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Antonio Alfieri

Publications and source records attributed to Antonio Alfieri.

15 recordsLinked to original sources
arXiv2026

Brieskorn spheres and rational homology ball symplectic fillings

Given a canonically oriented Brieskorn sphere $Y=Σ(a_1,...,a_n)$, we confirm some statements conjectured by Gompf. More specifically, we obstruct the existence of rational homology ball symplectic fillings for any contact structure on $-Y$ if $n=3$, and when there is no half convex Giroux torsion for $n>3$. Furthermore, we show that the same result holds for the Milnor fillable structure on $Y$ with the possible exception of $Σ(3,4,5),$ $Σ(2,5,7)$ and $Σ(2,3,6k+1)$ for $k\geq1$. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.

math.GT↗
arXiv2025

Holomorphic curves in Stein domains and the tau-invariant

The scope of the paper is threefold. First, we build on recent work by Hayden to compute Hedden's tau-invariant $τ_ξ(L)$ in the case when $ξ$ is a Stein fillable contact structure on a rational homology sphere, and $L$ is a transverse link arising as the boundary of a pseudo-holomorphic curve. This leads to a new proof of the relative Thom conjecture for Stein domains. Secondly, we compare the invariant $τ_ξ$ to the Grigsby-Ruberman-Strle topological tau-invariant $τ_{\mathfrak s}$, associated to the $\text{Spin}^c$-structure $\mathfrak s=\mathfrak s_ξ$ of the contact structure $ξ$, to obtain topological obstructions for a link type to admit a holomorphically fillable transverse representative. Finally, we use our main result together with methods from lattice cohomology to compute the $τ_{\mathfrak s}$-invariants of certain links in lens spaces, and estimate their PL slice genus.

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arXiv2025

Heegaard Floer theory and pseudo-Anosov flows II: Differential and Fried pants

In earlier work, relying on work of Agol-Guéritaud and Landry-Minsky-Taylor, we showed that given a pseudo-Anosov flow $(Y,ϕ)$ and a collection of closed orbits $\mathcal{C}$ satisfying the `no perfect fit' condition, one can construct a special Heegaard diagram for the link complement $Y^\sharp= Y \setminus ν(\mathcal{C})$ framed by the degeneracy curves. In this paper, we demonstrate how the special combinatorics of this diagram can be used to understand the differential of the associated Heegaard Floer chain complex. More specifically, we introduce a refinement of the $\text{spin}^\text{c}$-grading obstructing two Heegaard states from being connected by an effective domain. We describe explicitly the subcomplexes in the refined gradings that represent irreducible multi-orbits, in the sense that they contain states corresponding to multi-orbits which cannot be resolved along Fried pants. In particular we show that the homology of these subcomplexes are 1-dimensional. When specialized to the case of suspension flows our arguments prove some results in the spirit of Ni, Ghiggini, and Spano: the next-to-top non-zero sutured Floer group counts the number of periodic points of least period.

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arXiv2025

Heegaard Floer theory and pseudo-Anosov flows I: Generators and categorification of the zeta function

We bring to light a new connection between dynamics and Heegaard Floer homology. On a closed 3-manifold $Y$ we consider a pseudo-Anosov flow $ϕ$ with no perfect fits with respect to its singularity locus $L \subset Y$, or perhaps a larger collection of closed orbits. Using work of Agol and Guéritaud on veering branched surfaces we produce a chain complex computing the link Floer homology of $L$ in the framing specified by the degeneracy curves of the flow. Using work of Landry, Minsky, and Taylor we show that the generators of the chain complex correspond to certain closed multi-orbits of $ϕ$. We prove that two canonical generators $\mathbf{x}^\mathrm{top}$ and $\mathbf{x}^\mathrm{bot}$ determine non-trivial homology classes located in the $\text{spin}^\text{c}$-grading of the flow, and its opposite. Finally, we observe that our specific model of the chain complex for link Floer homology naturally supports a grading with dynamical significance. This grading, a modification of the regular Maslov grading, is shown to categorify a suitable normalization of the zeta function associated to $ϕ$.

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arXiv2024

Is the geography of Heegaard Floer homology restricted or the $L$-space conjecture false?

In a recent note F. Lin showed that if a rational homology sphere $Y$ admits a taut foliation then the Heegaard Floer module $HF^-(Y)$ contains a copy of $\mathbf{F}[U]/U$ as a summand (arXiv:2309.01222). This implies that either the $L$-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. We verify that Lin's geography restriction holds for a wide class of rational homology spheres. Indeed, we show that the Heegaard Floer module $HF^-(Y)$ may satisfy a stronger geography restriction.

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arXiv2023

Involutions and the Chern-Simons filtration in instanton Floer homology

Building on the work of Nozaki, Sato and Taniguchi, we develop an instanton-theoretic invariant aimed at studying strong corks and equivariant bounding. Our construction utilizes the Chern-Simons filtration and is qualitatively different from previous Floer-theoretic methods used to address these questions. As an application, we give an example of a cork whose boundary involution does not extend over any 4-manifold $X$ with $H_1(X, \mathbb{Z}_2) = 0$ and $b_2(X) \leq 1 $, and a strong cork which survives stabilization by either of $n\smash{\mathbb{CP}^2}$ or $n\smash{\overline{\mathbb{CP}}}^2$. We also prove that every nontrivial linear combination of $1/n$-surgeries on the strongly invertible knot $\smash{\overline{9}_{46}}$ constitutes a strong cork. Although Yang-Mills theory has been used to study corks via the Donaldson invariant, this is the first instance where the critical values of the Chern-Simons functional have been utilized to produce such examples. Finally, we discuss the geography question for nonorientable surfaces in the case of extremal normal Euler number.

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arXiv2022

Instanton Floer homology of almost-rational plumbings

We show that if $Y$ is the boundary of an almost-rational plumbing, then the framed instanton Floer homology $\smash{I^\#(Y)}$ is isomorphic to the Heegaard Floer homology $\smash{\widehat{\mathit{HF}}(Y; \mathbb{C})}$. This class of 3-manifolds includes all Seifert fibered rational homology spheres with base orbifold $S^2$ (we establish the isomorphism for the remaining Seifert fibered rational homology spheres$\unicode{x2014}$with base $\mathbb{RP}^2$$\unicode{x2014}$directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recently established by Baldwin and Sivek.

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arXiv2021

Upsilon invariants from cyclic branched covers

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $Υ_m^C (K)$ of knots in $S^3$. We give computations of these invariants for some families of alternating knots and reprove some independence results in the smooth concordance group.

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arXiv2020

Deformations of lattice cohomology and the upsilon invariant

We introduce deformations of lattice cohomology corresponding to the knot homologies found by Ozsv\' ath, Stipsicz and Szab\' o in \cite{OSS4}. By means of holomorphic triangles counting, we prove equivalence with the analytic theory for a wide class of knots. This yields combinatorial formulae for the upsilon invariant.

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arXiv2019

Connected Floer homology of covering involutions

Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.

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arXiv2019

On sums of torus knots concordant to alternating knots

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

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arXiv2019

An introduction to knot Floer homology and curved bordered algebras

We survey Ozsváth-Szabó's bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts ($\mathcal{A}_\infty$-modules, type $D$-structures, box tensor, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsváth and Szabó's analytic construction of the type $D$-structure associated to an upper diagram. Finally we give an explicit description of the structure maps of the $DA$-bimodules of some elementary partial diagrams. These can be used to perform explicit computations of the knot Floer differential of any knot in $S^3$. The boundary DGAs $\mathcal{B}(n,k)$ and $\mathcal{A}(n,k)$ of [7] are replaced here by an associative algebra $\mathcal{C}(n)$. These are the notes of two lecture series delivered by Peter Ozsváth and Zoltán Szabó at Princeton University during the summer of 2018.

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arXiv2017

Upsilon type concordance invariants

To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant $Υ^C$. For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's $h_i$ invariants, and the Ozsv\' ath-Stipsicz-Szab\' o upsilon invariant. Furthermore, to three such regions $C$, $C^+$ and $C^- $ we associate invariants $Υ_{C^\pm, C}$ generalising Kim-Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use them to obstruct concordances to Floer thin knots and algebraic knots.

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