Related papers: Classifying compact 4-manifolds via generalized re…
In this article, we introduce two new PL-invariants: weighted regular genus and weighted G-degree for manifolds with boundary. We first prove two inequalities involving some PL-invariants which state that for any PL-manifold $M$ with non…
Within crystallization theory, two interesting PL invariants for $d$-manifolds have been introduced and studied, namely {\it gem-complexity} and {\it regular genus}. In the present paper we prove that, for any closed connected PL…
The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called {\it semi-simple} and {\it weak semi-simple crystallizations},…
For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular…
A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it…
The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed…
The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization…
The idea of studying trisections of closed smooth $4$-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and…
Gems are a particular type of edge-colored graphs, dual to colored triangulations, which represent compact PL-manifolds of arbitrary dimension, both in the closed and boundary case. In the present paper, gem theory is used to approach…
A crystallization of a PL manifold is an edge-colored graph encoding a contracted triangulation of the manifold. The concept of regular genus generalizes the notions of surface genus and Heegaard genus for 3-manifolds to higher-dimensional…
Let $M$ be a connected compact PL 4-manifold with boundary. In this article, we have given several lower bounds for regular genus and gem-complexity of the manifold $M$. In particular, we have proved that if $M$ is a connected compact…
We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is ageneralization to arbitrary…
We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank…
We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct…
On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathcal{E}$ of rank $r \leq n-2$ and an ample line bundle $H$. A numerical character $m_2=m_2(X,\mathcal{E},H)$ of the triplet…
This article focuses on a class of properly edge-colored graphs, which arise from topological combinatorics, and investigates their embeddings onto surfaces. Specifically, these graphs are known as the dual graphs of balanced normal…
Let $N_{g,n}$ be a genus $g$ compact non-orientable surface with $n$ boundaries. We explain about relations on the level $d$ mapping class group $\mathcal{M}_d(N_{g,0})$ of $N_{g,0}$ and the level $d$ principal congruence subgroup…
It is well-known that in dimension 4 any framed link $(L,c)$ uniquely represents the PL 4-manifold $M^4(L,c)$ obtained from $\mathbb D^4$ by adding 2-handles along $(L,c)$. Moreover, if trivial dotted components are also allowed (i.e. in…
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…
Discrete d-manifolds are classes of finite simple graphs which can triangulate classical manifolds but which are defined entirely within graph theory. We show that the chromatic number X(G) of a discrete d-manifold G is sandwiched between…