Holomorphic curves in Stein domains and the tau-invariant (2024)

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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 τξ(L)subscript𝜏𝜉𝐿\tau_{\xi}(L)italic_τ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ( italic_L ) in the case when ξ𝜉\xiitalic_ξ is a Stein fillable contact structure on a rational hom*ology sphere, and L𝐿Litalic_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 τξsubscript𝜏𝜉\tau_{\xi}italic_τ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT to the Grigsby-Ruberman-Strle topological tau-invariant τ𝔰subscript𝜏𝔰\tau_{\mathfrak{s}}italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT, associated to the SpincsuperscriptSpin𝑐\operatorname{Spin}^{c}roman_Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT-structure 𝔰=𝔰ξ𝔰subscript𝔰𝜉\mathfrak{s}=\mathfrak{s}_{\xi}fraktur_s = fraktur_s start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT of the contact structure ξ𝜉\xiitalic_ξ, 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 τ𝔰subscript𝜏𝔰\tau_{\mathfrak{s}}italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT-invariants of certain links in lens spaces, and estimate their PL slice genus.

1. Introduction

A knot K𝐾Kitalic_K in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT is called slice if it bounds a smooth disk ΔΔ\Deltaroman_Δ in D4superscript𝐷4D^{4}italic_D start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT. 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 τ𝜏\tauitalic_τ-invariant [43], and the Rasmussen s𝑠sitalic_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 L𝐿Litalic_L in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT is called quasi-positive if it can be expressed as the closure of an n𝑛nitalic_n-braid β𝛽\betaitalic_β that has a factorisation of the form:

β=(w1σj1w11)(wbσjbwb1),𝛽subscript𝑤1subscript𝜎subscript𝑗1superscriptsubscript𝑤11subscript𝑤𝑏subscript𝜎subscript𝑗𝑏superscriptsubscript𝑤𝑏1\beta=(w_{1}\sigma_{j_{1}}w_{1}^{-1})\cdot...\cdot(w_{b}\sigma_{j_{b}}w_{b}^{-%1})\ ,italic_β = ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ⋅ … ⋅ ( italic_w start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ,

where the σisubscript𝜎𝑖\sigma_{i}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT’s are the Artin generators of the n𝑛nitalic_n-braid group. It was shown [26, 8, 9] that the identity

τ(L)=w(β)n+|L|2𝜏𝐿𝑤𝛽𝑛𝐿2\tau(L)=\frac{w(\beta)-n+\left|L\right|}{2}italic_τ ( italic_L ) = divide start_ARG italic_w ( italic_β ) - italic_n + | italic_L | end_ARG start_ARG 2 end_ARG(1.1)

holds for any quasi-positive link L𝐿Litalic_L in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, where |L|𝐿|L|| italic_L | denotes the number of components of the link L𝐿Litalic_L, and w(β)𝑤𝛽w(\beta)italic_w ( italic_β ) denotes the writhe of β𝛽\betaitalic_β, which is defined as the difference between the number of positive and negative generators in the factorisation of β𝛽\betaitalic_β. Quasi-positive links are exactly those links that arise as boundary of holomorphic curves properly embedded in D4superscript𝐷4D^{4}italic_D start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT [6, 50].This paper is devoted to the study of links that can be realised as boundary of J𝐽Jitalic_J-holomorphic curves in Stein domains.

Rational slice genus

To obstruct a knot K𝐾Kitalic_K in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT from being slice one can look at its double branched cover Σ(K)Σ𝐾\Sigma(K)roman_Σ ( italic_K ). If the knot K𝐾Kitalic_K is slice then Σ(K)Σ𝐾\Sigma(K)roman_Σ ( italic_K ) bounds a rational hom*ology ball X𝑋Xitalic_X, namely the double branched cover of D4superscript𝐷4D^{4}italic_D start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT 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 K𝐾Kitalic_K, the double branched cover Σ(K)Σ𝐾\Sigma(K)roman_Σ ( italic_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 K~Σ(K)~𝐾Σ𝐾\widetilde{K}\subset\Sigma(K)over~ start_ARG italic_K end_ARG ⊂ roman_Σ ( italic_K ). Indeed, if a knot is slice, not only Σ(K)=XΣ𝐾𝑋\Sigma(K)=\partial Xroman_Σ ( italic_K ) = ∂ italic_X for some rational hom*ology ball X𝑋Xitalic_X but one can also find a smooth disk ΔXΔ𝑋\Delta\subset Xroman_Δ ⊂ italic_X with Δ=K~Δ~𝐾\partial\Delta=\widetilde{K}∂ roman_Δ = over~ start_ARG italic_K end_ARG. This leads to the following question.

Question 1.1.

Suppose that K𝐾Kitalic_K is a knot, in a rational hom*ology sphere Y𝑌Yitalic_Y, and that Y=X𝑌𝑋Y=\partial Xitalic_Y = ∂ italic_X for some rational hom*ology ball X𝑋Xitalic_X. Then is it possible to find a smooth disk ΔXΔ𝑋\Delta\subset Xroman_Δ ⊂ italic_X with Δ=KΔ𝐾\partial\Delta=K∂ roman_Δ = italic_K?

If the answer to Question 1.1 is affirmative for a knot K𝐾Kitalic_K in Y𝑌Yitalic_Y we say that K𝐾Kitalic_K is rationally slice in X𝑋Xitalic_X. Of course if Y=X𝑌𝑋Y=\partial Xitalic_Y = ∂ italic_X for some rational hom*ology ball X𝑋Xitalic_X we can consider the obvious notion of genus:

g4X(K)=minF{g(F):FXoriented surface,F=K},subscriptsuperscript𝑔𝑋4𝐾subscript𝐹:𝑔𝐹𝐹𝑋oriented surface,𝐹𝐾g^{X}_{4}(K)=\min_{F}\left\{g(F):F\subset X\text{ oriented surface, }\partial F%=K\right\},italic_g start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_K ) = roman_min start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT { italic_g ( italic_F ) : italic_F ⊂ italic_X oriented surface, ∂ italic_F = italic_K } ,

and define g4(K)=minX{g4X(K):Xrational hom*ology ball withX=Y}subscript𝑔4𝐾subscript𝑋:subscriptsuperscript𝑔𝑋4𝐾𝑋rational hom*ology ball with𝑋𝑌g_{4}(K)=\displaystyle\min_{X}\left\{g^{X}_{4}(K):X\text{ rational hom*ology %ball with }\partial X=Y\right\}italic_g start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_K ) = roman_min start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT { italic_g start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_K ) : italic_X rational hom*ology ball with ∂ italic_X = italic_Y }.

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 χ(F)𝜒𝐹-\chi(F)- italic_χ ( italic_F ), to measure the complexity of surfaces. We define

χ4X(L,Σ)=maxF{χ(F):FXoriented surface,F=L,[F]=ΣH2(X,X;)}.superscriptsubscript𝜒4𝑋𝐿Σsubscript𝐹:𝜒𝐹𝐹𝑋oriented surface,𝐹𝐿delimited-[]𝐹Σsubscript𝐻2𝑋𝑋\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\}\ .italic_χ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ( italic_L , roman_Σ ) = roman_max start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT { italic_χ ( italic_F ) : italic_F ⊂ italic_X oriented surface, ∂ italic_F = italic_L , [ italic_F ] = roman_Σ ∈ italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_X , ∂ italic_X ; blackboard_Z ) } .

If X𝑋Xitalic_X is a rational hom*ology ball then the slice genus of L𝐿Litalic_L in X𝑋Xitalic_X is denoted by χ4X(L)superscriptsubscript𝜒4𝑋𝐿\chi_{4}^{X}(L)italic_χ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ( italic_L ), and again we set χ4(L)subscript𝜒4𝐿\chi_{4}(L)italic_χ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_L ) to be the maximum of χ4X(L)superscriptsubscript𝜒4𝑋𝐿\chi_{4}^{X}(L)italic_χ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ( italic_L ) over all rational balls X𝑋Xitalic_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 K𝐾Kitalic_K in a rational hom*ology sphere Y𝑌Yitalic_Y, equipped with a SpincsuperscriptSpin𝑐\operatorname{Spin}^{c}roman_Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT-structure 𝔰𝔰\mathfrak{s}fraktur_s, a numerical invariant τ𝔰(K)subscript𝜏𝔰𝐾\tau_{\mathfrak{s}}(K)italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_K ), and they show that if K𝐾Kitalic_K bounds a smooth disk in a rational hom*ology ball X𝑋Xitalic_X then τ𝔰(K)=0subscript𝜏𝔰𝐾0\tau_{\mathfrak{s}}(K)=0italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_K ) = 0 for all SpincsuperscriptSpin𝑐\operatorname{Spin}^{c}roman_Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT-structures 𝔰𝔰\mathfrak{s}fraktur_s extending over X𝑋Xitalic_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*ology111Available at https://sites.google.com/view/danieleceloria/programs/grid-hom*ology. that can be used to compute the τ𝔰subscript𝜏𝔰\tau_{\mathfrak{s}}italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT-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 τ𝔰subscript𝜏𝔰\tau_{\mathfrak{s}}italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT relate to contact geometry.

Main results

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

Theorem 1.3.

Suppose that M𝑀Mitalic_M is a rational hom*ology sphere equipped with a contact structure ξ𝜉\xiitalic_ξ, and let (W,J)𝑊𝐽(W,J)( italic_W , italic_J ) be a Stein filling of (M,ξ)𝑀𝜉(M,\xi)( italic_M , italic_ξ ). If T𝑇Titalic_T is a transverse link in (M,ξ)𝑀𝜉(M,\xi)( italic_M , italic_ξ ) which is the boundary of a properly embedded J𝐽Jitalic_J-holomorphic curve CW𝐶𝑊C\subset Witalic_C ⊂ italic_W then

τξ(T)=sl(T)+|T|2=χ(C)|T|+c1(J)[C]+[C]22,subscript𝜏𝜉𝑇subscriptsl𝑇𝑇2𝜒𝐶𝑇subscript𝑐1𝐽delimited-[]𝐶superscriptdelimited-[]𝐶22\tau_{\xi}(T)=\frac{\operatorname{sl}_{\mathbb{Q}}(T)+|T|}{2}=-\frac{\chi(C)-|%T|+c_{1}(J)[C]+[C]^{2}}{2}\ ,italic_τ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ( italic_T ) = divide start_ARG roman_sl start_POSTSUBSCRIPT blackboard_Q end_POSTSUBSCRIPT ( italic_T ) + | italic_T | end_ARG start_ARG 2 end_ARG = - divide start_ARG italic_χ ( italic_C ) - | italic_T | + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_J ) [ italic_C ] + [ italic_C ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,

where sl(T)subscriptsl𝑇\operatorname{sl}_{\mathbb{Q}}(T)roman_sl start_POSTSUBSCRIPT blackboard_Q end_POSTSUBSCRIPT ( italic_T ) denotes the rational self-linking number of T𝑇Titalic_T in (M,ξ)𝑀𝜉(M,\xi)( italic_M , italic_ξ ).

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)𝑊𝐽(W,J)( italic_W , italic_J ) is a Stein filling of a rational hom*ology sphere then a properly embedded J𝐽Jitalic_J-holomrphic curve in W𝑊Witalic_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 L𝐿Litalic_L be a link in a rational hom*ology sphere M𝑀Mitalic_M bounding a rational hom*ology ball W𝑊Witalic_W, and n=|H1(M;)|𝑛subscript𝐻1𝑀n=|H_{1}(M;\mathbb{Z})|italic_n = | italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_M ; blackboard_Z ) |. Suppose that W𝑊Witalic_W has a Stein structure J𝐽Jitalic_J, and that L𝐿Litalic_L bounds a properly embedded J𝐽Jitalic_J-holomorphic curve in W𝑊Witalic_W. Then the invariant τ𝔰(L)subscript𝜏𝔰𝐿\tau_{\mathfrak{s}}(L)italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_L ), associated to the SpincsuperscriptSpin𝑐\operatorname{Spin}^{c}roman_Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT-structure specified by the complex tangencies of J𝐽Jitalic_J, attains the maximum value within the tau-invariants associated to the n𝑛\sqrt{n}square-root start_ARG italic_n end_ARG SpincsuperscriptSpin𝑐\operatorname{Spin}^{c}roman_Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT-structures of M𝑀Mitalic_M extending over W𝑊Witalic_W. Furthermore, 2τ𝔰(L)|L|=χ4W(L)2subscript𝜏𝔰𝐿𝐿subscriptsuperscript𝜒𝑊4𝐿2\tau_{\mathfrak{s}}(L)-|L|=-\chi^{W}_{4}(L)2 italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_L ) - | italic_L | = - italic_χ start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_L ).

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

Theorem 1.6.

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

sl(T)2τ𝔰(T)|T|,subscriptsl𝑇2subscript𝜏𝔰𝑇𝑇\operatorname{sl}_{\mathbb{Q}}(T)\leqslant 2\tau_{\mathfrak{s}}(T)-|T|\ ,roman_sl start_POSTSUBSCRIPT blackboard_Q end_POSTSUBSCRIPT ( italic_T ) ⩽ 2 italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_T ) - | italic_T | ,

where 𝔰=𝔰ξ𝔰subscript𝔰𝜉\mathfrak{s}=\mathfrak{s}_{\xi}fraktur_s = fraktur_s start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT denotes the SpincsuperscriptSpin𝑐\operatorname{Spin}^{c}roman_Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT-structure associated to ξ𝜉\xiitalic_ξ.

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 J𝐽Jitalic_J-holomorphic curve in some Stein filling. We state two corollaries in this direction. In the following two statements, L𝐿Litalic_L denotes a link in a contact three-sphere (M,ξ)𝑀𝜉(M,\xi)( italic_M , italic_ξ ), and 𝔰𝔰\mathfrak{s}fraktur_s the SpincsuperscriptSpin𝑐\text{Spin}^{c}Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT-structure associated to ξ𝜉\xiitalic_ξ.

Corollary 1.8.

Suppose H1(M;)subscript𝐻1𝑀H_{1}(M;\mathbb{Z})italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_M ; blackboard_Z ) does not contain a metaboliser G𝐺Gitalic_G such that |τ𝔰+α(L)|τ𝔰(L)subscript𝜏𝔰𝛼𝐿subscript𝜏𝔰𝐿|\tau_{\mathfrak{s}+\alpha}(L)|\leqslant\tau_{\mathfrak{s}}(L)| italic_τ start_POSTSUBSCRIPT fraktur_s + italic_α end_POSTSUBSCRIPT ( italic_L ) | ⩽ italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_L ), for every αG𝛼𝐺\alpha\in Gitalic_α ∈ italic_G. Then L𝐿Litalic_L does not bound a J𝐽Jitalic_J-holomorphic curve in any rational hom*ology ball Stein filling of (M,ξ)𝑀𝜉(M,\xi)( italic_M , italic_ξ ).

The following corollary also stroke our attention.

Corollary 1.9.

Suppose (W,J)𝑊𝐽(W,J)( italic_W , italic_J ) is a rational hom*ology ball Stein filling of (M,ξ)𝑀𝜉(M,\xi)( italic_M , italic_ξ ), and that τ𝔰(L)τ𝔰¯(L)subscript𝜏𝔰𝐿subscript𝜏¯𝔰𝐿\tau_{\mathfrak{s}}(L)\neq\tau_{\overline{\mathfrak{s}}}(L)italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_L ) ≠ italic_τ start_POSTSUBSCRIPT over¯ start_ARG fraktur_s end_ARG end_POSTSUBSCRIPT ( italic_L ). Then L𝐿Litalic_L bounds a holomorphic curve in W𝑊Witalic_W, with respect to at most one of the two Stein structures J𝐽Jitalic_J and J𝐽-J- italic_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 L𝐿Litalic_L in a rational hom*ology three-sphere Y𝑌Yitalic_Y and a four-manifold X𝑋Xitalic_X bounding Y𝑌Yitalic_Y we define

τmaxX(L)=max{τ𝔰(L):𝔰Spinc(Y)extending to the four-manifoldX}.subscriptsuperscript𝜏𝑋𝐿:subscript𝜏𝔰𝐿𝔰superscriptSpin𝑐𝑌extending to the four-manifold𝑋\tau^{X}_{\max}(L)=\max\{\tau_{\mathfrak{s}}(L):\mathfrak{s}\in\text{Spin}^{c}%(Y)\text{ extending to the four-manifold }X\}\ .italic_τ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( italic_L ) = roman_max { italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( italic_L ) : fraktur_s ∈ Spin start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_Y ) extending to the four-manifold italic_X } .

Similarly, we define the invariant τminX(L)subscriptsuperscript𝜏𝑋𝐿\tau^{X}_{\min}(L)italic_τ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_L ) as the minimum value achieved.

Theorem 1.10.

Suppose that L𝐿Litalic_L is a link in a rational hom*ology sphere Y𝑌Yitalic_Y, and that X𝑋Xitalic_X is a rational hom*ology ball bounding Y𝑌Yitalic_Y. Then

|τmaxX(L)τminX(L)|2gPL(L),subscriptsuperscript𝜏𝑋𝐿subscriptsuperscript𝜏𝑋𝐿2subscriptsuperscript𝑔PL𝐿\dfrac{\left|\tau^{X}_{\max}(L)-\tau^{X}_{\min}(L)\right|}{2}\leqslant g^{*}_{%\text{PL}}(L)\ ,divide start_ARG | italic_τ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( italic_L ) - italic_τ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_L ) | end_ARG start_ARG 2 end_ARG ⩽ italic_g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT PL end_POSTSUBSCRIPT ( italic_L ) ,

where gPL(L)subscriptsuperscript𝑔PL𝐿g^{*}_{\text{PL}}(L)italic_g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT PL end_POSTSUBSCRIPT ( italic_L ) denotes the minimum genus of a PL𝑃𝐿PLitalic_P italic_L surface FX𝐹𝑋F\subset Xitalic_F ⊂ italic_X with |L|𝐿|L|| italic_L | connected components each bounding a different component of the link L𝐿Litalic_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 Ansubscript𝐴𝑛A_{n}italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-realisation problem [4]. The second family, depicted in Figure 2, is a family of quasi-positive links in the lens space L(9,2)𝐿92L(9,2)italic_L ( 9 , 2 ), and we construct it by playing around with the Nagata transform relating the Hirzebruch surface F1=P2#P2¯subscript𝐹1superscript𝑃2#¯superscript𝑃2F_{1}=\mathbb{C}P^{2}\#\overline{\mathbb{C}P^{2}}italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = blackboard_C italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT # over¯ start_ARG blackboard_C italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG to the Hirzebruch surface F2subscript𝐹2F_{2}italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Here is one of our results.

Proposition 1.11.

Let Nksubscript𝑁𝑘N_{k}italic_N start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT be the link in Figure 10. There exists a rational hom*ology ball Stein filling (W,J)𝑊𝐽(W,J)( italic_W , italic_J ) of L(9,2)𝐿92L(9,2)italic_L ( 9 , 2 ) such that: τmax(Nk)=k2+3k9subscript𝜏subscript𝑁𝑘superscript𝑘23𝑘9\tau_{\max}(N_{k})=\frac{k^{2}+3k}{9}italic_τ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 3 italic_k end_ARG start_ARG 9 end_ARG, and τmin(Nk)=k23k9subscript𝜏subscript𝑁𝑘superscript𝑘23𝑘9\tau_{\min}(N_{k})=\frac{k^{2}-3k}{9}italic_τ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 3 italic_k end_ARG start_ARG 9 end_ARG. Consequently, Nksubscript𝑁𝑘N_{k}italic_N start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT does not bound a J𝐽Jitalic_J-holomorphic curve in (W,J)𝑊𝐽(W,J)( italic_W , italic_J ) for any k𝑘kitalic_k that is not divisible by 3333. Furthermore, Nksubscript𝑁𝑘N_{k}italic_N start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT does not bound a (J)𝐽(-J)( - italic_J )-holomorphic curve for any k𝑘kitalic_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 N3=M3subscript𝑁3subscript𝑀3N_{3}=M_{3}italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and that for this link we are able to construct a holomorphic curve in (W,J)𝑊𝐽(W,J)( italic_W , italic_J ). We conjecture that the link N3dsubscript𝑁3𝑑N_{3d}italic_N start_POSTSUBSCRIPT 3 italic_d end_POSTSUBSCRIPT does not bound holomorphic curves in (W,J)𝑊𝐽(W,J)( italic_W , italic_J ) for d2𝑑2d\geqslant 2italic_d ⩾ 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 K𝐾Kitalic_K is a quasi-alternating quasi-positive knot in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT then τ𝔰0(K~)=τ(K)subscript𝜏subscript𝔰0~𝐾𝜏𝐾\tau_{\mathfrak{s}_{0}}(\widetilde{K})=\tau(K)italic_τ start_POSTSUBSCRIPT fraktur_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG italic_K end_ARG ) = italic_τ ( italic_K ), where K~Σ(K)~𝐾Σ𝐾\widetilde{K}\subset\Sigma(K)over~ start_ARG italic_K end_ARG ⊂ roman_Σ ( italic_K ) denotes the fixed point set of the branched double-covering involution, and 𝔰0subscript𝔰0\mathfrak{s}_{0}fraktur_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the unique SpinSpin\operatorname{Spin}roman_Spin-structure on Σ(K)Σ𝐾\Sigma(K)roman_Σ ( italic_K ).

It is indeed conjectured by Grigsby that τ𝔰(K~)=τ(K)subscript𝜏𝔰~𝐾𝜏𝐾\tau_{\mathfrak{s}}(\widetilde{K})=\tau(K)italic_τ start_POSTSUBSCRIPT fraktur_s end_POSTSUBSCRIPT ( over~ start_ARG italic_K end_ARG ) = italic_τ ( italic_K ) for all alternating knots in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, and so far this was only confirmed for alternating torus knots T2,2n+1subscript𝑇22𝑛1T_{2,2n+1}italic_T start_POSTSUBSCRIPT 2 , 2 italic_n + 1 end_POSTSUBSCRIPT in [3], and by unpublished computer experiments performed by Celoria.

Acknowledgements

Holomorphic curves in Stein domains and the tau-invariant (2024)
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