Gennaro Amendola

A 3-manifold complexity via immersed surfaces

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on ℙ2-irreducible manifolds. Moreover, for ℙ2-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S3, the projective space ℝℙ3 and the lens space L4,1, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of triangulations, Heegaard splittings, surgery presentations and Matveev complexity.

Optimality in Social Choice

Marengo and Settepanella (2010) have developed a geometric model of social choice when it takes place among bundles of interdependent elements, showing that by bundling and unbundling the same set of constituent elements an authority has the power of determining the social outcome. In this article, we will tie the model above to tournament theory, solving some of the mathematical problems arising in their work and opening new questions which are interesting from both a mathematical and social choice point of view. In particular, we will introduce the notion of u-local optima and study it from both a theoretical and a numerically probabilistic point of view; we will also describe an algorithm that computes the universal basin of attraction of a social outcome in O(M 3log M) time (where M is the number of social outcomes).

Decomposition and Enumeration of Triangulated Surfaces

We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions, and genus surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory to the case that the number of vertices is at most 11, and we obtain theoretical restrictions on genus surfaces, allowing us to obtain a list of all triangulations of closed surfaces with at most 11 vertices.

An algorithm producing a standard spine of a 3-manifold presented by surgery along a link

We provide a simple algorithm which produces a (branched) standard spine of a 3-manifold presented by surgery along a framed link inS3, giving an explicit upper bound on the complexity of the spine in terms of the complexity of a diagram of the link. As a corollary, we get an easy constructive proof of Casler’s result on the existence of a standard spine for a closed 3-manifold. We also describe an o-graph which represents the spine.

Moves for standard skeleta of 3-manifolds with marked boundary

A 3-manifold with marked boundary is a pair (M, X), where M is a compact 3-manifold whose (possibly empty) boundary is made up of tori and Klein bottles, and X is a trivalent graph that is a spine of ∂M. A standard skeleton of a 3-manifold with marked boundary (M, X) is a standard sub-polyhedron P of M such that P ∩ ∂M coincides with X and with ∂P, and such that

Non-orientable 3-manifolds of complexity up to 7

{P \cup \partial M}

Decidability and manipulability in social choice

is a spine of