Associated to every knot in the 3-sphere is an number, called distance, which is a measure of how complicated the knot is. Knots of high distance (eg. greater than 3) are very well behaved (eg. see papers 9 and 10 below). Thus, we are motivated to study knots with low distance. In this paper, we give a description of all knots in the 3-sphere having a bridge sphere of distance exactly 2.

We revisit some arguments from classical combinatorial sutured manifold theory and show how they can be used to give bounds on the distance between the slope of an exceptional filling on a cusped hyperbolic manifold and the slope corresponding to a knot in a 3-manifold which has wrapping number different from winding number with respect to a non-trivial second homology class. The paper is written in the style of a survey article.

This is an expository article which shows how concepts from vector calculus are clarified by using combinatorial analogues of basic vector calculus concepts. It includes what we believe to be a new (more intuitive) proof of Green's Theorem and an introduction to cohomology theory.

If a knot has a bridge surface of distance at least 3 in its curve complex, then its bridge number is bounded by 4g + 2, where g is the knots Seifert genus. We also show that a minimal bridge sphere for any counterexample to the cabling conjecture either has distance at most 2 or has bridge number equal to 5.

It is shown that if a knot has either a non-hyperbolic surgery or a cosmetic surgery then the distance (in the sense of Hempel and Bachmann-Schleimer) of any bridge surface for the knot is small in the arc and curve complex for the bridge surface. There is a certain sense in which this should mean that such surgeries are rare.

This is a short, fun paper describing some connections between the painting In Blue by Terry Winters and the mathematical theory of knots.

This paper proves a host of result relating two 2-handle additions to the genus two boundary component of a 3-manifold. It significantly improves on most of the results in paper 2 concerning knots and links obtained by boring a split link or unknot. It also proves that knots obtained by attaching a ``complicated'' band to a 2-component link satisfy the cabling conjecture.

This paper develops sutured manifold technology for studying 2-handle addition to the the boundary component of a 3-manifold. The main result relates the euler characteristic of an essential surface in the 3-manifold to the number of times the boundary of the surface intersects the sutures. It is a 2-handle addition version of a theorem of Lackenby. As an application, it is proved that a tunnel for any tunnel number one knot or link (in any 3-manifold admitting such a knot or link) is disjoint from some generalized Seifert surface for the knot or link. This generalizes a result of Scharlemann-Thompson. More applications are given in paper 6.

This paper develops the theory of thin position for graphs in 3-manifolds, generalizing work of Hayashi-Shimokawa, Tomova, Scharlemann-Thompson, and Gabai. The technology allows a bridge surface for a graph to be untelescoped not just along compressing discs but also along cut discs. In many cases, if it is possible to untelescope a bridge surface then there is an incompressible and cut-incompressible meridional surface in the graph complement. The main theorem relies on the classification results from our previous paper.

This paper classifies bridge surfaces for the spine of a compressionbody. At the end there's a nice combinatorial lemma about trees embedded in discs, but otherwise the paper is rather technical. The techniques are based on those used by Hayashi-Shimokawa in their classification results.

Informally, if W is a genus 2 handlebody embedded in a 3-manifold with constituent knots or links K and L, we say that K and L are obtained by ``boring'' each other, if there is a spine for W that contains one of K or L as a constituent knot or link, but not the other. This paper proves a collection of results about knots and links obtained by boring an unknot or split link and, in particular, generalizes some theorems of Eudave-Muñoz for tangle sums.

Like compact 3-manifolds, every non-compact 3-manifold has a Heegaard splitting. This paper gives a number of examples of such splittings (and shows, for example, that not every non-compact Heegaard splitting can be destabilized). It also shows that the Heegaard splittings of an eventually end-irreducible 3-manifold are obtained by amalgamating compact Heegaard splittings of certain compact submanifolds in an exhausting sequence for the 3-manifold. As a consequence, Heegaard splittings of topologically tame non-compact 3-manifolds are classified. Some open questions are posed.