Write fⁱ:R²-{x₁,x₂}® R²-{x_i} for the inclusion of the plane minus two points into the plane minus a single point. These induce maps f_i*:p₁(R²-{x₁,x₂})®p₁(R²-{x_i}) on the fundamental groups of the twospaces. The original questionis asking for anon-zero element a in \Ker(f¹*, f²*). However, since R²-{x₁,x₂} is homotopically equivalent to a one point union of two circles it has fundamental group isomorphic to the free group on two generators, say where A corresponds to a loop around x₁ and B to a loop around x₂. So, for example, Olof's picture above is AB^-1A^-1B and is therefore one of the four simplest possible examples (another is mine which is A^-1BAB^-1). Now, writing p₁(R²-{x₁})=áAñ = Z and p₁(R²-{x₂}=áBñ = Z we get f¹*(A^n₁B^m₁¼ A^n_rB^m_r=n₁+¼+n_r and similarly f²*(A^n₁B^m₁¼ A^n_rB^m_r)=m₁+¼+m_r. So the kernel of (f¹*, f²*) is just the elements where the sum of the powers of the A components is 0 and the sum of the powers of the B components is 0. This allows us to construct examples of arbitrary complexity. For example, A³B²A^-2B^-1A^-1B^-1 (see the picture below, I hope I drew it correctly). Well, I hope that was interesting.