The HOMFLY Braidor Algebra - Further Computations

First Load "Programs"!

fs=FormalExp[8, 2, P[{1,2},2]] ** FormalExp[8, 2, P[{1,2},1]]

GeneratingSeries[{1,2}, fs]

gs2=GeneratingSeries[{2,1}, fs]

GeneratingSeries[{1,2}, FormalExp[8, 2, P[{1,2},1]] ** FormalExp[8, 2, P[{1,2},2]]]

CoefficientList[gs2 /. {x->-1/(t[1]-t[2]), t[i_] :> h t[i]}, h]*{1/2,1,3,12,60,360,2520,20160} //Factor //  Expand

(127 ((t[1])^6)+246 ((t[1])^5) t[2]+337 ((t[1])^4) ((t[2])^2)+372 ((t[1])^3) ((t[2])^3)+337 ((t[1])^2) ((t[2])^4)+246 t[1] ((t[2])^5)+127 ((t[2])^6))-Expand[127(t[1]+t[2])^6]+Expand[516t[1]t[2](t[1]+t[2])^4]-Expand[496t[1]^2 t[2]^2(t[1]+t[2])^2]

GeneratingSeries[{1,2}, FormalExp[8, 2, P[{1,2},1]+ P[{1,2},2]]]

GeneratingSeries[{2,1}, FormalExp[8, 2, P[{1,2},1]+ P[{1,2},2]]]

{k,l}={7,4}; ps=FormalPower[P[{1,2}, 2], k] ** FormalPower[P[{1,2}, 1], l]

ps12=GeneratingSeries[{1,2},ps] /. {t[1]->t1, t[2]->t2}

ps12 == t1^l t2^k

Expand[ps21=GeneratingSeries[{2,1},ps] /. {t[1]->t1, t[2]->t2}]

ps21==Expand[Factor[x((((t2^(k+l))-(t1^l)(t2^k)+(t1^(k+l))-(t1^k)(t2^l))/(t2-t1)))]]

FormalPower[P[{1,2}, 2], 7] ** FormalPower[P[{1,2}, 1], 1]

FormalPower[P[{1,2,3}, 2], 3]**FormalPower[P[{1,2,3}, 3],4] ** FormalPower[P[{1,2,3},2], 1]

d=10; ps12=GeneratingSeries[{1,2},Delta[ FormalPower[P[{1},1], d]]] /. {t[1]->t1, t[2]->t2}

CoefficientList[ps12, t1]

aps12 =Expand[t2^d+x(∑_(k=0)^(d-1))(1/2)(t1^(d-1-k))((((t2+x))^k)-(((t2-x))^k))]

bps12=Expand[Factor[(t2^d)+(x/2)((((((t2+x))^d)-(t1^d))/(t2+x-t1))-(((((t2-x))^d)-(t1^d))/(t2-x-t1)))]]

ps12-bps12

CoefficientList[aps12, t1]

ps21=GeneratingSeries[{2,1},Delta[ FormalPower[P[{1},1], d]]] /. {t[1]->t1, t[2]->t2}

CoefficientList[ps21, t1]

aps21 =Expand[x(∑_(k=0)^(d-1))((t1^(d-1-k))/2)((((t2+x))^k)+(((t2-x))^k))]

bps21=Expand[Factor[(x/2)((((((t2+x))^d)-(t1^d))/(t2+x-t1))+(((((t2-x))^d)-(t1^d))/(t2-x-t1)))]]

ps21-bps21

gs=AllGenerators[8,1]

d1 /@ gs

d2 /@ (d1 /@ gs)

d2ker[8]

d2ker[8] // Length