arXiv subjects

Alessandro V. Cigna

Publications and source records attributed to Alessandro V. Cigna.

4 recordsLinked to original sources

arXiv2026

A formula for the Euler class of foliations

Given a cooriented branched surface $\mathcal B$ fully carrying a foliation $\mathcal F$, we use the dual graph of $\mathcal B$ to define a simplicial 1-cycle $Γ_m(\mathcal B)$ representing the Poincaré dual of the Euler class of $\mathcal F$ relative to the boundary. As an example, we complete the classification of which homology classes in the Whitehead link exterior are realisable as relative Euler classes of taut foliations. We also show how our formula generalises previous results of Lackenby and Dunfield. Finally, we observe that cooriented branched surfaces whose complement is a union of balls satisfy a Combinatorial Transverse Surface Theorem, in the sense of Landry--Minsky--Taylor.

math.GT↗

arXiv2026

Sutured manifold hierarchies and the Thurston norm

By the classical work of Thurston and Gabai, the Thurston norm of any compact oriented irreducible $3$-manifold with toroidal boundary is determined by finitely many taut sutured manifold hierarchies. We give an explicit procedure to extract this information from such hierarchies. This is achieved via the maw dual graph construction, which can be incorporated into a general method for computing the Thurston norm of a manifold. As an application, we compute the Thurston norm of the exterior of all alternating and some non-alternating pretzel links with three components. Using these computations, we give a negative answer to a question of Baker--Taylor. Namely, we exhibit an infinite family of manifolds, and an infinite family of knots in each manifold, such that the wrapping number associated with these knots is not a seminorm on the respective second real homology group. Next, we show that if a taut surface in a link exterior is disjoint from a boundary torus and does not represent a corner of the Thurston norm ball, then the corresponding link component is algebraically split from the rest of the link. Moreover, the surface remains norm-minimising under every non-longitudinal Dehn filling on that boundary component and, under an assumption of atoroidality, is a leaf of a taut foliation transverse to the filling core.

math.GT↗

arXiv2025

The Thurston norm of graph manifolds

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 $\ge 3$ 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.

math.GT↗

arXiv2024

The Thurston norm of 2-bridge link complements

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. Unfortunately, the Thurston norm is generally quite hard to compute, and a long-standing problem is to understand which polyhedra are realised as the unit balls of the Thurston norms of $3$-manifolds. We show that, when $M$ is the complement of a $2$-bridge link $L$ with components $\ell_1$ and $\ell_2$, 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 $\ell_1$ and some meridians of $\ell_2$. 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.

math.GT↗