| Heegaard splittings are a classical method for decomposing a 3-manifold into simpler pieces. While they have always been of interest, a number of recent advances have made them a very active research topic. We will discuss Heegaard splittings of 3-manifolds, and what they look like for complements of knots in the 3-sphere. Then we will present a theory of genus-2 Heegaard splittings of knot complements developed in joint work with Sangbum Cho, and give some of its applications. |