¹Technische Universität Berlin, Berlin, Germany. Email: pommeren@math.tu-berlin.de
²Universidad Nacional de Colombia, Medellín, Colombia. Email: mmtoro@unal.edu.co

We consider homomorphisms H_t 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, H_t depends only on one complex parameter t. We study the possible relators, that is, the words w∈ F with w≠ 1 such that H_t(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.

Key words: Representation, Parabolic, Wirtinger presentation, Two-generated groups, Homomorphism, Longitude.

Consideramos homomorfismos H_t del grupo libre F de rango 2 sobre el subgrupo de SL(2,C) que es generado por dos matrices parabólicas. Salvo conjugación, H_t depende sólo de un parámetro complejo t. Estudiamos los posibles relatores, esto es, las palabras w∈ F con w≠ 1 tal que H_t(w)=I para algún t∈C.
Encontramos varias familias de relatores. De particular interés aquí son los relatores asociados con nudos de 2 puentes, los cuales consideramos de forma puramente algebraica. Describimos un algoritmo para determinar cuándo una palabra dada es un posible relator.

Palabras clave: Representación, parabólico, presentación de Wirtinger, grupos dos-generados, homomorfismos, longitud.

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