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The main subjects of my PhD project are the Thurston norm and taut foliations on 3-dimensional manifolds.
The main subjects of my PhD project are the Thurston norm and taut foliations on 3-dimensional manifolds.
The Thurston norm is a seminorm on the second real homology group of a compact orientable 3-manifold. The unit ball of this norm is a convex polyhedron, whose shape's data (e.g. number of vertices, regularity) measures the complexity of the surfaces sitting in the ambient 3-manifold. Some of the principal general problems I would like to address are:
The Thurston norm is a seminorm on the second real homology group of a compact orientable 3-manifold. The unit ball of this norm is a convex polyhedron, whose shape's data (e.g. number of vertices, regularity) measures the complexity of the surfaces sitting in the ambient 3-manifold. Some of the principal general problems I would like to address are:
- What polyhedra can be realised as the unit ball of the Thurston norm on some 3-manifold?
- Given a 3-manifold, compute the unit ball of its Thurston norm.
Preprints (click on titles for access to papers)
Preprints (click on titles for access to papers)
Journal of Knot Theory and Its Ramifications Vol. 34, No. 12, 2550059 (2025)
We show that, when M is the complement of a 2-bridge link L with components l_1 and l_2, the Thurston ball of M has at most 8 faces. The proof of this result strongly relies on a description of essential surfaces in 2-bridge link complements given by Floyd and Hatcher. Then, we exhibit norm-minimizing representatives for the integral classes of H_2(M,\partial M) and use them to compare the complexity of the Thurston ball with the complexities of L and of M.
As an example, we show that all the vertices of the Thurston ball lie on the bisectors if and only if M fibers over the circle with fiber a surface with boundary equal to a longitude of l_1 and some meridians of l_2. Finally, we use 2-bridge links in satellite constructions to find 2-component links whose complements in S^3 have Thurston balls with arbitrarily many vertices.
Accepted for publication in Journal of Topology and Analysis
The Thurston norm of a closed oriented graph manifold is a sum of absolute values of linear functionals, and either each or none of the top-dimensional faces of its unit ball are fibered. We show that, conversely, every norm that can be written as a sum of absolute values of linear functionals with rational coefficients is the nonvanishing Thurston norm of some graph manifold, with respect to a rational basis on its second real homology. Moreover, we can choose such graph manifold either to fiber over the circle or not.
In particular, every symmetric polygon with rational vertices is the unit polygon of the nonvanishing Thurston norm of a graph manifold fibering over the circle. In dimension 3 or more, many symmetric polyhedra with rational vertices are not realizable as nonvanishing Thurston norm ball of any graph manifold. However, given such a polyhedron, we show that there is always a graph manifold whose nonvanishing Thurston norm ball induces a finer partition into cones over the faces.
A formula for the Euler class of foliations (work in progress)
A formula for the Euler class of foliations (work in progress)
Given a cooriented branched surface B fully carrying a foliation F, we use the dual graph of B to define a simplicial 1-cycle Gamma(B) that represents the Poincaré dual to the Euler class of F. This construction easily allows one to keep track of how the Euler class of a foliation carried by a hierarchy changes once a decomposing disk is reversed orientation.
Taut branched surfaces in the exterior of even arborescent links
Taut branched surfaces in the exterior of even arborescent links
(work in progress)
(work in progress)
Let L be an even arborescent link with components L_1,...,L_n. For any permutation s in S_n, we construct a taut branched surface B_s in the link exterior M_L carrying taut representatives for the homology classes L_s(1), L_s(1)+L_s(2),...,L_s(1)+...+L_s(n). Consequently, the cone spanned by the flag L_s(1), L_s(1)+L_s(2),...,L_s(1)+...+L_s(n) in the homology of the link exterior H_2(M_L, dM_L; R) is contained in the closed cone over a top-dimensional face of the Thurston ball.
On a different note, we show that if a nonseparating surface S in a Haken manifold M with toroidal boundary does not intersect a boundary torus, then the class [S] in H_2(M,dM) is not in the interior of a top-dimensional cone of the Thurston norm.
For instance, if two distinct components L_i and L_j of a link L have zero linking number, then neither L_i nor L_j is interior to top-dimensional cones. If L has just two components, then these directions are aligned with vertices of the Thurston ball.
Other
Other
As Master thesis I studied a proof of Thurston's Theorem about the hyperbolization of the 3-manifolds that fiber over the circle. Unfortunately, the thesis is in Italian, but perhaps it will be in English one day too!
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