Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $\pi_{1}(W)$ is injective. In this paper, we prove that there are $4$-dimensional Haken cobordisms whose boundary consists of any two closed Haken $3$-manifolds. In particular, each closed Haken $3$-manifold is the $\pi_{1}$-injective boundary of some Haken $4$-manifold.

The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of small complexity: the punctured torus or thrice punctured sphere.

We show certain cocompact lattices in $\operatorname{SL}(3,\mathbf{R})$ contain closed surface groups. With further restrictions, we exhibit such lattices containing infinitely many commensurability classes of closed surface groups.

We give explicit equations that describe the character variety of the figure eight knot for the groups $\mathrm{SL}(3,\mathbb{C})$, $\mathrm{GL}(3,\mathbb{C})$ and $\mathrm{PGL}(3,\mathbb{C})$. For any of these $G$, it has five components of dimension $2$, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $\mathrm{SL}(2,\mathbb{C})$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties.

We prove that, for every integer $n\ge2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, that is, every quadratic equation over $G$ of length at most $n$ has a solution in $G$ if and only if this equation, considered as an equation over $H$, has a solution in $H$.

We say that a knot $k_{1}$ in the $3$-sphere $1$-dominates another $k_{2}$ if there is a proper degree 1 map $E(k_{1})\to E(k_{2})$ between their exteriors, and write $k_{1}\ge k_{2}$. When $k_{1}\ge k_{2}$ but $k_{1}\ne k_{2}$ we write $k_{1}>k_{2}$. One expects in the latter eventuality that $k_{1}$ is more complicated. In this paper, we produce various sorts of evidence to support this philosophy.

In Studies in modern topology (1968) 39–98 Prentice Hall, Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the “interesting” surface increases from zero, more complicated phenomena occur. Kobayashi (Osaka J. Math. 24 (1987) 173–215) showed that if a 3-manifold $M^{3}$ contains an essential torus $T$, then it contains one which can be isotoped to intersect a (strongly irreducible) Heegaard splitting surface $F$ in a collection of simple closed curves which are essential in $T$ and in $F$. In general, there is no global bound on the number of curves in this collection. We show that given a 3-manifold $M$, a minimal genus, strongly irreducible Heegaard surface $F$ for $M$, and an essential torus $T$, we can either restrict the number of curves of intersection of $T$ with $F$ (to four), find a different essential surface and minimal genus Heegaard splitting with at most four essential curves of intersection, find a thinner decomposition of $M$, or produce a small Seifert-fibered piece of $M$.

In 1852, Augustus De Morgan, Professor of Mathematics at University College, London, was asked: Can every map be colored with just four colors in such a way that neighboring countries are colored differently? Over a century later, in a controversial proof that made substantial use of a computer, Wolfgang Haken and Kenneth Appel of the University of Illinois answered the question in the affirmative. But how did Haken come to be involved with the problem, and what was his role in its solution?

This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the same authors. The proof is a rigorous calculation using Thurston’s spun-normal surfaces in the spirit of Haken’s original normal surface algorithms.

The Four Color Conjecture, which in 1977 became the Four Color Theorem of Kenneth Appel and Wolfgang Haken, is famous for the number of its reformulations. Three of them found by the author at different time are discussed in this paper.

In 1985, Akbulut and Kirby analyzed a homotopy $4$-sphere $\Sigma$ that was first discovered by Cappell and Shaneson, depicting it as a potential counterexample to three important conjectures, all of which remain unresolved. In 1991, Gompf’s further analysi showed that $\Sigma$ was one of an infinite collection of examples, all of which were (sadly) the standard $S^{4}$, but with an unusual handle structure.

Recent work with Gompf and Thompson, showed that the construction gives rise to a family $L_{n}$ of $2$-component links, each of which remains a potential counterexample to the generalized Property R Conjecture. In each $L_{n}$, one component is the simple square knot $Q$, and it was argued that the other component, after handle-slides, could in theory be placed very symmetrically. How to accomplish this was unknown, and that question is resolved here, in part by finding a symmetric construction of the $L_{n}$. In view of the continuing interest and potential importance of the Cappell-Shaneson-Akbulut-Kirby-Gompf examples (e.g., the original $\Sigma$ is known to embed very efficiently in $S^{4}$ and so provides unique insight into proposed approaches to the Schoenflies Conjecture) digressions into various aspects of this view are also included.

This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and we give a simple extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs presented as immersions in the plane.

It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte’s relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic (trivalent) graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte’s upper bound on the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The proofs of these results rely on the structure of the Temperley–Lieb algebra and more generally on methods of topological quantum field theory.

Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating series of normal surfaces associated to the ideal triangulation. This shows a connection of the 3D-index with classical normal surface theory, and fulfills a dream of constructing topological invariants of 3-manifolds using normal surfaces.

We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.