The Kinoshita graph is a particular embedding in the 3-sphere of a graph with three edges, two vertices, and no loops. It has the remarkable property that although the removal of any edge results in an unknotted loop, the Kinoshita graph is itself knotted. We use two classical theorems from knot theory to give two particularly simple proofs that the Kinoshita graph is knotted.

This is a survey article. I give an overview of known results on planar graphs embedded in space. Particular attention is paid to the special properties of Brunnian theta graphs: knotted theta graphs with the property that removing any edge results in an unknot.

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called thin position. Thin position is defined using orderings of the cells of a pseudomanifold. In addition to defining and finding combinatorial minimal surfaces, from thin orderings, we derive invariants of even-dimensional closed simplicial pseudomanifolds called width and trunk. We study the additivity properties of these invariants under connected sum and prove theorems analogous to theorems in knot theory and 3-manifold theory.

Distortion is a geometric invariant of knots and bridge number and bridge distance are topological invariants of knots. Combining arguments of Pardon with Heegaard splitting techniques, we produce a lower bound on distortion from the bridge number and bridge distance of a knot. We also provide an infinite family of alternating knots for which our lower bound is greater than Pardon's lower bound.

The Thurston norm measures the complexity of second homology classes of 3-dimensional manifolds. Dehn filling is an operation which converts a 3-manifold with toroidal boundary into a 3-manifold with one fewer boundary components. We show that (in most circumstances) for all but finitely many fillings the Thurston norms of the filled and unfilled manifolds are related in an expected way. We apply our results to showing that twisting a knot along an unknot with positive linking number will, in the limit, increase the Seifert genus of the knot without bound, unless the unknot is a meridian.

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai's width for knots in the 3-sphere. We give applications to the tunnel number and higher genus bridge number of connected sums of knots. The material in this paper relies heavily on our work in the paper Thin position... above.

We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants show up as thin levels. The material in this paper is a crucial ingrediant in the paper Additive invariants... above.

Crossing changes are the most basic, and most important, method for converting one knot into another, but their effects on the topological properties of a knot remain mysterious. In this paper we show that beginning with any knot we can perform a single crossing change to arrive at a knot of arbitrarily high bridge number and bridge distance. In particular, there are unknotting number one knots of arbitrarily high bridge number and arbitrarily high bridge distance.

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. Furthermore, for knots with bridge distance at least 3, four times the Seifert genus of the knot is an upper bound on bridge number. The current version incorporates our earlier paper arXiv:1211.4787.

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 paper 9 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.

The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by certain subgroup of the pure braid group on four strands. We prove that infinitely many of these diagrams give rise to distinct Brunnian theta graphs.

There is a standard way of decomposing the 3-sphere into two homeomorphic pieces by a genus g surface (called a Heegaard surface). Every knot K can be isotoped into ``bridge position'' with respect to a genus g Heegaard surface and the ``minimal'' way of doing this is measured by a natural number b_g(K). It is relatively easy to see that b_g+1(K) ≤ b_g(K) - 1. If strict inequality holds, we say that there is a ``gap'' at level g. In this paper, we produce an infinite family of twisted torus knots having two gaps in their bridge spectrum. Our examples also provide answers to two questions of Eudave-Muñoz.

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 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.