Narrow Results

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle–Hernandez and Némethi and is based on the Bézout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsváth–Szabó inequalities for $d$-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step, we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Némethi. We also show that the construction cannot always work in the reverse direction: in fact, the U-filtration depth of contact Ozsváth–Szabó invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsváth–Szabó invariants of those contact structures fillable by an AR plumbing graph, generalizing an earlier work of the first author.

We define a new combinatorial complex computing the hat version of link Floer homology over ℤ/2ℤ, which turns out to be significantly smaller than the Manolescu–Ozsváth–Sarkar one.

We use the Heegaard Floer obstructions defined by Grigsby, Ruberman, and Strle to show that forty-five of the sixty-six knots through eleven crossings whose concordance orders were previously unknown have infinite concordance order.

In this survey paper, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsváth and Szabó. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

Let $D(K)$ be the positively clasped untwisted Whitehead double of a knot $K$, and $T_{p,q}$ be the $(p,q)$ torus knot. We show that $D(T_{2,2m+1})$ and $D^2(T_{2,2m+1})$ are linearly independent in the smooth knot concordance group ${𝒞}$ for each $m\ge 2$. Further, $D(T_{2,5})$ and $D^2(T_{2,5})$ generate a $\mathbb{Z}\oplus \mathbb{Z}$ summand in the subgroup of ${𝒞}$ generated by topologically slice knots. We use the concordance invariant $\delta$ of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the $\tau$-invariant of knot Floer theory and the $s$-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of $D(T_{2,2m+1})$ for any $m\ge 1$ generalizing a result for $T_{2,3}$ of Hedden, Kim and Livingston.

In [D. Auckly, H. J. Kim, P. Melvin and D. Ruberman, Equivariant corks, Algebr. Geom. Topol.17(3) (2017) 1771–1783], Auckly–Kim–Melvin–Ruberman showed that for any finite subgroup $G$ of $SO(4)$ there exists a contractible smooth 4-manifold with an effective $G$-action on its boundary so that the twists associated to the non-trivial elements of $G$ don’t extend to diffeomorphisms of the entire manifold. We give a different proof of this phenomenon using the Heegaard Floer techniques in [S. Akbulut and Ç. Karakurt, Action of the cork twist on Floer homology, in Proc. Gökova Geometry-Topology Conf. 2011 (International Press, Somerville, MA, 2012), pp. 42–52].

In 2003, Ozsváth and Szabó defined the concordance invariant $\tau$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $\tau$ for knots in $S^3$ and a combinatorial proof that $\tau$ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in $S^3$, extending HFK for knots. We define a $\mathbb{Z}$-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined $\tau$ invariant for balanced spatial graphs generalizing the $\tau$ knot concordance invariant. In particular, this defines a $\tau$ invariant for links in $S^3$. Using techniques similar to those of Sarkar, we show that our $\tau$ invariant is an obstruction to a link being slice.

We compute the involutive knot invariants for pretzel knots of the form P(−2,m,n) for m and n odd and greater than or equal to 3.

In this paper, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsváth-Stipsicz-Szabó can be combinatorially computed. We rely on Plamenevskaya’s combinatorial description of the Heegaard Floer contact invariant.

We define a notion of Heegaard Floer homology for three-dimensional orbifolds with arbitrary cyclic singularities, generalizing the recent work of Biji Wong where the singular locus is assumed to be connected.

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens–Golla $\delta$-invariant to obstruct a knot from being a stabilization of another.

Dai, Hom, Stoffregen and Truong defined a family of concordance invariants ${\phi }_j$. The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots.

The present article aims to discuss the graded roots introduced by the author in his study of the topology of normal surface singularities. In the body of the paper we emphasize two aspects of them: their potential role in the classification of normal surface singularities, and also their connections with the Seiberg–Witten (and Heegaard–Floer) theory of rational homology sphere 3–manifolds. The article contains many non-trivial applications and examples.

We study when a lens space is homeomorphic to p-Dehn surgery of a knot in S³. Using correction term defined by P. Ozsváth and Z. Szabó, we will prove some obstructions of Alexander polynomial of K.