Sostenes Lins

GRAPH-ENCODED 3-MANIFOLDS

Abstract From each G in a certain class of 4-regular edge-colored graphs we obtain a ball complex whose underlying space | G | is a 3-manifold and which has G as its 1-skeleton. The construction of this ball complex B ( G ) is dual to the one of Crystallization Theory originated by [Pezzana, 1974] and [Ferri, 1976]. There a pseudo-complex K ( G ) is obtained from an edge-colored graph G . If | K ( G )| is a manifold, then K ( G ) and B ( G ) are dual cellular decomposition of the same manifold, | G |≅| K ( G )|. The proof that every 3-manifold can be represented by a | G | is somewhat simpler than the original based on K ( G ): it is a direct consequence of the triangulation theorem [Moise, 1977]. We present new crystallizations for some of the most commonly studied 3-manifolds. They have fewer vertices and more symmetry than previously published ones. A computer program was used to obtain them from bigger 3-gems and another to get symmetric drawings by using the algorithm in [Tutte, 1963]. A 4-parametric family of 3-manifolds which includes the lens spaces and Poincares sphere is introduced. A table of the homologies of an initial segment of this family which exhibits its richness appears in the Appendix.

A simple proof of Gagliardi's handle recognition theorem

Abstract Recently Gagliardi [2] has proved a theorem on recognizing handles in 3-manifolds crystallizations [3]. The purpose of this note is to present a short proof of this result which follows as a corollary of Theorem 2 and to introduce a more general definition of combinatorial handle, crucial for the proof. We also present a crystallization which settles two questions posed in [2].

Computing Turaev-Viro invariants for 3-manifolds

We give a simple surface interpretation for each summand in the evaluation of Turaev-Viro invariants, for the case of small (up to eighth) roots of unity. From this interpretation follows an efficient scheme to compute these invariants. Extensive tables relative to a rich variety of 3-manifolds are explicitly presented.

Twistors: bridges among 3-manifolds

Abstract The (bipartite) 3-gems are special edge-colored graphs which induce the (orientable) 3-manifolds. Each 3-manifold is induced in this way and there are simple combinatorial moves on 3-gems which replace topological homeomorphisms. These moves are named dipole moves. In this paper a simple configuration on bipartite 3-gems, named twistors is isolated. A twistor can be twisted in two different ways yielding other two twistors and providing simple moves internal to the class of 3-gems. We prove that by recoupling twistors and by 1- and 2-dipole moves we can transform any bipartite 3-gem into any other. Therefore, the twistors are like brigdes among all orientable 3-manifolds. There is an important connection (not treated here) between this result on these combinatorial twists and the basic theorem of Lickorish that the 3-sphere can be reached from any 3-manifold by removing a finite number of disjoint solid tori and pasting them back differently. This connection will be algorithmically explored elsewhere.

Decomposition of the vertex group of 3-manifolds

Abstract The vertex group ξ( M ) of a closed PL n -manifold is an invariant obtained from a crystallization representing M (see Ferri and Gagliardi (1982)). These groups are introduced by Lins (1989) where their topological invariance is proved. In this note we propose, for n = 3 to illuminate the connection of ξ and the fundamental group π 1 . We show, as suspected in [8], that for every closed 3-manifold M 3 , ξ( M 3 ) ≅ π 1 ( M 3 )∗π 1 ( M 3 )∗π 1 ( M 3 )∗ F , where F is a free group in one generator. The present proof does not seem to (directly) generalize for higher dimensions. The result also opens the possibility of searching for topologically invariant automorphisms of ξ. Such automorphisms could give significant information about the manifold M 3 .