We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i: S � M of a closed,

Every element in the first cohomology group of a 3–manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity ’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3–sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of Mc-Mullen and Turaev. Our bounds attain their most elegant form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. The bounds are very powerful and can be easily implemented with a computer. We show that these lower bounds determine the genus of all knots with 12 crossings or less, including the Conway knot and the Kinoshita–Terasaka knot which have trivial Alexander polynomial. We also give many examples of closed manifolds and link complements where twisted Alexander polynomials detect the correct Thurston norm. We also give obstructions to fibering 3–manifolds using twisted Alexander polynomials and detect all knots with 12 crossings or less that are not fibered. For some

Abstract. We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures. It has been known for some time that there are deep connections between the theory of taut foliations and tight contact structures due to the work of Eliashberg and Thurston [7]. In particular, they proved that a taut foliation can be perturbed into a (universally) tight contact structure. In a previous paper [19] we introduced the notion of convex decompositions and explained how convex decompositions can naturally be viewed as generalizations of sutured manifold decompositions introduced by Gabai in [8]. In this paper we take the viewpoint that convex decompositions are completely natural in 3-dimensional contact topology and that many theorems can be proven directly in the category of tight contact manifolds with convex splittings as morphisms. The first theorem of the paper is a version of a theorem by Gabai-Eliashberg-Thurston in the case of manifold with boundary:

In 1976, Atiyah [2] constructed the L2-Betti numbers of a compact Riemannian manifold. They are defined in terms of the spectrum of the Laplace operator on the universal covering of M. ByAtiyah’sL2-index theorem [2], they can be used e.g. to compute the Euler characteristic of M.

In this paper we describe the complement of real line arrangements in the complex plane in terms of the boundary three-manifold of the line arrangement. We show that the boundary manifold of any line arrangement is a graph manifold with Seifert fibered vertex manifolds, and depends only on the incidence graph of the arrangement. When the line arrangement is defined over the real numbers, we show that the homotopy type of the complement is determined by the incidence graph together with orderings on the edges emanating from each vertex. 1

Abstract. Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 ×N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing that their behavior is the same as of those of fibered 3–manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S 1 ×N. As an application of these results we will show that S 1 ×N(P) does not admit a symplectic structure, where N(P) is the 0–surgery along the pretzel knot P = (5, −3, 5), answering a question of Peter Kronheimer. 1.

Abstract. In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4–manifold with even intersection form and prescribed fundamental group. Furthermore, we introduce a new geometric group invariant and discuss some of its properties. 1.

Abstract. Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 × N admits a symplectic structure if and only if N admits a fibration over the circle. We will prove this conjecture in the case when N is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes 3–manifolds with vanishing Thurston norm, graph manifolds and 3–manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic 3–manifolds). Our result covers, in particular, the case of 0– framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the 3–manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered. Dedicated to the memory of Xiao-Song Lin 1.

This paper investigates properties of finite sheeted covering spaces of arithmetic hyperbolic 3-orbifolds (see §2). The main motivation is a central unresolved question in the theory of closed hyperbolic 3-manifolds; namely whether a closed