,4]))≡(EE^(σ[1,2,4,3]))),Arb];

Block[{$IterationLimit = ∞ },
EE@{EDegr};//Timing
]

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Syzygy ac)

Clear[EE,EEs]
EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧(ℏ^(-1))(b[DKS[t[2,3]],(EE^(σ[1,2,4,5]))+(EE^(σ[12,3,4,5]))]≡-b[DKS[t[4,5]],(EE^(σ[1,4,2,3]))+(EE^(σ[14,5,2,3]))]),Arb];

Block[{$IterationLimit = ∞ },
EE@{EDegr};//Timing
]

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

Syzygy bc)

Clear[EE,EEs];
EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,2,3,4]))≡(EE^(σ[1,2,4,3])))∧(ℏ^(-1))(b[DKS[t[2,3]],(EE^(σ[1,2,4,5]))+(EE^(σ[12,3,4,5]))]≡-b[DKS[t[4,5]],(EE^(σ[1,4,2,3]))+(EE^(σ[14,5,2,3]))]),Arb];

Block[{$IterationLimit = ∞ },
EE@{EDegr};//Timing
]

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]

All Syzygies

Clear[EE,EEs];
EE=DKS[4,EEs];
SyzInfo=SeriesSolve[EE,((EE^(σ[1,3,4,5]))+(EE^(σ[13,2,4,5]))-(EE^(σ[1,2,4,5]))-(EE^(σ[12,3,4,5]))≡DKS[0])∧((EE^(σ[1,2,3,4]))≡(EE^(σ[1,2,4,3])))∧(ℏ^(-1))(b[DKS[t[2,3]],(EE^(σ[1,2,4,5]))+(EE^(σ[12,3,4,5]))]≡-b[DKS[t[4,5]],(EE^(σ[1,4,2,3]))+(EE^(σ[14,5,2,3]))]),Arb];

Block[{$IterationLimit = ∞ },
EE@{EDegr};//Timing
]

ArbSyz = Length[Last[#]] & /@ Read[SyzInfo]