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Antonio Alfieri and Alberto CavalloCentre de recherches mathématiques (CRM), Montréal (QC) H3C 3J7, Canadaantonioalfieri90@gmail.comInstitute of Mathematics of the Polish Academy of Sciences (IMPAN), Warsaw 00-656, Polandacavallo@impan.pl

###### Abstract.

The scope of the paper is threefold. First, we build on recent work by Hayden to compute Hedden’s tau-invariant $\tau_{\xi}(L)$ in the case when $\xi$ is a Stein fillable contact structure on a rational hom*ology sphere, and $L$ is a transverse link arising as the boundary of a holomorphic curve. This leads to a new proof of the relative Thom conjecture for Stein domains. Secondly, we compare the invariant $\tau_{\xi}$ to the Grigsby-Ruberman-Strle topological tau-invariant $\tau_{\mathfrak{s}}$, associated to the $\operatorname{Spin}^{c}$-structure $\mathfrak{s}=\mathfrak{s}_{\xi}$ of the contact structure $\xi$, to obtain topological obstructions for a link type to admit a holomorphically fillable transverse representative. We combine the latter with a result of Mark and Tosun to confirm a conjecture of Gompf: no standardly-oriented Brieskorn hom*ology sphere admits a rational hom*ology ball Stein filling. Finally, we use our main result together with methods from lattice cohom*ology to compute the $\tau_{\mathfrak{s}}$-invariants of certain links in lens spaces, and estimate their PL slice genus.

## 1. Introduction

A knot $K$ in $S^{3}$ is called slice if it bounds a smooth disk $\Delta$ in $D^{4}$. The question of characterising slice knots was first asked by Fox in 1985 and lead to a lot of interesting mathematics in recent years.

To obstruct a knot from being slice one can look at some classical invariants like the Arf invariant or the Tristram-Levine signatures. If these invariants do not work, one can resort to sophisticated knot invariants like the Ozsváth-Szabó $\tau$-invariant [43], and the Rasmussen $s$-invariant [49]. These invariants contain a lot of information, but unlike classical invariants they can be extremely hard to compute. Nevertheless, computations can be performed for some particular classes of knots and links with nice combinatorics. This is the case of alternating knots [35], links of plane curves singularities [22], and *quasi-positive* links.

Recall that a link $L$ in $S^{3}$ is called quasi-positive if it can be expressed as the closure of an $n$-braid $\beta$ that has a factorisation of the form:

$\beta=(w_{1}\sigma_{j_{1}}w_{1}^{-1})\cdot...\cdot(w_{b}\sigma_{j_{b}}w_{b}^{-%1})\ ,$ |

where the $\sigma_{i}$’s are the Artin generators of the $n$-braid group. It was shown [26, 8, 9] that the identity

$\tau(L)=\frac{w(\beta)-n+\left|L\right|}{2}$ | (1.1) |

holds for any quasi-positive link $L$ in $S^{3}$, where $|L|$ denotes the number of components of the link $L$, and $w(\beta)$ denotes the writhe of $\beta$, which is defined as the difference between the number of positive and negative generators in the factorisation of $\beta$. Quasi-positive links are exactly those links that arise as boundary of holomorphic curves properly embedded in $D^{4}$ [6, 50].This paper is devoted to the study of links that can be realised as boundary of $J$-holomorphic curves in Stein domains.

### Rational slice genus

To obstruct a knot $K$ in $S^{3}$ from being slice one can look at its double branched cover $\Sigma(K)$. If the knot $K$ is slice then $\Sigma(K)$ bounds a rational hom*ology ball $X$, namely the double branched cover of $D^{4}$ along a slice disk. This observation has been used by Lisca [31] to confirm the slice-ribbon conjecture for many interesting classes of knots. His work uses Donaldson’s theorem and relies on some heavy, but elementary arithmetics.

One of the main limitations of Lisca’s technique is that, for reasons that are not related to the topology of the knot $K$, the double branched cover $\Sigma(K)$ may bound a rational hom*ology ball. This is the case for the Conway knot for example, whose double cover can be obtained by surgery on a slice knot. To obtain further information one can look at the pull-back knot $\widetilde{K}\subset\Sigma(K)$. Indeed, if a knot is slice, not only $\Sigma(K)=\partial X$ for some rational hom*ology ball $X$ but one can also find a smooth disk $\Delta\subset X$ with $\partial\Delta=\widetilde{K}$. This leads to the following question.

###### Question 1.1.

Suppose that $K$ is a knot, in a rational hom*ology sphere $Y$, and that $Y=\partial X$ for some rational hom*ology ball $X$. Then is it possible to find a smooth disk $\Delta\subset X$ with $\partial\Delta=K$?

If the answer to Question 1.1 is affirmative for a knot $K$ in $Y$ we say that $K$ is *rationally slice* in $X$. Of course if $Y=\partial X$ for some rational hom*ology ball $X$ we can consider the obvious notion of genus:

$g^{X}_{4}(K)=\min_{F}\left\{g(F):F\subset X\text{ oriented surface, }\partial F%=K\right\},$ |

and define $g_{4}(K)=\displaystyle\min_{X}\left\{g^{X}_{4}(K):X\text{ rational hom*ology %ball with }\partial X=Y\right\}$.

###### Remark 1.2.

One can also consider the notion of slice genus of links in rational hom*ology spheres. In the case of links we use the negative of the Euler characteristics $-\chi(F)$, to measure the complexity of surfaces. We define

$\chi_{4}^{X}(L,\Sigma)=\max_{F}\left\{\chi(F):F\subset X\text{ oriented %surface, }\partial F=L,[F]=\Sigma\in H_{2}(X,\partial X;\mathbb{Z})\right\}\ .$ |

If $X$ is a rational hom*ology ball then the slice genus of $L$ in $X$ is denoted by $\chi_{4}^{X}(L)$, and again we set $\chi_{4}(L)$ to be the maximum of $\chi_{4}^{X}(L)$ over all rational balls $X$ bounding the underlying three-manifold.

In [23] Grigsby, Ruberman and Strle proposed to study Question 1.1 with the methods of Heegaard Floer hom*ology. They associate to a knot $K$ in a rational hom*ology sphere $Y$, equipped with a $\operatorname{Spin}^{c}$-structure $\mathfrak{s}$, a numerical invariant $\tau_{\mathfrak{s}}(K)$, and they show that if $K$ bounds a smooth disk in a rational hom*ology ball $X$ then $\tau_{\mathfrak{s}}(K)=0$ for all $\operatorname{Spin}^{c}$-structures $\mathfrak{s}$ extending over $X$. An alternative version of this invariant has been defined in [48] by Raoux.

While this approach towards answering Question 1.1 sounds very promising it has been very little explored. One of the main issues is that there are not many computational techniques available. We note that Celoria [13] developed a software based on grid hom*ology^{1}^{1}1Available at https://sites.google.com/view/danieleceloria/programs/grid-hom*ology. that can be used to compute the $\tau_{\mathfrak{s}}$-invariant of knots and links in lens spaces.Furthermore, the first author [2] found a formula based on lattice cohom*ology which applies to certain knots in almost rational plumbed three-manifolds. In this paper we perform computations in the case of quasi-positive links, and investigate how the invariants $\tau_{\mathfrak{s}}$ relate to contact geometry.

### Main results

If $(M,\xi)$ is a contact three-manifold then the hom*otopy class of $\xi$ specifies a canonical $\operatorname{Spin}^{c}$-structure $\mathfrak{s}=\mathfrak{s}_{\xi}$ and thus a somewhat canonical tau-invariant $\tau_{\mathfrak{s}}$. Alternatively, Hedden [25] constructed another link invariant $\tau_{\xi}$. This is defined using the contact invariant of Ozsváth and Szabó [44], and unlike $\tau_{\mathfrak{s}}$ depends on the geometry of $\xi$: hom*otopic contact plane distributions may have different $\tau_{\xi}$-invariants. We compare the definition of the two invariants $\tau_{\mathfrak{s}}$ and $\tau_{\xi}$ in Section 2 below. Our main result can be stated as follows.

###### Theorem 1.3.

Suppose that $M$ is a rational hom*ology sphere equipped with a contact structure $\xi$, and let $(W,J)$ be a Stein filling of $(M,\xi)$. If $T$ is a transverse link in $(M,\xi)$ which is the boundary of a properly embedded $J$-holomorphic curve $C\subset W$ then

$\tau_{\xi}(T)=\frac{\operatorname{sl}_{\mathbb{Q}}(T)+|T|}{2}=-\frac{\chi(C)-|%T|+c_{1}(J)[C]+[C]^{2}}{2}\ ,$ |

where $\operatorname{sl}_{\mathbb{Q}}(T)$ denotes the rational self-linking number of $T$ in $(M,\xi)$.

As a corollary we get a new proof of the Relative Thom conjecture for Stein domains.

###### Theorem 1.4 (Relative Thom conjecture for Stein fillings).

If $(W,J)$ is a Stein filling of a rational hom*ology sphere then a properly embedded $J$-holomrphic curve in $W$ maximises the Euler characteristic within its relative hom*ology class.

This was first proved in [17]. Their proof builds on the proof of the symplectic Thom conjecture for closed manifolds [41] and uses the existence of symplectic caps [14]. Another proof was suggested by Kronheimer based on some work of Mrowka and Rollin [38]. Our proof is based on Heegaard Floer hom*ology instead, and is closer in spirit to Rasmussen’s original proof of the Milnor conjecture [49]. The main ingredient of our proof of Theorem 1.3 is some recent work by Hayden [24] that we combine with ideas by Hedden, Plamenevskaya, and the second author [26, 47, 9].

We state the following corollary explicitly since it was the conjecture that originally motivated our work.

###### Corollary 1.5.

Let $L$ be a link in a rational hom*ology sphere $M$ bounding a rational hom*ology ball $W$, and $n=|H_{1}(M;\mathbb{Z})|$. Suppose that $W$ has a Stein structure $J$, and that $L$ bounds a properly embedded $J$-holomorphic curve in $W$. Then the invariant $\tau_{\mathfrak{s}}(L)$, associated to the $\operatorname{Spin}^{c}$-structure specified by the complex tangencies of $J$, attains the maximum value within the tau-invariants associated to the $\sqrt{n}$ $\operatorname{Spin}^{c}$-structures of $M$ extending over $W$. Furthermore, $2\tau_{\mathfrak{s}}(L)-|L|=-\chi^{W}_{4}(L)$.

Our corollaries regarding rational hom*ology ball Stein fillings are based on the inequality in Theorem 1.6 below.

###### Theorem 1.6.

Let $T$ be a transverse link in a contact three-manifold $(M,\xi)$. If $(M,\xi)$ admits a rational hom*ology ball Stein filling then:

$\operatorname{sl}_{\mathbb{Q}}(T)\leqslant 2\tau_{\mathfrak{s}}(T)-|T|\ ,$ |

where $\mathfrak{s}=\mathfrak{s}_{\xi}$ denotes the $\operatorname{Spin}^{c}$-structure associated to $\xi$.

This can be used to obstruct the existence of a Stein filling which is a rational hom*ology ball, see the work of Bhupal and Stipsicz [5] for the basic literature.In particular, Theorem 1.6 and the work of Mark and Tosun [37] give a positive answer to the following conjecture of Gompf [21].

###### Theorem 1.7 (Gompf Conjecture [21]).

No standardly-oriented Brieskorn integer hom*ology sphere admits a rational hom*ology ball Stein filling.

The methods of this paper can also be used to find topological obstructions for a link type to be the boundary of a $J$-holomorphic curve in some Stein filling. We state two corollaries in this direction. In the following two statements, $L$ denotes a link in a contact three-sphere $(M,\xi)$, and $\mathfrak{s}$ the $\text{Spin}^{c}$-structure associated to $\xi$.

###### Corollary 1.8.

Suppose $H_{1}(M;\mathbb{Z})$ does not contain a metaboliser $G$ such that $|\tau_{\mathfrak{s}+\alpha}(L)|\leqslant\tau_{\mathfrak{s}}(L)$, for every $\alpha\in G$. Then $L$ does not bound a $J$-holomorphic curve in any rational hom*ology ball Stein filling of $(M,\xi)$.

The following corollary also stroke our attention.

###### Corollary 1.9.

Suppose $(W,J)$ is a rational hom*ology ball Stein filling of $(M,\xi)$, and that $\tau_{\mathfrak{s}}(L)\neq\tau_{\overline{\mathfrak{s}}}(L)$. Then $L$ bounds a holomorphic curve in $W$, with respect to at most one of the two Stein structures $J$ and $-J$.

### Further applications

We conclude with a couple of extra applications and some examples.First we observe that our results can be applied to the study of the PL genus of knots and links. To make our statement precise we use the following notation: given a link $L$ in a rational hom*ology three-sphere $Y$ and a four-manifold $X$ bounding $Y$ we define

$\tau^{X}_{\max}(L)=\max\{\tau_{\mathfrak{s}}(L):\mathfrak{s}\in\text{Spin}^{c}%(Y)\text{ extending to the four-manifold }X\}\ .$ |

Similarly, we define the invariant $\tau^{X}_{\min}(L)$ as the minimum value achieved.

###### Theorem 1.10.

Suppose that $L$ is a link in a rational hom*ology sphere $Y$, and that $X$ is a rational hom*ology ball bounding $Y$. Then

$\dfrac{\left|\tau^{X}_{\max}(L)-\tau^{X}_{\min}(L)\right|}{2}\leqslant g^{*}_{%\text{PL}}(L)\ ,$ |

where $g^{*}_{\text{PL}}(L)$ denotes the minimum genus of a $PL$ surface $F\subset X$ with $|L|$ connected components each bounding a different component of the link $L$.

To illustrate Theorem 1.10 and our other results we study two families of quasi-positive links. The first family is shown on the left-hand side of Figure 1, and we stumbled into it while thinking about the $A_{n}$-realisation problem [4]. The second family, depicted in Figure 2, is a family of quasi-positive links in the lens space $L(9,2)$, and we construct it by playing around with the Nagata transform relating the Hirzebruch surface $F_{1}=\mathbb{C}P^{2}\#\overline{\mathbb{C}P^{2}}$ to the Hirzebruch surface $F_{2}$. Here is one of our results.

###### Proposition 1.11.

Let $N_{k}$ be the link in Figure 10. There exists a rational hom*ology ball Stein filling $(W,J)$ of $L(9,2)$ such that: $\tau_{\max}(N_{k})=\frac{k^{2}+3k}{9}$, and $\tau_{\min}(N_{k})=\frac{k^{2}-3k}{9}$. Consequently, $N_{k}$ does not bound a $J$-holomorphic curve in $(W,J)$ for any $k$ that is not divisible by $3$. Furthermore, $N_{k}$ does not bound a $(-J)$-holomorphic curve for any $k$.

The computations for the proof of Proposition 1.11 are based on an adaptation of the methods of lattice cohom*ology, a technique that was first explored in [2]. Note that $N_{3}=M_{3}$ and that for this link we are able to construct a holomorphic curve in $(W,J)$. We conjecture that the link $N_{3d}$ does not bound holomorphic curves in $(W,J)$ for $d\geqslant 2$. We believe that it would be interesting to address similar questions for the other families we describe in the paper.

Finally, we mention an application that came up during a conversation with Boyer. The proof is based on some celebrated results by Loi and Piergallini [34].

###### Theorem 1.12.

If $K$ is a quasi-alternating quasi-positive knot in $S^{3}$ then $\tau_{\mathfrak{s}_{0}}(\widetilde{K})=\tau(K)$, where $\widetilde{K}\subset\Sigma(K)$ denotes the fixed point set of the branched double-covering involution, and $\mathfrak{s}_{0}$ is the unique $\operatorname{Spin}$-structure on $\Sigma(K)$.

It is indeed conjectured by Grigsby that $\tau_{\mathfrak{s}}(\widetilde{K})=\tau(K)$ for all alternating knots in $S^{3}$, and so far this was only confirmed for alternating torus knots $T_{2,2n+1}$ in [3], and by unpublished computer experiments performed by Celoria.