"SO(3,1) parabolics could be put into the following form";

A={{1,0,1,a14},{0,1,1,a24},{0,0,1,a34},{0,0,0,1}};

B={{1,0,0,0},{b21,1,0,0},{b31,1,1,0},{1,1,0,1}};

Map[MatrixForm,{MatrixPower[A-IdentityMatrix[4],2],MatrixPower[B-IdentityMatrix[4],2]}]

"So we see that in order to preserve the same Jordan type we must insist that a34≠0 and b21≠0";

"Now we can compute some examples";

"In what follows W will be the word that conjugates A to B and when neccesary L will be a longitude";

"First the figure 8 knot";
W=B.Inverse[A].Inverse[B].A;
L=W.A.Inverse[B].Inverse[A].B;

Reduce[A.W-W.B==0&&a34≠0&&b21≠0]

"The fact that there is a 1 parameter family of deformations comes from the fact that for the figure 8 the meridian is not a rigid slope";
"If we also insist that the trace of the longitude stay fixed then we do not have any local deformations";

Reduce[A.W-W.B==0&&Tr[L]==4&&a34≠0&&b21≠0]

"These solutions form a discrete set and so we see that the hyperbolic structure is locally rigid.";

"The discrete faithful representation occurs when b21=2";

"Now 5_2";

W=B.A.Inverse[B].Inverse[A].B.A;

list=Reduce[A.W-W.B==0&&a34≠0&&b21≠0]

"Next we do 6_1";

W=B.Inverse[A].Inverse[B].A.B.Inverse[A].Inverse[B].A;

list=Reduce[A.W-W.B==0&&a34≠0&&b21≠0];

list[[2]]

"Again the solutions form a discrete set and so the hyperbolic structure is locally rigid";

"Finally, we try the Whitehead link";

W=B.A.Inverse[B].Inverse[A].Inverse[B].A.B;

Reduce[A.W-W.A==0&&a34≠0&&b21≠0]

"So again we see that the hyperbolic structure is rigid";