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

PL 4-manifolds admitting simple crystallizations: framed links and regular genus

Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present paper, we prove that any (simply-connected) PL $4$-manifold $M$ admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, $M$ may be represented by a framed link yielding $\mathbb S^3$, with exactly $\beta_2(M)$ components ($\beta_2(M)$ being the second Betti number of $M$). As a consequence, the regular genus of $M$ is proved to be the double of $\beta_2(M)$. Moreover, the characterization of any such PL $4$-manifold by $k(M)= 3 \beta_2(M)$, where $k(M)$ is the gem-complexity of $M$ (i.e. the non-negative number $p-1$, $2p$ being the minimum order of a crystallization of $M$) implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL $4$-manifolds admitting simple crystallizations (in particular: within the class of all "standard" simply-connected PL 4-manifolds).Comment: 14 pages, no figures; this is a new version of the former paper "A characterization of PL 4-manifolds admitting simple crystallizations

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

Computing Matveev's complexity via crystallization theory: the boundary case

The notion of Gem-Matveev complexity has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base $\mathbb D^2$ and two exceptional fibers and, therefore, for all torus knot complements.Comment: 27 pages, 14 figure

TOPOLOGY IN COLORED TENSOR MODELS

From a “geometric topology” point of view, the theory of manifold representation by means of edge-colored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL d-manifold by means of a totally combinatorial tool. Edge-colored graphs also play an important rôle within colored tensor models theory, considered as a possible approach to the study of Quantum Gravity: the key tool is the G-degree of the involved graphs, which drives the 1/N expansion in the higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the G-degree, have specific relevance in the tensor models framework, show- ing a direct fruitful interaction between tensor models and discrete geometry, via edge-colored graphs. In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edge-colored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds - or, at least, singular d-manifolds, if d ≥ 4 - represented by graphs of a given G-degree. In dimension 4, the existence of colored graphs encoding different PL manifolds with the same underlying TOP manifold, suggests also to investigate the ability of ten- sor models to accurately reflect geometric degrees of freedom of Quantum Gravity

Uniform random colored complexes

We present here random distributions on $(D+1)$-edge-colored, bipartite graphs with a fixed number of vertices $2p$. These graphs are dual to $D$-dimensional orientable colored complexes. We investigate the behavior of quantities related to those random graphs, such as their number of connected components or the number of vertices of their dual complexes, as $p \to \infty$. The techniques involved in the study of these quantities also yield a Central Limit Theorem for the genus of a uniform map of order $p$, as $p \to \infty$.Comment: 36 pages, 9 figures, minor additions and correction

Integrated optical source of polarization entangled photons at 1310 nm

We report the realization of a new polarization entangled photon-pair source based on a titanium-indiffused waveguide integrated on periodically poled lithium niobate pumped by a CW laser at $655 nm$. The paired photons are emitted at the telecom wavelength of $1310 nm$ within a bandwidth of $0.7 nm$. The quantum properties of the pairs are measured using a two-photon coalescence experiment showing a visibility of 85%. The evaluated source brightness, on the order of $10^5$ pairs $s^{-1} GHz^{-1} mW^{-1}$, associated with its compactness and reliability, demonstrates the source's high potential for long-distance quantum communication.Comment: There is a typing mistake in the previous version in the visibility equation. This mistake doesn't change the result