Colored HOMFLY polynomial of the trefoil with horizontal double boxes

define the quantum number where t^2 = q

qno[n_]=(t^n-t^(-n))/(t-t^(-1));

the quantum deimension of representation with horizontal double boxes

qdimR2=qno[N]*qno[N+1]/qno[2];

Braiding eigenvalue with prallel orientation

Array[lambdaplus, 3];
lambdaplus[1]=t^(2*(N+2));
lambdaplus[2]=-t^(2*(N+1));
lambdaplus[3]=t^(2*(N-1));

quantum 6j symbols of second type

Array[b,{3,3}];
b[1,1]=(qno[N]/qno[2])*qno[N-1]^(1/2)*qno[N+1]^(1/2)/(qno[3]^(1/2));
b[2,1]=-(qno[N-1]^(1/2)*qno[N]^(1/2)*qno[N+1]^(1/2)*qno[N+2]^(1/2))/(qno[4]^(1/2)*qno[2]^(1/2));
b[3,1]=(qno[N]^(1/2)*qno[N+1]^(1/2)*qno[N+2]^(1/2)*qno[N+3]^(1/2))/(qno[4]^(1/2)*qno[3]^(1/2)*qno[2]^(1/2));
b[1,2]=-(qno[N]*qno[N+1]/qno[2])/((qno[3])^(1/2));
b[2,2]=(qno[N+3]*qno[N]/qno[N+2]-qno[N-1])*(qno[N]^(1/2)*qno[N+2]^(1/2)/(qno[4]^(1/2)*qno[2]^(1/2)));
b[3,2]=(qno[N+1]*qno[2]/qno[N+2])*((qno[N-1]^(1/2)*qno[N]^(1/2)*qno[N+2]^(1/2)*qno[N+3]^(1/2))/(qno[4]^(1/2)*qno[3]^(1/2)*qno[2]^(1/2)));
b[1,3]=(qno[N]/qno[2])*qno[N+3]^(1/2)*qno[N+1]^(1/2)/qno[3]^(1/2);
b[2,3]=(qno[N]/qno[N+2])((qno[N]^(1/2)*qno[N+1]^(1/2)*qno[N+2]^(1/2)*qno[N+3]^(1/2))/(qno[4]^(1/2)*qno[2]^(1/2)));
b[3,3]=(qno[2]*qno[N+1]/qno[N+2]-1)*((qno[N-1]^(1/2)*qno[N]^(1/2)*qno[N+1]^(1/2)*qno[N+2]^(1/2))/(qno[4]^(1/2)*qno[3]^(1/2)*qno[2]^(1/2)));

SU(N) Chern-Simons invariants of of the trefoil

H31=Sum[b[x,1]^2*(lambdaplus[x])^(3)/qdimR2,{x,3}];
Simplify[H31]/.t->q^(1/2)

Replacing q^N by a

%/.N->Log[q,a]//FullSimplify

Colored HOMFLY polynomial of the trefoil with horizontal double boxes

HTorusp=Sum[b[x,1]^2*(lambdaplus[x])^(2p+1)/qdimR2,{x,3}];
Simplify[HTorusp]/.t->q^(1/2)

Replacing q^N by a

%/.N->Log[q,a]//FullSimplify

Colored HOMFLY polynomial of the figure-eight with horizontal double boxes

Braiding eigenvalue with prallel orientation

Array[lambdaminus, 3];
lambdaminus[1]=1;
lambdaminus[2]=-t^(N);
lambdaminus[3]=t^(2(N+1));

quantum 6j symbols of first type

Array[a,{3,3}];
a[1,1]=1;
a[1,2]=-(qno[N+1])^(1/2)*(qno[N-1])^(1/2);
a[1,3]=qno[N]/qno[2]*(qno[N-1])^(1/2)*(qno[N+3])^(1/2);
a[2,1]=-(qno[N+1])^(1/2)*(qno[N-1])^(1/2);
a[2,2]=(qno[N]/qno[2])^2*(qno[N-1])*(qno[N+3])/(qno[N]-1)-1;
a[2,3]=(qno[N+1])^(1/2)*(qno[N-1])^(1/2)*qno[N]/qno[2]*(qno[N-1])^(1/2)*(qno[N+3])^(1/2)/(qno[N]-1);
a[3,1]=a[1,3]=qno[N]/qno[2]*(qno[N-1])^(1/2)*(qno[N+3])^(1/2);
a[3,2]=(qno[N+1])^(1/2)*(qno[N-1])^(1/2)*qno[N]/qno[2]*(qno[N-1])^(1/2)*(qno[N+3])^(1/2)/(qno[N]-1);
a[3,3]=(qno[N+1])*(qno[N-1])/(qno[N]-1)-1;

SU(N) Chern-Simons invariants of of the figure-eight

H41:=-Sum[a[0,k]*a[0,l]*lambdaminus[k]^2*lambdaminus[l]^(-2)*a[k,l]/qdimR2^2,{k,1,3},{l,1,3}]//Simplify

Simplify[H31]/.t->q^(1/2)

%/.N->Log[q,a]//FullSimplify

Kauffmann polynomial of the trefoil with a single box

the quantum deimension of representation with horizontal double boxes

dimR=qno[N-1]+1;

Braiding eigenvalue with prallel orientation

Array[Klambdaplus,3];
Klambdaplus[1]=(t^((N-1)))(t^(-1));
Klambdaplus[2]=-(t^((N-1)))t;
Klambdaplus[3]=(t^(2*(N-1)));

Quantumd dimension of irreducible representations

Array[R,3];
R[1]=qno[N-1]+((qno[N-1]qno[N])/(qno[2]));
R[2]=qno[N-1]+((qno[N-1]qno[N-2])/(qno[2]));
R[3]=1;

SO(N) Chern-Simons invariants of of the trefoil

KTorus[p_]:=Sum[R[k]*(Klambdaplus[k])^p,{k,1,3}]/dimR;

KTorus[3]/.t->q//FullSimplify

FullSimplify[%/.N->Log[q,λ]+1,q>0]

Refined Chern-Simons

Littlewood-Richardson coefficients

SLR[r_,l_]:=Product[((1-q1^(j-1)q2)/(1-q1^(j))),{j,1,r-l}]Product[((1-q1^(j-1)q2^2)/(1-q1^(j)q2)),{j,2l+1,r+l}]Product[(((1-q1^(j))/(1-q1^(j-1)q2)))^2,{j,l+1,r}]

ALR[r_,l_]:=Product[((1-q1 q2^(i-1))/(1-q2^(i))),{i,1,l}]Product[((1-q2^(i))/(1-q1 q2^(i-1))),{i,l+1,2l}]

Definition of Macdonald functions

DualPartition[l_]:=Module[{i},Table[Length[Select[l,(#>=i)&]],{i,1,l[[1]]}]]

get[Y_,i_]:=If[i>Length[Y],0,Y[[i]]]

row[Y_,i_]:=get[Y,i]

rowt[Y_,i_]:=get[DualPartition[Y],i]

boxes[Y_]:=Join@@Module[{i,j},Table[Table[{i,j},{j,1,Y[[i]]}],{i,1,Length[Y]}]]

e[N_,Y_,{i_,j_}]:=((q2^(((N-i+1)/2))q1^(((j-1)/2))-q2^(-((N-i+1)/2))q1^(-((j-1)/2)))/(q2^(((rowt[Y,j]-i+1)/2))q1^(((row[Y,i]-j)/2))-q2^(-((rowt[Y,j]-i+1)/2))q1^(-((row[Y,i]-j)/2))))

mac[N_,Y_]:=(Times @@ ( (e[N,Y,#])&/@ boxes[Y]))

Braiding Eigenvalues

absR[Y_]:=Sum[get[Y,i],{i,1,Length[Y]}]

absRt[Y_]:=Sum[get[DualPartition[Y],i],{i,1,Length[DualPartition[Y]]}]

squareR[Y_]:=Sum[get[Y,i]^2,{i,1,Length[Y]}]

squareRt[Y_]:=Sum[get[DualPartition[Y],i]^2,{i,1,Length[DualPartition[Y]]}]

lambda[N_,Y1_,Y2_]:=((q1^(squareR[Y1])q2^(N absR[Y1])q1^(-(1/4)squareR[Y2])q2^(-(N/4)absR[Y2]))/(q2^(squareRt[Y1])q1^( (1/N)absR[Y1]^2)q2^(-(1/4)squareRt[Y2])q1^(-(1/(4 N))absR[Y2]^2)))

Gamma Factors

Sgamma[r_,l_]:=Product[((q1^(((r-i)/2))q2^((1/2))-q1^(-((r-i)/2))q2^(-(1/2)))/(q1^(((r-i+1)/2))-q1^(-((r-i+1)/2)))),{i,1,r-l}]

Agamma[r_,l_]:=Product[((q2^((i/2))-q2^(-(i/2)))/(q2^(((i-1)/2))q1^((1/2))-q2^(-((i-1)/2))q1^(-(1/2)))),{i,1,l}]

Khovanov polynomial of the Hopf link

KRHopf[N_,r_]:=Sum[SLR[r,l]mac[N,{r+l,r-l}](lambda[N,{r},{r+l,r-l}])^2,{l,0,r}]

KRHopf[2,1]//FullSimplify

%/.{q1->q^2 t^2,q2->q^2}

Superpolynomial of the Trefoil

KRTrefoil[N_,r_,p_]:=Sum[Sgamma[r,l] SLR[r,l]mac[N,{r+l,r-l}]((-1)^l lambda[N,{r},{r+l,r-l}])^p,{l,0,r}]/mac[N,{r}]

KRTrefoil[N,1,3]//FullSimplify

%/.{N->Log[q2,A]}//FullSimplify

%/.{A-> -a t^3,q1->q^(-1)t^(-2),q2->q^(-1)}//FullSimplify