Below are notes and slides that I've written, but that aren't quite appropriate for the arXiv or other forms of publication.
I made two attempts at writing a thorough introduction to Heegaard splittings. The second attempt was more detailed and used more recent techniques, but I only made it through the first four chapters. On the other hand, the first attempt but contains all the basics of Casson and Gordon's theory of weak reduction. For an even broader overview, you should read Scharlemann's survey of Heegaard splittings.
An introduction to thin position data clustering is a short, relatively non-technical account of the theory behind the TILO/PRC algorithms. These uses ideas from thin position for knots and three-dimensional manifolds to find intrinsic clusters in large data sets. The technical account is in my paper Topological graph clustering with thin position, or you can also find more information in a blog post I wrote about the algorithm.
I gave a series of four workshop talks at the Summer Topology and its Applications conference in Mankato, MN 2012. Here are annotated slides from the presentations.