Cataloguing PL 4-manifolds by gem-complexity

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n=4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices). Possible interactions with the (not completely known) relationship among different classification in TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.Comment: 25 pages, 5 figures. Improvements suggested by the refere

A note about complexity of lens spaces

Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p,q), with p greater than 2, is bounded by S(p,q)-3, where S(p,q) denotes the sum of all partial quotients in the expansion of q/p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide. Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2 c(L(p,q)), where c(M) denotes the complexity of a 3-manifold M and k(M)+1 is half the minimum order of a crystallization of M.Comment: 14 pages, 2 figures; v2: we improved the paper (changes in Proposition 10; Corollary 9 and Proposition 11 added) taking into account Theorem 2.6 of arxiv:1310.1991v1 which makes use of our Prop. 6(b) (arxiv:1309.5728v1). Minor changes have been done, too, in particular to make references more essentia

Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra

We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices.Comment: 18 pages, 3 figure

Local order in aqueous solutions of rare gases and the role of the solute concentration: a computer simulation study with a polarizable potential

Aqueous solutions of rare gases are studied by computer simulation employing a polarizable potential for both water and solutes. The use of a polarizable potential allows to study the systems from ambient to supercritical conditions for water. In particular the effects of increasing the concentration and the size of the apolar solutes are considered in an extended range of temperatures. By comparing the results at increasing temperature it appears clearly the change of behaviour from the tendency to demix at ambient conditions to a regime of complete solubility in the supercritical region. In this respect the role of the hydrogen bond network of water is evidenced.Comment: Accepted for publication in Molecular Physics 2004. 19 pages, 10 figure

4-colored graphs and knot/link complements

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.Comment: 19 pages, 6 figures, 3 tables; changes in Lemma 6, Corollaries 7 and

The double of the doubles of Klein surfaces

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface $X$ is a Klein surface $Y$ such that there is a degree two morphism (of Klein surfaces) $Y\rightarrow X$. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if $X$ is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: "the double of doubles" denoted by $DX$. We describe how to use the double of doubles in the study of both moduli spaces and automorphisms of Klein surfaces. Furthermore, we show that the morphism from $DX$ to $X$ is not given by the action of an isometry group on classical surfaces.Comment: 14 pages; more details in the proof of theorem