The SL(2, C) character variety is a 1-dimensional complex curve associated to the fundamental group of a hyperbolic knot complement. The work of Thurston and Culler-Shalen established this as an important tool in studying the topology of hyperbolic 3-manifolds. In particular, Culler-Shalen described a process which associates an essential surface to a "point at infinity" of this curve. We show that certain infinite families of essential once- and twice-punctured tori in hyperbolic knot complements are associated to such a point. We go on to discuss consequences that relate the algebra of the character variety to the topology of the 3-manifold. If there is time we will also discuss some ongoing projects stemming from this work.