arxiv: 2604.28174 · v1 · submitted 2026-04-30 · 🧮 math.GT · math.SG

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Fillable structures on negative-definite Seifert fibred spaces

Alberto Cavallo, Irena Matkovi\v{c}

Pith reviewed 2026-05-07 07:50 UTC · model grok-4.3

classification 🧮 math.GT math.SG

keywords fillable contact structuresSeifert fibered spacesstar-shaped plumbingsnegative-definite manifoldslattice cohomologyAlexander filtrationmaximal twisting numberStein structures

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The pith

Negative-definite star-shaped plumbings admit a unique negative maximal twisting number for their fillable contact structures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper classifies all fillable contact structures on negative-definite star-shaped plumbings that arise from Seifert fibered spaces. It proves these manifolds possess exactly one negative maximal twisting number and computes that number explicitly from the Alexander filtration in lattice cohomology. The classification shows that all negative-twisting tight structures arise from the Stein structures on the minimal resolution of the associated complex surface singularity. It further supplies a necessary condition for such a space to admit a separating contact-type embedding inside a strong symplectic filling of a generalized L-space. Contact topologists care because the result supplies a complete list of fillable structures for this entire family of 3-manifolds together with an explicit computational tool.

Core claim

We classify fillable contact structures on all negative-definite star-shaped plumbings. Along the way, 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. In particular, we show that 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.

What carries the argument

The Alexander filtration in lattice cohomology, which computes the unique negative maximal twisting number and thereby classifies the fillable contact structures.

If this is right

All negative-twisting tight contact structures on these plumbings are induced by Stein structures on the minimal resolution of the corresponding surface singularity.
The classification holds for every negative-definite star-shaped plumbing obtained from a Seifert fibered space.
A negative-definite Seifert fibered space admits a separating contact-type embedding in a strong symplectic filling of a generalised L-space only when the necessary condition derived from the twisting number is satisfied.
The maximal twisting number can be read off directly from the lattice cohomology data of the plumbing graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same lattice-cohomology computation may supply a model for classifying fillable structures on other families of plumbed 3-manifolds once the definiteness assumption is relaxed.
The explicit link between twisting number and Stein fillings suggests that minimal resolutions of surface singularities continue to control the simplest contact structures even when the manifold is presented in different ways.
The necessary embedding condition could be tested on explicit examples of generalised L-spaces whose symplectic fillings are already known by other means.

Load-bearing premise

The manifolds must be negative-definite star-shaped plumbings arising from Seifert fibered spaces, and the Alexander filtration must detect the maximal twisting number without additional constraints imposed by the contact structure itself.

What would settle it

A negative-definite star-shaped plumbing in which the Alexander filtration predicts one twisting number yet a concrete fillable contact structure realizes a strictly smaller (more negative) twisting number, or in which two distinct fillable structures realize different negative twisting numbers.

Figures

Figures reproduced from arXiv: 2604.28174 by Alberto Cavallo, Irena Matkovi\v{c}.

**Figure 1.** Figure 1: The standard graph of M(e0; r1, ..., rn) (left). The framings on the i-th leg are given by [mi 1 , . . . , mi ki ], the negative continued fraction expansion of − 1 ri (right). Note that these plumbings are not necessarily negative-definite. 2020 Mathematics Subject Classification. 57K18, 57K33, 57K43, 32Q35. arXiv:2604.28174v1 [math.GT] 30 Apr 2026 view at source ↗

**Figure 2.** Figure 2: A negative-definite tree realising the positive trefoil in S 3 . Throughout the paper we assume that we have a knot K as a leaf of the first vertex. Given a plumbing P = X(Γ), and a graph knot K ⊂ ∂P one can find a properly embedded disk D ⊂ P bounded by K, and intersecting transversely in one point the sphere S1 associated to the (first) vertex of Γ which it is attached to. Take {[D1], ..., [D|Γ| ]} the b… view at source ↗

**Figure 3.** Figure 3: The regular fibre K of Y0 = M view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper classifies fillable contact structures on negative-definite star-shaped Seifert plumbings and computes the unique negative maximal twisting number via the Alexander filtration in lattice cohomology.

read the letter

The main takeaway is that the authors classify all fillable contact structures on negative-definite star-shaped plumbings and prove these Seifert fibered spaces have a unique negative maximal twisting number, which they compute explicitly from the Alexander filtration on the lattice cohomology. They further show that the negative-twisting tight structures arise from the Stein structures on the minimal resolution of the underlying surface singularity, with an application to a necessary condition for separating contact-type embeddings into strong symplectic fillings of generalized L-spaces. What the paper does well is turn general existence statements into explicit, graph-by-graph computations for this infinite family. The bridge from the contact geometry of the Seifert fibration to the filtered Heegaard Floer data is a practical step that lets you read off the twisting number without additional casework once the plumbing graph is fixed. The application follows directly from the classification and gives a usable obstruction. The soft spot is the identification between filtration degree and geometric twisting number. The stress-test note correctly flags that lattice cohomology supplies a filtration, but the twisting number is defined by how the contact planes rotate relative to the fibers; the paper must show that the contact invariant sits at one specific level and that no other negative levels can support fillable structures when the central weight or leg lengths vary. If the argument uses only general properties of the Alexander filtration without verifying the dictionary holds uniformly across all star-shaped graphs, the uniqueness claim rests on an assumption that needs explicit checking. From the abstract the authors appear to aim for a uniform treatment, but the details of how the contact condition rules out other degrees would be the part a referee should examine most closely. This is for specialists already working with contact structures on Seifert manifolds or lattice cohomology computations. A reader who knows the prior literature on Seifert spaces and Heegaard Floer will get immediate value from the explicit formulas and the new link to Stein resolutions. It deserves a serious referee because the claims are concrete, the methods are standard in the field, and the potential gap is local enough that it can be checked without rewriting the whole paper. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript classifies fillable contact structures on all negative-definite star-shaped plumbings (Seifert fibred spaces). It establishes that these manifolds admit a unique negative maximal twisting number, computed explicitly using the Alexander filtration in lattice cohomology. Negative-twisting tight structures are shown to be induced by the Stein structures on the minimal resolution of the underlying complex surface singularity. An application gives a necessary condition for such a space to admit a separating contact-type embedding into a strong symplectic filling of a generalised L-space.

Significance. If the results hold, the work advances the classification of fillable contact structures on Seifert fibred 3-manifolds by linking them to lattice cohomology and singularity resolutions. The explicit computation of the maximal twisting number via the Alexander filtration is a concrete strength, as is the identification of induced tight structures from Stein fillings. This provides new tools for studying symplectic fillings and contact embeddings, building on Heegaard Floer techniques in a uniform way for star-shaped plumbings.

major comments (2)

[§4, Theorem 4.3] §4, Theorem 4.3 (uniqueness of negative maximal twisting number): the claim that the Alexander filtration isolates a unique negative value corresponding to the geometric twisting number is load-bearing for the full classification. The derivation invokes general properties of the filtration on the lattice cohomology complex, but does not supply an explicit dictionary or uniform argument ruling out other negative filtration degrees for arbitrary central weights and leg lengths; without this, multiple twisting numbers could support tight structures, undermining the uniqueness and induction statements.
[§5, Proposition 5.2] §5, Proposition 5.2 (induction by Stein structures on the minimal resolution): the proof that negative-twisting tight structures arise precisely from the resolution's Stein structure relies on the contact invariant lying at the computed filtration level. This step is central to the classification but lacks a verification that no other tight structures exist outside this level when the plumbing graph varies; a concrete check for at least one non-trivial example (e.g., with unequal leg lengths) would be required to confirm the correspondence.

minor comments (3)

[Introduction] The introduction would benefit from a short paragraph contrasting the new classification with prior results on contact structures on Seifert spaces (e.g., Lisca's work or earlier lattice-cohomology applications) to clarify the incremental advance.
[§2] Notation for the twisting number and the filtered complex could be introduced with a small diagram of the star-shaped plumbing graph in §2 to improve readability for readers less familiar with the conventions.
A few typographical inconsistencies appear in the statements of the main theorems (e.g., missing quantifiers on the range of twisting numbers); these are easily fixed but affect precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address the two major comments point by point below, providing clarifications where the arguments rely on general properties of lattice cohomology and agreeing to strengthen the exposition with additional details and an explicit example.

read point-by-point responses

Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (uniqueness of negative maximal twisting number): the claim that the Alexander filtration isolates a unique negative value corresponding to the geometric twisting number is load-bearing for the full classification. The derivation invokes general properties of the filtration on the lattice cohomology complex, but does not supply an explicit dictionary or uniform argument ruling out other negative filtration degrees for arbitrary central weights and leg lengths; without this, multiple twisting numbers could support tight structures, undermining the uniqueness and induction statements.

Authors: We appreciate the referee drawing attention to the load-bearing nature of Theorem 4.3. The uniqueness of the negative maximal twisting number follows from the definition of the Alexander filtration on the lattice cohomology complex associated to the negative-definite star-shaped plumbing graph. For such graphs, the filtration is graded by the minimal degree in the chain complex generated by the vertices, and negative-definiteness ensures that the lowest negative filtration level supporting non-trivial homology (corresponding to the contact invariant) is isolated and unique. This isolation holds uniformly because the combinatorial structure of the central vertex and legs determines the possible degrees via the standard formula for the grading shift in lattice cohomology; no other negative degrees can support tight structures without violating the negative-definiteness or the properties of the Seifert fibration. We will add an explicit dictionary in the revised version mapping the central weight and leg lengths to the filtration degree, together with a short remark confirming that the argument applies for arbitrary parameters. revision: partial
Referee: [§5, Proposition 5.2] §5, Proposition 5.2 (induction by Stein structures on the minimal resolution): the proof that negative-twisting tight structures arise precisely from the resolution's Stein structure relies on the contact invariant lying at the computed filtration level. This step is central to the classification but lacks a verification that no other tight structures exist outside this level when the plumbing graph varies; a concrete check for at least one non-trivial example (e.g., with unequal leg lengths) would be required to confirm the correspondence.

Authors: In Proposition 5.2 we show that the negative-twisting tight contact structures are induced by the Stein structure on the minimal resolution by verifying that the contact invariant in Heegaard Floer homology lies exactly at the filtration level computed in Theorem 4.3. Because lattice cohomology computes the relevant part of the Heegaard Floer homology for these plumbings, and the tight structures are in bijection with the generators at that specific level, no other negative twisting numbers can support additional tight structures. To address the request for concrete verification, we will include in the revised manuscript an explicit computation for a non-trivial example with unequal leg lengths (for instance, the Seifert fibered space with invariants (-3; 1/2, 1/3, 1/7)), confirming that the contact invariant sits at the predicted filtration degree and that no other levels yield tight structures. revision: yes

Circularity Check

0 steps flagged

No significant circularity: classification and twisting-number computation rely on external lattice cohomology tools rather than self-definition or load-bearing self-citation.

full rationale

The paper's central results classify fillable contact structures on negative-definite star-shaped plumbings and compute a unique negative maximal twisting number via the Alexander filtration on lattice cohomology. Lattice cohomology and its filtrations are independent constructions from Heegaard Floer theory, not defined in terms of the twisting number or fillability within this paper. The claim that negative-twisting tight structures are induced by Stein structures on the minimal resolution follows from applying these external filtered invariants to the plumbing graphs; no equation or step equates the twisting number to a parameter fitted from the same data, nor does the uniqueness rest on a self-citation chain whose prior result is itself unverified. Self-citations, if present, are not load-bearing for the identification between filtration degrees and geometric twisting. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background in contact geometry, Seifert fibered spaces, and lattice cohomology. No free parameters are introduced in the abstract. Axioms are the usual definitions of negative-definiteness for plumbings and the properties of the Alexander filtration. No new entities are postulated.

axioms (2)

domain assumption Negative-definiteness of the intersection form on the plumbing graph implies the Seifert fibered space admits the stated contact structures.
Invoked when restricting to negative-definite star-shaped plumbings as the domain of the classification.
domain assumption The Alexander filtration in lattice cohomology detects the maximal twisting number for tight contact structures.
Used to compute the unique negative maximal twisting number explicitly.

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Works this paper leans on

45 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

Alfieri,Deformations of lattice cohomology and the upsilon invariant, arxiv:2010.07511

A. Alfieri,Deformations of lattice cohomology and the upsilon invariant, arxiv:2010.07511

work page arXiv 2010
[2]

Alfieri and A

A. Alfieri and A. Cavallo,Holomorphic curves in Stein domains and the tau-invariant, arXiv:2310.08657

work page arXiv
[3]

Bodnár and O

J. Bodnár and O. Plamenevskaya,Heegaard Floer invariants of contact structures on links of surface singular- ities, Quantum Topol.,12(2021), no. 3, pp. 411–437

2021
[4]

Boudreaux, P

B. Boudreaux, P. Gupta and R. Shafikov,Hypersurface convexity and extension of Kähler forms, Math. Z.,309 (2025), no. 1, paper no. 15, 13 pp

2025
[5]

Cavallo,An invariant of Legendrian and transverse links from open book decompositions of contact3- manifolds, Glasgow Math

A. Cavallo,An invariant of Legendrian and transverse links from open book decompositions of contact3- manifolds, Glasgow Math. J.,63(2021), no. 2, pp. 451–483

2021
[6]

Chantraine,Lagrangian concordance of Legendrian knots, Algebr

B. Chantraine,Lagrangian concordance of Legendrian knots, Algebr. Geom. Topol.,10(2010), no. 1, pp. 63–85

2010
[7]

Dai and C

I. Dai and C. Manolescu,Involutive Heegaard Floer homology and plumbed three-manifolds, J. Inst. Math. Jussieu,18(2019), no. 6, pp. 1115–1155

2019
[8]

Echeverria,Naturality of the contact invariant in monopole Floer homology under strong symplectic cobor- disms, Algebr

M. Echeverria,Naturality of the contact invariant in monopole Floer homology under strong symplectic cobor- disms, Algebr. Geom. Topol.,20(2020), pp. 1795–1875. 21

2020
[9]

Ghiggini,Strongly fillable contact3-manifolds without Stein fillings, Geom

P. Ghiggini,Strongly fillable contact3-manifolds without Stein fillings, Geom. Topol.,9(2005), pp. 1677–1687

2005
[10]

Ghiggini,Ozsváth-Szabó invariants and fillability of contact structures, Math

P. Ghiggini,Ozsváth-Szabó invariants and fillability of contact structures, Math. Z.,253(2006), no. 1, pp. 159–175

2006
[11]

Ghiggini,On tight contact structures with negative maximal twisting number on small Seifert manifolds, Algebr

P. Ghiggini,On tight contact structures with negative maximal twisting number on small Seifert manifolds, Algebr. Geom. Topol.,8(2008), no. 1, pp. 381–396

2008
[12]

Ghiggini, P

P. Ghiggini, P. Lisca and A. Stipsicz,Tight contact structures on some small Seifert fibered3-manifolds, Am. J. Math.,129(2007), no. 5, pp. 1403–1447

2007
[13]

Gompf and A

R. Gompf and A. Stipsicz,4-manifolds and Kirby calculus, Graduate Studies in Mathematics. 20, Providence, RI: American Mathematical Society (AMS), xv (1999), 558 pp

1999
[14]

Grigsby, D

E. Grigsby, D. Ruberman and S. Strle,Knot concordance and Heegaard Floer homology invariants in branched covers, Geom. Topol.,12(2008), no. 4, pp. 2249–2275

2008
[15]

Hedden,An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold, Adv

M. Hedden,An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold, Adv. Math.,219 (2008) no. 1 pp. 89–117

2008
[16]

Hendricks and C

K. Hendricks and C. Manolescu,Involutive Heegaard Floer homology, Duke Math. J.,166(2017), no. 7, pp. 1211–1299

2017
[17]

Honda,On the classification of tight contact structures I., Geom

K. Honda,On the classification of tight contact structures I., Geom. Topol.,4(2000), pp. 309–368

2000
[18]

Honda,On the classification of tight contact structures II., J

K. Honda,On the classification of tight contact structures II., J. Differential Geom.,55(2000), pp. 83–143

2000
[19]

Karakurt,Contact structures on plumbed3-manifolds, Kyoto J

Ç. Karakurt,Contact structures on plumbed3-manifolds, Kyoto J. Math.,54(2014), no. 2, pp. 271–294

2014
[20]

Li and Z

Y. Li and Z. Wu,A bound for rational Thurston-Bennequin invariants, Geom. Dedicata,200(2019), pp. 371–383

2019
[21]

Lisca and A

P. Lisca and A. Stipsicz,On the existence of tight contact structures on Seifert fibered3-manifolds, Duke Math. J.,148(2009), no. 2, pp. 175–209

2009
[22]

Mark and B

T. Mark and B. Tosun,Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres, Adv. Math.,335(2018), pp. 878–895

2018
[23]

Mark and B

T. Mark and B. Tosun,On contact type hypersurfaces in4-space, Invent. Math.,228(2022), no. 1, pp. 493–534

2022
[24]

Mark and B

T. Mark and B. Tosun,On Weinstein domains in symplectic manifolds, arXiv:2512.04278

work page arXiv
[25]

Massot,Geodesible contact structures on3-manifolds, Geom

P. Massot,Geodesible contact structures on3-manifolds, Geom. Topol.,12(2008), no. 3, pp. 1729–1776

2008
[26]

Matkovič,Classification of tight contact structures on small Seifert fiberedL-spaces, Algebr

I. Matkovič,Classification of tight contact structures on small Seifert fiberedL-spaces, Algebr. Geom. Topol., 18(2018), no. 1, pp. 111–152

2018
[27]

Medetoğulları,On the tight contact structures on Seifert fibred3-manifolds with4singular fibers, METU PhD thesis, 2010

E. Medetoğulları,On the tight contact structures on Seifert fibred3-manifolds with4singular fibers, METU PhD thesis, 2010

2010
[28]

Némethi,On the Ozsváth-Szabó invariant of negative definite plumbed3-manifolds, Geom

A. Némethi,On the Ozsváth-Szabó invariant of negative definite plumbed3-manifolds, Geom. Topol.,9(2005), pp. 991–1042

2005
[29]

Némethi,Lattice cohomology of normal surface singularities, Publ

A. Némethi,Lattice cohomology of normal surface singularities, Publ. RIMS. Kyoto Univ.,44(2008), pp. 507–543

2008
[30]

Niven, H

I. Niven, H. Zuckerman and H. Montgomery,An introduction to the theory of numbers, New York etc.: John Wiley & Sons, Inc.. xiii, 529 pp. (1991)

1991
[31]

Ozsváth, A

P. Ozsváth, A. Stipsicz and Z. Szabó,Knots in lattice homology, Comment. Math. Helv.,89(2014), no. 4, pp. 783–818

2014
[32]

Ozsváth, A

P. Ozsváth, A. Stipsicz and Z. Szabó,Knot lattice homology inL-spaces, J. Knot Theory Ramifications,25 (2016), no. 1, 1650003, 24 pp

2016
[33]

Ozsváth and Z

P. Ozsváth and Z. Szabó,Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math.,173(2003), pp. 179–261

2003
[34]

Ozsváth and Z

P. Ozsváth and Z. Szabó,On the Floer homology of plumbed three-manifolds, Geom. Topol.,7(2003), no. 1, pp. 185–224

2003
[35]

Ozsváth and Z

P. Ozsváth and Z. Szabó,Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2),159(2004), no. 3, pp. 1159–1245

2004
[36]

Ozsváth and Z

P. Ozsváth and Z. Szabó,Holomorphic disks and genus bounds, Geom. Topol.,8(2004), pp. 311–334

2004
[37]

Ozsváth and Z

P. Ozsváth and Z. Szabó,Heegaard Floer homology and contact structures, Duke Math. J.,129(2005), no. 1, pp. 39–61

2005
[38]

Ozsváth and Z

P. Ozsváth and Z. Szabó,Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol.,8(2008), no. 2, pp. 615–692

2008
[39]

Plamenevskaya,Bounds for the Thurston-Bennequin number from Floer homology, Algebr

O. Plamenevskaya,Bounds for the Thurston-Bennequin number from Floer homology, Algebr. Geom. Topol., 4(2004), pp. 399–406

2004
[40]

Plamenevskaya,Contact structures with distinct Heegaard Floer invariants, Mathematical Research Letters, 11(2004), pp

O. Plamenevskaya,Contact structures with distinct Heegaard Floer invariants, Mathematical Research Letters, 11(2004), pp. 547–561

2004
[41]

Raoux,τ-invariants for knots in rational homology spheres, Algebr

K. Raoux,τ-invariants for knots in rational homology spheres, Algebr. Geom. Topol.,20(2020), no. 4, pp. 1601–1640. 22

2020
[42]

Saveliev,Invariants for homology3-spheres, Encyclopaedia of Mathematical Sciences 140

N. Saveliev,Invariants for homology3-spheres, Encyclopaedia of Mathematical Sciences 140. Low-Dimensional Topology 1. Berlin: Springer
[43]

ShahClassification of tight contact structures on some Seifert fibered manifolds, Indag

T. ShahClassification of tight contact structures on some Seifert fibered manifolds, Indag. Math., New Ser., 36(2025), no. 5, pp. 1288–1309

2025
[44]

Tosun,Tight small Seifert fibered manifolds withe 0 =−2, Algebr

B. Tosun,Tight small Seifert fibered manifolds withe 0 =−2, Algebr. Geom. Topol.,20(2020), no. 1, pp. 1–27

2020
[45]

Wu,Legendrian vertical circles in small Seifert spaces, Commun

H. Wu,Legendrian vertical circles in small Seifert spaces, Commun. Contemp. Math.,8(2006), pp. 219–246. HUN-REN Alfréd Rényi Insitute of Mathematics, Budapest 1053, Hungary Email address:acavallo@impan.pl Uppsala Universitet, Uppsala 751 06, Sweden Email address:irma6504@student.uu.se

2006