Cavallo (2026) determined the complete list of Brieskorn spheres carrying exactly two symplectically fillable contact structures up to isotopy, compatible with a given orientation: the two infinite families of small spheres (three singular fibers) Σ(2,3,6k+1)\Sigma(2,3,6k+1) for k≥1k\geq 1 and Σ(2,3,6k−1)\Sigma(2,3,6k-1) for k≥2k\geq 2, together with finitely many further small spheres, all of which carry no other tight structures, and the four-fiber spheres Σ(2,3,7,41)\Sigma(2,3,7,41) and Σ(2,3,11,13)\Sigma(2,3,11,13), which also carry non-fillable tight structures with Giroux torsion. The tight-structure classification on these two four-fiber spheres nevertheless remains incomplete: since Cavallo and Matkovič (2026) proved that every negative-twisting tight structure on a Seifert fibred space over S2S^2 is symplectically fillable, any further tight structure on these spheres must be zero-twisting, and no classification theorem covers that regime, even for small Seifert fibred spaces. The source-faithful open problem posed here is to determine the full set of tight contact structures on Σ(2,3,7,41)\Sigma(2,3,7,41) and Σ(2,3,11,13)\Sigma(2,3,11,13): whether every tight structure is one of the two fillable ones or obtained from one of them by inserting Giroux torsion along an incompressible torus, which torsion amounts occur (in particular whether there are infinitely many isotopy classes), and whether tight non-fillable structures with zero Giroux torsion exist on these spheres.

Background

A Brieskorn sphere Σ(a1,…,an)\Sigma(a_1,\dots,a_n), for pairwise coprime integers ai≥2a_i\geq 2, is a 3-manifold, namely the boundary of the standard star-shaped plumbing, equivalently the Seifert fibred space over S2S^2 with nn exceptional fibers of multiplicities a1,…,ana_1,\dots,a_n (for n=3n=3 it coincides with the intersection of the complex hypersurface z1a1+z2a2+z3a3=0z_1^{a_1}+z_2^{a_2}+z_3^{a_3}=0 with a small sphere centered at the origin in C3\mathbb{C}^3); it is a rational homology 3-sphere carrying a natural Seifert fibration (a decomposition of the 3-manifold into disjoint circles) over the base orbifold S2S^2 with nn singular fibers. Such a sphere is called small when n=3n=3, i.e. when its standard star-shaped plumbing graph — the tree with one central vertex and nn legs encoding the manifold as the boundary of a plumbed 4-manifold — has three legs. As the link of an isolated complex surface singularity, Σ(a1,…,an)\Sigma(a_1,\dots,a_n) carries the Milnor fillable contact structure ξcan\xi_{\mathrm{can}}, obtained by restricting the field of complex tangent planes to the link; a contact structure is a maximally non-integrable cooriented 2-plane field on an oriented 3-manifold, and it is (strongly) symplectically fillable when it arises as the boundary of the field of complex lines of a compact Stein domain (a compact complex manifold with a strictly plurisubharmonic potential), or more generally of a symplectic 4-manifold with convex boundary. By Eliashberg–Gromov theory, fillable implies tight, where tight means the absence of an overtwisted disk (a disk whose Legendrian boundary — a boundary curve everywhere tangent to the contact planes — forces the plane field to twist past itself); classifying tight structures and comparing tightness with fillability is a central program in 3-dimensional contact topology.

Two invariants organize the tight structures on Seifert fibred spaces. The maximal twisting number of a tight structure records, roughly, how closely a regular fiber can follow the contact planes, and is never positive for a tight structure; a tight structure is called negative-twisting or zero-twisting according to whether this maximum is negative or zero. Giroux torsion is the phenomenon of a submanifold T2×[0,1]T^2\times[0,1], with incompressible boundary tori (tori injective on π1\pi_1), on whose slices the dividing curves of the characteristic foliation (the singular foliation a contact structure induces on a surface) rotate through at least one full turn; the torsion amount counts these turns. Positive Giroux torsion obstructs strong symplectic fillability and forces the Ozsváth–Szabó Heegaard Floer contact invariants c^\hat c and c+c^+ — elements of Heegaard Floer homology, a 3-manifold invariant built from counts of holomorphic disks, which detect tightness and fillability — to vanish (Ghiggini, building on Ozsváth–Szabó). Seifert fibred spaces over S2S^2 with four or more singular fibers contain incompressible tori, so Giroux torsion can occur there.

The prior state of the art is as follows. Etnyre and Honda proved that the oppositely oriented Poincaré sphere −Σ(2,3,5)-\Sigma(2,3,5) admits no tight contact structure at all. Complete classifications of tight structures exist for lens spaces, solid tori and T2×IT^2\times I (Honda), for the Brieskorn spheres −Σ(2,3,6n−1)-\Sigma(2,3,6n-1) (Ghiggini–Van Horn-Morris), for small Seifert fibred manifolds with orbifold Euler number e0=−2e_0=-2 — e0e_0 being the Euler number of the Seifert fibration, i.e. the framing of the central vertex of the standard graph — (Tosun), and for small Seifert fibred L-spaces — manifolds whose Heegaard Floer homology is as simple as possible — in the zero-twisting case (Matkovič). On torus bundles Giroux and Honda classified the tight structures: one Stein fillable structure plus infinitely many obtained by adding Giroux torsion, which Ding and Geiges showed are weakly but not strongly symplectically fillable; this is the closest complete analogue. For four singular fibers the only classification is due to Shah, who determined the tight structures with zero Giroux torsion on the Seifert fibred manifolds M(e0;r1,r2,r3,r4)M(e_0;r_1,r_2,r_3,r_4) with e0≤−4e_0\leq -4 — all of them Stein fillable — and constructed on those manifolds tight, not even weakly fillable structures with every Giroux torsion amount; her convex-surface argument requires e0≤−4e_0\leq -4, whereas the canonically oriented four-fiber Brieskorn spheres have e0=−3e_0=-3 (their orientation reversals have e0=−1e_0=-1), so no comparable result covers them. In 2026 Cavallo and Matkovič completed the negative-twisting picture: they classified all negative-twisting tight structures on any Seifert fibred space over S2S^2, proved that each of them is symplectically fillable, and distinguished them by the contact invariant c+c^+. Building on this, Cavallo determined all Brieskorn spheres carrying exactly two fillable structures up to isotopy, compatible with a given orientation: the two infinite families of small spheres Σ(2,3,6k+1)\Sigma(2,3,6k+1) for k≥1k\geq 1 and Σ(2,3,6k−1)\Sigma(2,3,6k-1) for k≥2k\geq 2, together with finitely many further small spheres (on which ξcan\xi_{\mathrm{can}} and its conjugate ξˉcan\bar\xi_{\mathrm{can}} are then the only tight structures, by convex surface theory), and the two four-fiber spheres Σ(2,3,7,41)\Sigma(2,3,7,41) and Σ(2,3,11,13)\Sigma(2,3,11,13), which in addition carry non-fillable tight structures with Giroux torsion, obtained by inserting torsion along the incompressible torus. In a companion work, Cavallo and Matkovič constructed the first known tight zero-twisting structures on Brieskorn spheres (on Σ(8,13,105)\Sigma(8,13,105) and −Σ(8,13,103)-\Sigma(8,13,103), where c^≠0\hat c\neq 0 but c+=0c^+=0), and state explicitly that a complete classification of zero-twisting structures is still missing even on small Seifert fibred spaces, outside the fillable and L-space cases. Thus, on Σ(2,3,7,41)\Sigma(2,3,7,41) and Σ(2,3,11,13)\Sigma(2,3,11,13) every tight structure other than the two fillable ones must be zero-twisting, and nothing is currently known about the zero-twisting structures on these two spheres beyond the existence of some with Giroux torsion.

Problem Statement

Determine, up to isotopy, the complete set of tight contact structures on the canonically oriented (i.e. oriented as the link of the complex surface singularity) four-fiber Brieskorn spheres Σ(2,3,7,41)\Sigma(2,3,7,41) and Σ(2,3,11,13)\Sigma(2,3,11,13) — the only non-small Brieskorn spheres carrying exactly two symplectically fillable contact structures, by Cavallo's Theorem 1.1. Since the two fillable structures ξcan\xi_{\mathrm{can}} and ξˉcan\bar\xi_{\mathrm{can}} are the only negative-twisting tight structures on these spheres, the problem is equivalent to classifying the zero-twisting tight structures on them. On each sphere, determine the full set of tight contact structures up to isotopy: (i) whether every tight structure on each sphere is ξcan\xi_{\mathrm{can}}, ξˉcan\bar\xi_{\mathrm{can}}, or obtained from one of them by inserting Giroux torsion along an incompressible torus; (ii) which Giroux torsion amounts occur, and in particular whether there are infinitely many tight contact structures up to isotopy; and (iii) whether there exist tight, non-fillable structures with zero Giroux torsion on these spheres. Resolving this completes the classification of tight contact structures on the Brieskorn spheres with exactly two fillable structures, the small cases of which are already settled.

References

  1. Alberto Cavallo, "Brieskorn spheres with two fillable contact structures", arXiv:2605.17567 (2026), https://arxiv.org/abs/2605.17567
  2. Alberto Cavallo and Irena Matkovič, "Heegaard Floer homology and maximal twisting numbers", arXiv:2604.28162 (2026), https://arxiv.org/abs/2604.28162
  3. Alberto Cavallo and Irena Matkovič, "Fillable structures on negative-definite Seifert fibred spaces", arXiv:2604.28174 (2026), https://arxiv.org/abs/2604.28174
  4. Alberto Cavallo and Irena Matkovič, "The hat and plus version of the Heegaard Floer contact invariant are not equivalent", arXiv:2604.28170 (2026), https://arxiv.org/abs/2604.28170
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