On the Two-Parabolic Subgroups of SL(2,\mathbbC)

we consider homomorphisms ht from the free group f of rank 2 onto the subgroup of sl(2,c) that is generated by two parabolic matrices. up to conjugation, ht depends only on one complex parameter t. we study the possible relators, that is, the words w∈ f with w≠ 1 such that ht(w)=i for some t∈c. we find several families of relators. of particular interest here are relators connected with 2-bridge knots, which we consider in a purely algebraic setting. we describe an algorithm to determine whether a given word is a possible relator.