arXiv subjects
Publications and source records attributed to Alberto Cavallo.
We prove that an $n$-component fibered link $L$ in $S^3$ is strongly quasi-positive if and only if $τ(L)=g_3(L)+n-1$, where $g_3(L)$ denotes the Seifert genus and $τ$ is the Ozsváth-Szabó concordance invariant. We also provide a table which contains a list of some fibered prime links with at most 9 crossings; and we explicitly determine the ones that are strongly quasi-positive and their maximal self-linking number.
We use the Heegaard Floer homology cobordism maps to obstruct the existence of a symplectic structure on the Akbulut-Kirby Mazur manifolds whose boundary is a Brieskorn sphere $Y$ among $Σ(2,3,13),$ $Σ(2,5,7)$ and $Σ(3,4,5)$. Furthermore, we describe how they imply the existence of exotic pairs of simply connected 4-manifolds, with definite intersection form, whose boundary is a Brieskorn sphere bounding a Mazur manifold.
We adapt the Ozsváth-Szabó full path algorithm to every star-shaped graph and establish a correspondence between negative-twisting tight contact structures on any Seifert fibred space over $S^2$, and its Heegaard Floer homology groups equipped with the Alexander filtration induced by the regular fibre. This provides the complete classification of negative-twisting structures on these manifolds; in particular, we distinguish them by their contact invariant $c^+$. We prove that every such structure is symplectically fillable and extend a known obstruction to Stein fillability. In addition, we show that the number of negative-twisting structures can be expressed combinatorially in terms of the Seifert coefficients of the star-shaped graph, while their $d_3$-invariant and homotopy type are determined explicitly through our correspondence. Our results also complete the classification of fillable structures on any small Seifert fibred space.
We classify fillable contact structures on all negative-definite star-shaped plumbings. We show that such Seifert fibred spaces admit a unique negative maximal twisting number and compute it explicitly using the Alexander filtration in lattice cohomology, providing its first Floer-theoretic interpretation. In addition, we show that all the negative-twisting tight structures on these manifolds are induced by the Stein structures on the minimal resolution of the underlying complex surface singularity. As an application, we provide a necessary condition for a negative-definite Seifert fibred space to admit a separating contact-type embedding in a strong symplectic filling of a generalised $L$-space.
Applying our recent classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere, we provide the complete list of all the Brieskorn spheres carrying at most two symplectically fillable structures, up to isotopy, compatible with a given orientation.
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.
We advance Matkovič ideas, originally applied to complete the classification of tight structures on small Seifert fibred $L$-spaces, to show the existence of contact structures on Brieskorn spheres which are tight and zero-twisting. This uncovers a phenomenon that has never appeared in literature before: namely, that a contact structure $ξ$ on a 3-manifold can be such that $\widehat c(ξ)$ is non-vanishing, but $c^+(ξ)$ is zero.
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.
We study in details how and when the radical $\sqrt[3]{a+b\sqrt p}$ with rational numbers $a,b$ and $p$ positive can be simplified, providing a complete answer to the problem; furthermore, a program that computes the result is also made available. The solution highlights an interesting connection with the cubic formula.
We collect some observations about Legendrian links with non-vanishing contact invariants, mostly concerning the non-loose realizations of links and the addition of boundary-parallel half Giroux torsion. In particular, we show that every null-homologous link with irreducible complement admits a non-loose Legendrian realization with non-zero (at least) invariant $\text{EH}$ in $\text{SFH}$, in some overtwisted contact structure (determined by its gradings); for many links these come from Gabai's work, for others the existence follows from the sutured interpretation of link Floer invariants. We reveal that separating half Giroux torsion does not necessary cause Legendrian invariants to vanish. Furthermore, we propose a conjectural characterization of links with non-vanishing $\widehat{\mathfrak L}$ in $\widehat{\text{HFL}}$ among links with non-zero $\mathfrak L$ in $c\text{HFL}^-$.
We classify all the $n$-component links in the $3$-sphere that bound a Thurston norm minimizing Seifert surface $Σ$ with Euler characteristic $χ(Σ)=n-2$ and that are nearly fibered, which means that their rank of the maximal (collapsed) Alexander grading $s_{\text{top}}$ of the link Floer homology group $\widehat{HFL}$ is equal to two. In other words, such a link $L$ satisfies $s_{\text{top}}=\frac{n-χ(Σ)}{2}=1$, and in addition $\text{rk}\:\widehat{HFL}_{*}(L)[1]=2$ and $\text{rk}\:\widehat{HFL}_{*}(L)[s]=0$ for every $s>1$. The proof of the main theorem is inspired by the one of a similar recent result for knots by Baldwin and Sivek; and involves techniques from sutured Floer homology. Furthermore, we also compute the group $\widehat{HFL}$ for each of these links.
We show that the local equivalence class of the collapsed link Floer complex $cCFL^\infty(L)$, together with many $Υ$-type invariants extracted from this group, is a concordance invariant of links. In particular, we define a version of the invariants $Υ_L(t)$ and $ν^+(L)$ when $L$ is a link and we prove that they give a lower bound for the slice genus $g_4(L)$. Furthermore, in the last section of the paper we study the homology group $HFL'(L)$ and its behaviour under unoriented cobordisms. We obtain that a normalized version of the $\upsilon$-set, introduced by Ozsváth, Stipsicz and Szabó, produces a lower bound for the 4-dimensional smooth crosscap number $γ_4(L)$.
We study the set $\widehat{\mathcal S}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold $M$. We show that $\widehat{\mathcal S}_M$ is well-defined and describe how it relates to exotic phenomena in dimension four. In particular, in the case when $X$ is smooth, with a handle decompositions with no 1-handles and homeomorphic to but not smoothly embeddable in $D^4$, our results tell us that $X$ is exotic if and only if there is a link $L\hookrightarrow S^3$ which is smoothly slice in $X$, but not in $D^4$. Furthermore, we extend the notion of high genus 2-handle attachment, introduced by Hayden and Piccirillo, to prove that exotic 4-disks that are smoothly embeddable in $D^4$, and therefore possible counterexamples to the smooth 4-dimensional Schönflies conjecture, cannot be distinguished from $D^4$ only by comparing the slice genus functions of links.
We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases: whether or not the cubic roots of unity belong to $K$. We also give a criterion to determine whether a polynomial as $f(x)$ is irreducible, when $F$ is a finite field. Moreover, at the end of the paper we also give a complete list of all the possible subfields of $F$.
Every $L$-space knot is fibered and strongly quasi-positive, but this does not hold for $L$-space links. In this paper, we use the so called H-function, which is a concordance link invariant, to introduce a subfamily of fibered strongly quasi-positive $L$-space links. Furthermore, we present an infinite family of $L$-space links which are not quasi-positive.
We introduce a generalization of the Lisca-Ozsváth-Stipsicz-Szabó Legendrian invariant $\mathfrak L$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link $L$ in a contact 3-manifold $(M,ξ)$ with a diagram $D$, given by an open book decomposition of $(M,ξ)$ adapted to $L$, and we construct a chain complex $cCFL^-(D)$ with a special cycle in it denoted by $\mathfrak L(D)$. Then, given two diagrams $D_1$ and $D_2$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complexes, that induces an isomorphism in homology and sends $\mathfrak L(D_1)$ into $\mathfrak L(D_2)$. Moreover, a connected sum formula is also proved and we use it to give some applications about non-loose Legendrian links; that are links such that the restriction of $ξ$ on their complement is tight.
We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere $(M,ξ)$, whenever $ξ$ is tight. More specifically, we show that the self-linking number of a transverse link $T$ in $(M,ξ)$, such that the boundary of its tubular neighbourhood consists of incompressible tori, is bounded by the Thurston norm $||T||_T$ of $T$. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off. We apply this bound to compute the tau-invariant for every strongly quasi-positive link in $S^3$. This is done by proving that our inequality is sharp for this family of smooth links. Moreover, we use a stronger Bennequin inequality, for links in the tight 3-sphere, to generalize this result to quasi-positive links and determine their maximal self-linking number.
The weak splitting number $wsp(L)$ of a link $L$ is the minimal number of crossing changes needed to turn $L$ into a split union of knots. We describe conditions under which certain $\mathbb{R}$-valued link invariants give lower bounds on $wsp(L)$. This result is used both to obtain new bounds on $wsp(L)$ in terms of the multivariable signature and to recover known lower bounds in terms of the $τ$ and $s$-invariants. We also establish new obstructions using link Floer homology and apply all these methods to compute $wsp$ for all but two of the $130$ prime links with $9$ or fewer crossings.