Abstract. Using the fundamental group of a punctured torus, a free group F of rank two, and the fact that the natural eipmorphism from AutF onto Aut(F/F ) has as kernel the group of inner automorphisms of F , we describe representatives of the conjugacy classes of generating pairs of F and give explicit relations between them.Let F = F (S, T ) be the free group on S and T . By a theorem of Nielsen [N] (see [LS, p. 25]) the natural epimorphism from AutF onto Aut(F/F ) ( = GL(2, Z)) has as kernel the group of inner automorphisms of F . From this it follows easily that, if α is the abelianization homomorphism from F onto F/F (= Z 2 ) and a ∈ Z 2 is primitive 1 , then the inverse image of a under α is a conjugacy class of primitive elements. Also, if (a 1 , a 2 ) is a basis of Z 2 , then, up to conjugacy, there is a unique basis ( , 2)). In the important paper [OZ], Osborne and Zieschang define explicitly primitive words W m,n ∈ F (S, T ), where m and n are relatively prime integers, such that (W m,n )α = (m, n). They also state that if mn − pq = 1, then (W m,n , W p,q ) is a basis of F ; this, while correct for nonnegative values of m, n, p, q, is not valid in general (for example W −2,−3 and W 1,1 do not generate F ). A composition formula is also stated in [OZ, Thm. 3.5] but this, even with the correction of indices in [LTZ, 2.1.3], is incorrect in general.In the present article we consider elements V ε a of F for a = (m, n) ∈ Z 2 and ε ∈ D ⊂ R 2 where D is the complement of the union of all the lines that intersect, and obtain in Theorem 1.ii) a composition formula. Everything is obtained by applying the fundamental group functor π to the punctured torus.Denote by T the torus R 2 / Z 2 , by T 0 the punctured torus (R 2 − Z 2 ) / Z 2 and by ρ : R 2 − Z 2 → T 0 the natural projection. If a ∈ Z 2 and ε ∈ D, then denote (ε)ρ by ε and define γ ε a ∈ π(T 0 , ε) as the homotopy class of the loop (ε+ta)ρ, t ∈ [0, 1]. Denote γ ε (1,0) (resp. γ ε (0,1) ) by S ε (resp. T ε ). There is an isomorphism