arXiv subjects

Joan Licata

Publications and source records attributed to Joan Licata.

8 recordsLinked to original sources

arXiv2026

Morse diagrams, Murasugi sums, and the mapping class group

A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition.

math.GT↗

arXiv2026

The Giroux Correspondence in dimension 3

This paper proves the Giroux Correspondence in dimension three using Heegaard splittings of contact manifolds. In two of the authors earlier paper they proved the Giroux Correspondence for tight contact 3-manifolds via convex Heegaard surfaces, and simultaneously, Honda, Breen and Huang gave an alldimensions proof of the Giroux Correspondence by generalising convex surface theory to higher dimensions. This paper extends the Heegaard splitting approach to arbitrary (not necessarily tight) contact 3-manifolds in order to provide a proof accessible to a low-dimensional audience. The proof assumes classification moves relating bypass decompositions for isotopic contact structures on cobordisms that are topological products; in the Appendix, we prove this result in the 3- dimensional setting.

math.GT↗

arXiv2025

Common positive stabilisation of open book decompositions

The Giroux Correspondence states that two open book decompositions supporting the same contact structure are related by a sequence of positive open book stabilisations and destabilisations. In this note we show that any two open book decompositions supporting isotopic contact structures admit a common positive stabilisation.

math.GT↗

arXiv2025

Liftable braids and the coloured braid groupoid

When $π:\widetildeΣ\rightarrow D^2$ is a cover of the disc branched over $n$ marked points, the braid group $B_n$ acts on the disc by homeomorphisms fixing the marked points setwise. A braid $β$ \textit{lifts} if there is a homeomorphism $\widetildeβ\in \textit{Mod}(\widetildeΣ)$ such that $β\circ π=π\circ \widetildeβ$. For arbitrary covers, the \textit{lifting homomorphism} taking $β$ to $\widetildeβ$ is only defined on a proper subgroup of the braid group. This paper extends the lifting homomorphism to a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc. We characterise the lift of every coloured braid, recovering the classical lifting homomorphism on the liftable braid group.

math.GT↗

arXiv2024

Heegaard splittings and the tight Giroux Correspondence

This paper presents a new proof of the Giroux Correspondence for tight contact $3$-manifolds using techniques from Heegaard splittings and convex surface theory. We introduce tight Heegaard splittings, which generalise the Heegaard splittings naturally induced by an open book decomposition of a contact manifold. Via a process called refinement, any tight Heegaard splitting determines an open book, up to positive open book stabilisation. This allows us to translate moves relating distinct tight Heegaard splittings into moves relating their associated open books. We use these tools to show that every Heegaard splitting of a contact 3-manifold may be stabilised to a splitting associated to a supporting open book decomposition. Finally, we prove the tight Giroux Correspondence, showing that any pair of open book decompositions compatible with isotopic contact structures become isotopic after a sequence of positive open book stabilisations.

math.GT↗

arXiv2023

Arc diagrams on 3-manifold spines

We develop a theory of link projections to trivalent spines of 3-manifolds. We prove a Reidemeister Theorem providing a set of combinatorial moves sufficient to relate the projections of isotopic links. We also show that any link admits a crossingless projection to any special spine and we refine our theorem to provide a set of combinatorial moves sufficient to relate crossingless diagrams. Finally, we discuss the connection to Turaev's shadow world, interpreting our result as a statement about shadow equivalence of a class of 4-manifolds.

math.GT↗

arXiv2023

Bordered Floer homology and contact structures

We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $(M, ξ, \mathcal{F})$ whose convex boundary is equipped with a signed singular foliation $\mathcal{F}$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux Correspondence, as described in earlier work of Licata and Vértesi. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $c_D$ and $c_A$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $\mathcal{F}$ in favor of the dividing set, and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda-Kazez-Matić.

math.GT↗

arXiv2022

A friendly introduction to the bordered contact invariant

We give a short introduction to the contact invariant in bordered Floer homology defined by Földvári, Hendricks, and the authors. The construction relies on a special class of foliated open books. We discuss a procedure to obtain such a foliated open book and present a definition of the contact invariant. We also provide a "local proof", through an explicit bordered computation, of the vanishing of the contact invariant for overtwisted structures.

math.GT↗