Brieskorn spheres with two fillable contact structures

Alberto Cavallo

arxiv: 2605.17567 · v1 · pith:MVB2RPWWnew · submitted 2026-05-17 · 🧮 math.GT

Brieskorn spheres with two fillable contact structures

Alberto Cavallo This is my paper

Pith reviewed 2026-05-19 22:15 UTC · model grok-4.3

classification 🧮 math.GT

keywords Brieskorn spheressymplectically fillable contact structuresSeifert fibered spacestight contact structuresnegative-twisting3-manifoldssymplectic fillingscontact geometry

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The pith

All Brieskorn spheres with at most two symplectically fillable contact structures are listed up to isotopy for a fixed orientation.

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

The paper applies a prior classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere to produce an explicit enumeration. Brieskorn spheres are the specific 3-manifolds obtained as links of isolated surface singularities in complex space. A sympathetic reader would care because the enumeration isolates the cases where the number of symplectically fillable structures is small, thereby giving concrete data on how contact geometry behaves on this infinite family of manifolds. The count is performed separately for each orientation and structures are considered up to isotopy.

Core claim

By applying the recent classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere, the complete list of all the Brieskorn spheres carrying at most two symplectically fillable structures, up to isotopy, compatible with a given orientation, is provided.

What carries the argument

The classification of negative-twisting tight contact structures on Seifert fibered spaces with spherical base orbifold, which determines which of those structures can be symplectically filled when restricted to Brieskorn spheres.

Load-bearing premise

The recent classification of negative-twisting tight contact structures on Seifert fibered spaces with spherical base orbifold is complete and correctly identifies every structure that can appear on any Brieskorn sphere.

What would settle it

Exhibiting one Brieskorn sphere absent from the list yet possessing exactly two symplectically fillable contact structures, or one sphere on the list possessing three or more, would disprove the completeness of the enumeration.

Figures

Figures reproduced from arXiv: 2605.17567 by Alberto Cavallo.

**Figure 1.** Figure 1: The standard graph of Σ(3, 4, 5) (left), Σ(2, 5, 7) (middle) and Σ(2, 3, 6k + 1) with k ⩾ 1 (right). There are k − 1 vertices with framing −2 on the third leg of the graph of Σ(2, 3, 6k + 1). we proved in [2, Theorem 1.1] that all the fillable structures on a negative-definite Seifert fibered space are presented by Legendrian surgery on the link obtained by blowing down Γ; in the case we have here, each fr… view at source ↗

**Figure 2.** Figure 2: The standard graph Γ representing Y (left) and its dual graph (right) representing −Y , when e0 = −2. There are s vertices with framing −2 on the first leg before S, and t ones after S; moreover, the number of vertices with framing −2 in Γ is a − 1 on the second leg, and b − 1 on the third one. We now assume that e0 = −2, and that the vertex S with framing −3 appears on the first leg; the same argument of… view at source ↗

**Figure 3.** Figure 3: The standard graph of Σ(2, 7, 11) (left), Σ(3, 5, 7) (right) and Σ(3, 4, 11) (bottom). All of these Brieskorn spheres have positive correction term. Note that by reversing the orientation we obtain the Seifert fibred spaces M(−1; 4 11 , 1 2 , 1 7 ), M(−1; 2 3 , 1 5 , 1 7 ) and M(−1; 2 3 , 1 4 , 1 11 ). the manifolds Σ(2, 3, 7, 41) and Σ(2, 3, 11, 13), whose standard graphs have four legs with vertices wit… view at source ↗

**Figure 4.** Figure 4: The standard graph of Σ(3, 7, 10) (top), Σ(4, 5, 9) (left), Σ(3, 8, 11) (right), Σ(3, 7, 19) (middle) and Σ(3, 5, 14) (bottom). Note that by reversing the orientation we obtain the Seifert fibred spaces M(−1; 4 7 , 1 3 , 1 10 ), M(−1; 5 9 , 1 4 , 1 5 ), M(−1; 6 11 , 1 3 , 1 8 ), M(−1; 10 19 , 1 3 , 1 7 ) and M(−1; 3 5 , 1 3 , 1 14 ). Σ(2, 3, 6k − 1) with k ⩾ 2 and Σ(2, 5, 9), see [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 5.** Figure 5: The standard graph of Σ(2, 5, 9) (top) and Σ(2, 3, 6k−1) with k ⩾ 2 (bottom). There are k − 2 vertices with framing −2 after the one with −3 on the first leg of the graph of Σ(2, 3, 6k − 1). Note that by reversing the orientation we obtain the Seifert fibered spaces M(−1; 2 5 , 1 2 , 1 9 ) and M(−1; k 6k−1 , 1 2 , 1 3 ). We now consider an oppositely oriented Brieskorn sphere −Y . We proved in [3, Corollar… view at source ↗

read the original abstract

Applying our recent classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere, we provide the complete list of all the Brieskorn spheres carrying at most two symplectically fillable structures, up to isotopy, compatible with a given orientation.

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 produces the explicit list of Brieskorn spheres with at most two symplectically fillable contact structures by applying the author's prior classification of tight structures on spherical Seifert spaces.

read the letter

The main point is that this gives the full list of Brieskorn spheres that have at most two symplectically fillable contact structures, up to isotopy and for a fixed orientation. It comes from applying the recent classification of negative-twisting tight structures on spherical-base Seifert manifolds to the Brieskorn case. The paper is basically a specialization: it takes the general list from before and sees which structures show up on Brieskorn spheres and which of those admit symplectic fillings. The result is a concrete enumeration that wasn't compiled before. It does the job cleanly because the method is already set up. The author just has to match the Seifert invariants for the Brieskorn spheres and apply the fillability test. That produces usable reference data for people listing contact structures on these manifolds. The soft spot is obvious: the whole thing rests on the prior classification being right and complete. If that earlier paper missed a tight structure that turns out to be fillable here, or if the fillability check has a gap for these specific manifolds, then the list isn't actually complete. The abstract doesn't add any new checks or examples to confirm the application, so the strength is tied directly to the previous work. This is for contact geometers who care about explicit lists on Seifert fibered 3-manifolds. Someone already familiar with the classification will find the Brieskorn case useful as a worked example. It is not changing the big picture but it closes a small enumeration question. I would send it out for peer review. A referee who knows the classification can verify the application without too much trouble, and the result is narrow enough that it should be straightforward to check.

Referee Report

1 major / 1 minor

Summary. The manuscript applies the authors' recent classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere to enumerate all Brieskorn spheres that admit at most two symplectically fillable contact structures up to isotopy, compatible with a fixed orientation.

Significance. If the prior classification is exhaustive and the restriction to Brieskorn spheres is applied correctly, the resulting explicit list would be a useful reference for contact geometers working on fillability questions for Seifert fibered 3-manifolds. The work converts an abstract classification into concrete topological examples with bounded numbers of fillable structures.

major comments (1)

[Abstract and the section applying the classification to Brieskorn spheres] The completeness assertion in the abstract rests entirely on the prior classification of negative-twisting tight structures being exhaustive for all spherical-base Seifert fibered spaces and on the fillability criterion being applied without post-hoc omissions when restricting to Brieskorn invariants. The manuscript should contain an explicit verification (e.g., a table or case-by-case check in the section applying the classification) that every Brieskorn sphere appears among the classified structures and that no fillable structure is excluded by the negative-twisting restriction.

minor comments (1)

Add a short summary paragraph recalling the precise statement of the prior classification (including the range of twisting numbers and the base orbifold conditions) so that readers need not consult the earlier paper to follow the application.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

Referee: [Abstract and the section applying the classification to Brieskorn spheres] The completeness assertion in the abstract rests entirely on the prior classification of negative-twisting tight structures being exhaustive for all spherical-base Seifert fibered spaces and on the fillability criterion being applied without post-hoc omissions when restricting to Brieskorn invariants. The manuscript should contain an explicit verification (e.g., a table or case-by-case check in the section applying the classification) that every Brieskorn sphere appears among the classified structures and that no fillable structure is excluded by the negative-twisting restriction.

Authors: We agree that an explicit verification would strengthen the clarity of the presentation. Our prior work provides an exhaustive classification of all negative-twisting tight contact structures on Seifert fibered spaces with spherical base orbifold. Brieskorn spheres are the Seifert fibered spaces with precisely three exceptional fibers over the sphere and are therefore included without omission in that classification. The fillability criterion is applied uniformly to the structures arising from the classification. To address the suggestion directly, we will add a short verification subsection (including a summary table of the relevant Brieskorn parameters) in the revised manuscript confirming coverage and that the negative-twisting condition captures all fillable structures on these manifolds. revision: yes

Circularity Check

1 steps flagged

Completeness of the list depends on the authors' prior classification of negative-twisting tight structures being exhaustive

specific steps

self citation load bearing [Abstract]
"Applying our recent classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere, we provide the complete list of all the Brieskorn spheres carrying at most two symplectically fillable structures, up to isotopy, compatible with a given orientation."

The claimed complete enumeration is produced by restricting the authors' prior classification to Brieskorn spheres and identifying which of those structures are symplectically fillable. Because the classification itself is cited as 'our recent' work by the same author, the list and its 'at most two' bound are forced by the self-citation rather than derived from independent data or theorems external to the author group.

full rationale

The paper's central result is the complete list of Brieskorn spheres with at most two symplectically fillable contact structures. The abstract states this list is obtained by applying the authors' own recent classification of negative-twisting tight contact structures on spherical-base Seifert fibered spaces. This classification is the sole foundation for enumerating and restricting to fillable structures on Brieskorn spheres (which are a subclass of such Seifert spaces). No independent external verification, machine-checked proof, or parameter-free external benchmark is supplied in the abstract or described derivation chain; the completeness claim therefore reduces directly to the correctness and exhaustiveness of the self-cited prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result depends on the completeness of the prior classification of negative-twisting tight contact structures on spherical Seifert spaces; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere is complete and applies without gaps to Brieskorn spheres.
The abstract states that the list is obtained by applying this classification.

pith-pipeline@v0.9.0 · 5556 in / 1260 out tokens · 49145 ms · 2026-05-19T22:15:39.199074+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Applying our recent classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere, we provide the complete list of all the Brieskorn spheres carrying at most two symplectically fillable structures
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the number of structures is determined combinatorially by the Seifert coefficients as explained in [2, Section 5] and it is equal to |e0 + 1| · ∏ ki ∏ |mi j + 1|

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 3 internal anchors

[1]

Brieskorn spheres and rational homology ball symplectic fillings

A. Alfieri, A. Cavallo and I. Matkovič,Brieskorn spheres and rational homology ball symplectic fillings, arXiv:2605.13812

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Fillable structures on negative-definite Seifert fibred spaces

A. Cavallo and I. Matkovič,Fillable structures on negative-definite Seifert fibred spaces, arXiv:2604.28174

work page internal anchor Pith review Pith/arXiv arXiv
[3]

Heegaard Floer homology and maximal twisting numbers

A. Cavallo and I. Matkovič,Heegaard Floer homology and maximal twisting numbers, arXiv:2604.28162

work page internal anchor Pith review Pith/arXiv arXiv
[4]

I. Dai, M. Hedden and A. Mallick,Corks, involutions, and Heegaard Floer homology, J. Eur. Math. Soc.,25 (2023), no. 6, pp. 2319–2389

work page 2023
[5]

Etnyre and K

J. Etnyre and K. Honda,On the nonexistence of tight contact structures, Ann. Math. (2),153(2001), no. 3, pp. 749–766

work page 2001
[6]

Fintushel and R

R. Fintushel and R. Stern,Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc.,61(1990), pp. 109–137

work page 1990
[7]

Ghiggini and J

P. Ghiggini and J. Van Horn-Morris,Tight contact structures on the Brieskorn spheres−Σ(2,3,6n−1)and contact invariants, J. Reine Angew. Math.,718(2016), pp. 1–24

work page 2016
[8]

Issa and D

A. Issa and D. McCoy,On Seifert fibered spaces bounding definite manifolds, Pac. J. Math.,304(2020), no. 2, pp. 463–480

work page 2020
[9]

Lisca and G

P. Lisca and G. Matić,Tight contact structures and Seiberg-Witten invariants, Invent. Math.,129(1997), pp. 509–525

work page 1997
[10]

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

work page 2003
[11]

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

work page 2003
[12]

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

work page
[13]

Şavk,More Brieskorn spheres bounding rational balls, Topology Appl.,286(2020), 107400, 11 pp

O. Şavk,More Brieskorn spheres bounding rational balls, Topology Appl.,286(2020), 107400, 11 pp

work page 2020
[14]

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. HUN-REN Alfréd Rényi Insitute of Mathematics, Budapest 1053, Hungary Email address:acavallo@impan.pl

work page 2020

[1] [1]

Brieskorn spheres and rational homology ball symplectic fillings

A. Alfieri, A. Cavallo and I. Matkovič,Brieskorn spheres and rational homology ball symplectic fillings, arXiv:2605.13812

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Fillable structures on negative-definite Seifert fibred spaces

A. Cavallo and I. Matkovič,Fillable structures on negative-definite Seifert fibred spaces, arXiv:2604.28174

work page internal anchor Pith review Pith/arXiv arXiv

[3] [3]

Heegaard Floer homology and maximal twisting numbers

A. Cavallo and I. Matkovič,Heegaard Floer homology and maximal twisting numbers, arXiv:2604.28162

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

I. Dai, M. Hedden and A. Mallick,Corks, involutions, and Heegaard Floer homology, J. Eur. Math. Soc.,25 (2023), no. 6, pp. 2319–2389

work page 2023

[5] [5]

Etnyre and K

J. Etnyre and K. Honda,On the nonexistence of tight contact structures, Ann. Math. (2),153(2001), no. 3, pp. 749–766

work page 2001

[6] [6]

Fintushel and R

R. Fintushel and R. Stern,Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc.,61(1990), pp. 109–137

work page 1990

[7] [7]

Ghiggini and J

P. Ghiggini and J. Van Horn-Morris,Tight contact structures on the Brieskorn spheres−Σ(2,3,6n−1)and contact invariants, J. Reine Angew. Math.,718(2016), pp. 1–24

work page 2016

[8] [8]

Issa and D

A. Issa and D. McCoy,On Seifert fibered spaces bounding definite manifolds, Pac. J. Math.,304(2020), no. 2, pp. 463–480

work page 2020

[9] [9]

Lisca and G

P. Lisca and G. Matić,Tight contact structures and Seiberg-Witten invariants, Invent. Math.,129(1997), pp. 509–525

work page 1997

[10] [10]

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

work page 2003

[11] [11]

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

work page 2003

[12] [12]

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

work page

[13] [13]

Şavk,More Brieskorn spheres bounding rational balls, Topology Appl.,286(2020), 107400, 11 pp

O. Şavk,More Brieskorn spheres bounding rational balls, Topology Appl.,286(2020), 107400, 11 pp

work page 2020

[14] [14]

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. HUN-REN Alfréd Rényi Insitute of Mathematics, Budapest 1053, Hungary Email address:acavallo@impan.pl

work page 2020