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

We consider homomorphisms $H_{t}$ from the free group $F$ of rank $2$ onto the subgroup of SL$(2,\mathbb{C})$ that is generated by two parabolic matrices. Up to conjugation, $H_{t}$ depends only on one complex parameter $t$. We study the possible relators, that is, the words $w\in F$ with $w\neq 1$ such that $H_{t}(w)=I$ for some $t\in\mathbb{C}$. 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.