arXiv subjects

Carlo Petronio

Publications and source records attributed to Carlo Petronio.

47 records · Page 3Linked to original sources

arXiv2002

Small hyperbolic 3-manifolds with geodesic boundary

We classify the orientable finite-volume hyperbolic 3-manifolds having non-empty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, we describe their canonical Kojima decomposition, and we discuss manifolds having cusps. The manifolds built from one or two tetrahedra were previously known. There are 151 different manifolds built from three tetrahedra, realizing 18 different volumes. Their Kojima decomposition always consists of tetrahedra (but occasionally requires four of them). And there is a single cusped manifold, that we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), or of two pyramids, or of a single octahedron. There are 30 manifolds having a single cusp, and one having two cusps. Our results were obtained with the aid of a computer. The complete list of manifolds (in SnapPea format) and full details on their invariants are available on the world wide web.

math.GT↗

arXiv2002

Complexity and Heegaard genus of an infinite class of compact 3-manifolds

Using the theory of hyperbolic manifolds with totally geodesic boundary, we provide for every integer n greater than 1 a class of such manifolds all having Matveev complexity equal to n and Heegaard genus equal to n+1. All the elements of this class have a single boundary component of genus n, and the numbers of distinct members of the class grows at least exponentially with n.

math.GT↗

arXiv2001

A New Decomposition Theorem for 3-Manifolds

Let M be a (possibly non-orientable) compact 3-manifold with (possibly empty) boundary consisting of tori and Klein bottles. Let $X\subset\partial M$ be a trivalent graph such that $\partial M\setminus X$ is a union of one disc for each component of $\partial M$. Building on previous work of Matveev, we define for the pair (M,X) a complexity c(M,X) and show that, when M is closed, irreducible and P^2-irreducible, $c(M,\emptyset)$ is the minimal number of tetrahedra in a triangulation of M. Moreover c is additive under connected sum, and, given any n>=0, there are only finitely many irreducible and P^2-irreducible closed manifolds having complexity up to n. We prove that every irreducible and P^2-irreducible pair (M,X) has a finite splitting along tori and Klein bottles into pairs having the same properties, and complexity is additive on this splitting. As opposed to the JSJ decomposition, our splitting is not canonical, but it involves much easier blocks than all Seifert and simple manifolds. In particular, most Seifert and hyperbolic manifolds appear to have non-trivial splitting. In addition, a given set of blocks can be combined to give only a finite number of pairs (M,X). Our splitting theorem provides the theoretical background for an algorithm which classifies 3-manifolds of any given complexity. This algorithm has been already implemented and proved effective in the orientable case for complexity up to 9.

math.GT↗

arXiv2000

3-Manifolds with complexity at most 9

We describe an algorithm which has enabled us to give a complete list, without repetitions, of all closed oriented irreducible 3-manifolds of complexity up to 9. More interestingly, we have actually been able to give a "name" to each such manifold, by recognizing its canonical decomposition into Seifert fibred spaces and hyperbolic manifolds. The algorithm relies on the extension of Matveev's theory of complexity to the case of manifolds bounded by suitably marked tori, and on the notion of assembling of two such manifolds. We show that every manifold is an assembling of manifolds which cannot be further deassembled, and we prove that there are surprisingly few such manifolds up to complexity 9. Our most interesting experimental discovery is the following: there are 4 hyperbolic manifolds having complexity 9, and they turn out to be precisely the 4 closed hyperbolic manifolds of least known volume. All other manifolds having complexity at most 9 are graph manifolds.

math.GT↗

arXiv2000

Feynman Diagrams of Generalized Matrix Models and the Associated Manifolds in Dimension 4

The problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which relying on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central role is played by combinatorial topology, often used to recover the space-time manifold from the other structures involved. An extremely attractive possibility is that of encoding all possible space-times as specific Feynman diagrams of a suitable field theory. In this work we analyze how exactly one can associate combinatorial 4-manifolds to the Feynman diagrams of certain tensor theories.

gr-qc↗

arXiv2000

Reidemeister Torsion of 3-Dimensional Euler Structures with Simple Boundary Tangency and Pseudo-Legendrian Knots

We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M). Namely, we allow M to have arbitrary boundary and x to have simple (convex and/or concave) tangency circles to the boundary. We prove that Turaev's H_1(M)-equivariance formula holds also in our generalized context. Our torsions apply in particular to (the exterior of) pseudo-Legendrian knots (i.e. knots transversal to a given vector field), and hence to Legendrian knots in contact 3-manifolds. We show that torsion, as an absolute invariant, contains a lifting to pseudo-Legendrian knots of the classical Alexander invariant. We also precisely analyze the information carried by torsion as a relative invariant of pseudo-Legendrian knots which are framed-isotopic. Using branched standard spines to describe vector fields we show how to explicitly invert Turaev's reconstruction map from combinatorial to smooth Euler structures, thus making the computation of torsions a more effective one. As an example we work out a specific calculation.

math.GT↗

arXiv2000

Combed 3-Manifolds with Concave Boundary, Framed Links, and Pseudo-Legendrian Links

We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to the boundary, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M,v,L), with (M,v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transversality-preserving simultaneous homotopy of v and isotopy of L). All realizations are based on the notion of branched standard spine, and build on results previously obtained. Links are encoded by means of diagrams on branched spines, where the diagram is smooth with respect to the branching. Several motivations for being interested in combinatorial realizations of the topological categories considered in this paper are given in the introduction. The encoding of links is suitable for the comparison of the framed and the pseudo-Legendrian categories, and some applications are given in connection with contact structures, torsion and finite-order invariants. An estension of Trace's notion of winding number of a knot diagram is introduced and discussed.

math.GT↗

arXiv1999

Reidemeister Torsion of 3-Dimensional Euler Structures with Simple Boundary Tangency and Legendrian Knots

We generalize Turaev's definition of torsion invariants of pairs $(M,ξ)$, where $M$ is a 3-dimensional manifold and $ξ$ is an Euler structure on $M$ (a non-singular vector field up to homotopy relative to the boundary of $M$ and local modifications in the interior of $M$). Namely, we allow $M$ to have arbitrary boundary and $ξ$ to have simple (convex and/or concave) tangency circles to the boundary. We prove that Turaev's $H_1(M)$-equivariance formula holds also in our generalized context. Our torsions apply in particular to (the exterior of) Legendrian links (in particular, knots) in contact 3-manifolds, and we prove that they can distinguish knots which are isotopic as framed knots but not as Legendrian knots. Using the combinatorial encoding of vector fields based on branched standard spines we show how to explicitly invert Turaev's reconstruction map from combinatorial to smooth Euler structures, thus making the computation of torsions a more effective one. As an example we work out a specific computation.

math.GT↗

arXiv1999

Negatively Oriented Ideal Triangulations and a Proof of Thurston's Hyperbolic Dehn Filling Theorem

We give a complete proof of Thurston's celebrated hyperbolic Dehn filling theorem, following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. This forces us to deal with negatively oriented tetrahedra. Our analysis of the set of hyperbolic Dehn filling coefficients is elementary and self-contained. In particular, it does not assume smoothness of the complete point in the variety of deformations.

math.GT↗

arXiv1998

Branched Spines and Contact Structures on 3-manifolds

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in a given homotopy class of plane fields.

math.GT↗

arXiv1998

Torsion Invariants of Combed 3-Manifolds with Boundary Pattern and Legendrian Links

We extend Turaev's definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in contact 3-manifolds. Our approach uses a combinatorial encoding of vector fields, based on standard spines. In this paper we extend this encoding from closed manifolds to manifolds with boundary.

math.GT↗