All the shapes of spaces: a census of small 3-manifolds

Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth of a compact, connected 3-manifold $M$ defined by \[ \operatorname{tw}(M) = \min\{\operatorname{tw}(\Gamma(\mathcal{T})):\mathcal{T}~\text{is a triangulation of }M\}, \] where $\Gamma(\mathcal{T})$ denotes the dual graph of $\mathcal{T}$. In this setting the relationship between the topology of a 3-manifold and its treewidth is of particular interest. First, as a cor...

In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 surface.

In this work we ask when a group is a 3-manifold group, or more specifically, when does a group presentation come naturally from a Heegaard diagram for a 3-manifold? We will give some conditions for partial answers to this form of the Isomorphism Problem by addressing how the presentation associated to a diagram for a splitting is related to the fundamental group of a 3-manifold. In the process, we determine an invariant of groups (by way of group presentations) for how far s...

This is a survey paper on algorithms for solving problems in 3-dimensional topology. In particular, it discusses Haken's approach to the recognition of the unknot, and recent variations.

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive ...

Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. The goal of this review article is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for ...

By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly pseudoconvex 5-spheres. These manifolds not only inherit interesting intrinsic structures (eg, they have canonical Stein-fillable contact structures), they also have extrinsic structures of a knot-theoretical nature (eq, the 3-sphere arise...

In this survey article, we are interested on minimal triangulations of closed pl manifolds. We present a brief survey on the works done in last 25 years on the following: (i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers $n$ and $d$, construction of $n$-vertex triangulations of different $d$-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of ver...

This is lecture notes of a talk I gave at the Morningside Center of Mathematics on June 20, 2006. In this talk, I survey on Poincare and geometrization conjecture.

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400 orientable manifolds, only 149 of them have hyperbolic torsion linking forms and are thus candi...