The HOMFLY Braidor Algebra - Braidor Computations

First Load "Programs"!

The "Exponential" Solution:

{R3[B[0]],  R3[B[1]],  R3[Append[B[1],  0]]}

d = 2;  d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]

d = 2;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 2;  R3[Append[B[d],  0]]

d = 3;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 3;  R3[Append[B[d],  0]]

d = 3;  d3[R3[Append[B[d],  0]]  //  Last]

d = 4;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 4;  R3[Append[B[d],  0]]

d = 5;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 5;  R3[Append[B[d],  0]]

d = 5;  d3[R3[Append[B[d],  0]]  //  Last]

d = 6;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 7;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 8;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 9;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

d = 10;  
    B[d] = Append[B[d - 1],  
        x^d P[RotateLeft[{2, 1},  d]]/d! - 
          d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]]

f12  =  
      GeneratingSeries[{1, 2},  B[10]]  /.  {t[i_]  ->  h (Subscript[t, i]),  
          x  ->  h x}

f21  =  
      GeneratingSeries[{2, 1},  B[10]]  /.  {t[i_]  ->  h (Subscript[t, i]),  
          x  ->  h x}

f12  /.  h ->  1 //  TeXForm

f21  /.  h  ->  1  //  TeXForm

The "Non-Exponential" Solution:

{R3[B[0]],  R3[B[1]],  R3[Append[B[1],  0]]}

d = 2;  d2inv[d,  Last[R3[Append[B[d - 1],  0]]]]

d = 2;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 2;  R3[Append[B[d],  0]]

d = 3;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 3;  R3[Append[B[d],  0]]

d = 3;  d3[R3[Append[B[d],  0]]  //  Last]

d = 4;  
    B[d] = Append[
        B[d - 1], -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 4;  R3[Append[B[d],  0]]

d = 5;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 5;  R3[Append[B[d],  0]]

d = 5;  d3[R3[Append[B[d],  0]]  //  Last]

d = 6;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 7;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 8;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 9;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 10;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 11;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

d = 12;  
    B[d] = Append[
        B[d - 1],  -d2inv[d,  
            Last[R3[Append[B[d - 1],  0]]]]]

f12  =  
      GeneratingSeries[{1, 2},  B[12]]  /.  {t[i_]  ->  h (Subscript[t, i]),  
          x  ->  h x}

f21  =  
      GeneratingSeries[{2, 1},  B[12]]  /.  {t[i_]  ->  h (Subscript[t, i]),  
          x  ->  h x}

CoefficientList[f12,  h]  //  TeXForm

CoefficientList[f21,  h]  //  TeXForm