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Irena Matkovič

Publications and source records attributed to Irena Matkovič.

10 recordsLinked to original sources
arXiv2026

Heegaard Floer homology and maximal twisting numbers

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.

math.GT↗
arXiv2026

Fillable structures on negative-definite Seifert fibred spaces

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.

math.GT↗
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↗
arXiv2026

The hat and plus version of the Heegaard Floer contact invariant are not equivalent

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.

math.GT↗
arXiv2023

Legendrian invariants and half Giroux torsion

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 $c\text{HFL}^-$.

math.GT↗
arXiv2023

Nearly fibered links with genus one

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.

math.GT↗
arXiv2022

On contact surgery and knot Floer invariants

We establish some general relations between Heegaard Floer based contact invariants. In particular, we observe that if the contact invariant of large negative, respectively positive, contact surgeries along a Legendrian knot does not vanish, then the Legendrian invariant, respectively the Legendrian inverse limit invariant, of that knot is non-zero. We use sutured Floer homology, and the limit constructions due to Golla, and Etnyre, Vela-Vick and Zarev.

math.GT↗
arXiv2022

Non-loose negative torus knots

We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. Additionally, we show that the strongly non-loose transverse realizations $T$ are classified by their non-zero invariants $\mathfrak T(T)$ in the minus version of the knot Floer homology. However, not all the elements of $HFK^-(T_{(p,q)})$ can be realized. Along the way, we relate our Legendrian realizations to the tight contact structures on the Legendrian surgeries along them. Specifically, we realize all tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in overtwisted structures to the non-fillable tight structures on the large negative surgeries along the underlying knots.

math.GT↗
arXiv2020

Fillability of small Seifert fibered spaces

On small Seifert fibered spaces $M(e_0;r_1,r_2,r_3)$ with $e_0\neq-1,-2,$ all tight contact structures are Stein fillable. This is not the case for $e_0=-1$ or $-2$. However, for negative twisting structures it is expected that they are all symplectically fillable. Here, we characterize fillable structures among zero-twisting contact structures on small Seifert fibered spaces of the form $M(-1;r_1,r_2,r_3)$. The result is obtained by analyzing monodromy factorizations of associated planar open books.

math.GT↗