Complete classification of tight contact structures on the remaining surgeries on the figure-eight knot and Stein fillability of the Phi-counted structures

Dehn surgery with rational coefficient $r$ on the figure-eight knot $K\subset S^3$ produces a closed 3-manifold $M(r)$ that is hyperbolic for all $r\notin\{0,\pm1,\pm2,\pm3,\pm4\}$. Conway and Min (Geom. Topol. 24, 2020) gave the first classification of tight contact structures on an infinite family of hyperbolic 3-manifolds: for $r$ in $\mathcal{R}=((-\infty,-4)\cup[-3,0)\cup[1,4)\cup[5,\infty))\cap\mathbb{Q}$ they proved that $M(r)$ carries exactly $2\Phi(r)$ (for $r\in[1,4)\cup[5,\infty)$) or $\Phi(r)+\Psi(r)$ (for $r\in(-\infty,-4)\cup[-3,0)$) isotopy classes of tight contact structures, all strongly symplectically fillable, and all Stein fillable when $r\geq-9$. The paper explicitly leaves two gaps, and no later work closing either has been identified. First (Remark 1.2): for $r\in(-4,-3)\cup(0,1)\cup(4,5)$ only lower bounds $2\Phi(r)$ (for $r>0$) and $\Phi(r)+\Psi(r)$ (for $r<0$) are constructed, the upper-bound method fails on these intervals, and the authors state they believe additional tight structures exist; a complete classification is missing. Second (Remark 1.5): for $r<-9$ it is open whether the $\Phi(r)$-counted tight structures on $M(r)$, which arise from negative contact surgery on non-loose Legendrian figure-eight knots in an overtwisted contact $S^3$, are Stein fillable. This problem asks to complete the program: classify the tight contact structures and their fillability on the three remaining slope intervals, and settle Stein fillability of the $\Phi$-counted structures for $r<-9$.

Background

A contact structure on an oriented 3-manifold $M$ is the kernel $\xi=\ker\alpha$ of a 1-form satisfying $\alpha\wedge d\alpha>0$. It is tight if it contains no overtwisted disk (an embedded disk whose boundary is tangent to $\xi$), and overtwisted otherwise. Overtwisted structures are flexible and classified by homotopy data, while tight structures are rigid and must be classified manifold by manifold. A contact 3-manifold is strongly symplectically fillable if it is the convex boundary of a compact symplectic 4-manifold, and Stein fillable if it is the boundary of a Stein domain (a complex manifold with a strictly plurisubharmonic exhaustion); in dimension three Stein fillability is strictly stronger than strong fillability. A basic question of the field is which closed 3-manifolds admit tight contact structures and, when they do, how many up to isotopy. Classifications were long known for lens spaces, solid tori, many small Seifert fibered spaces (3-manifolds admitting a fibration over the circle with at most three exceptional fibers), and torus bundles, but hyperbolic families resisted. By a theorem of Colin, Giroux, and Honda, a closed orientable irreducible 3-manifold (irreducible: every embedded 2-sphere bounds a 3-ball) carries infinitely many tight contact structures up to isotopy exactly when it is toroidal (it contains an embedded incompressible torus), so on the atoroidal (hyperbolic) surgeries below any classification is a finite list. Let $K$ be the figure-eight knot, the simplest hyperbolic knot; it is a genus-one fibered knot (its complement fibers over the circle with fiber a once-punctured torus), and the monodromy of that fibration is pseudo-Anosov (it stretches one of a pair of transverse measured foliations of the fiber and contracts the other). Let $M(r)$ denote Dehn surgery on $K$ with rational coefficient $r$ (remove a tubular neighborhood of $K$ and reglue it so that the meridian of the knot maps to the slope-$r$ curve on the boundary torus); by Thurston $M(r)$ is hyperbolic unless $r\in\{0,\pm1,\pm2,\pm3,\pm4\}$. For $r=0$ and $r=\pm4$ the manifold $M(r)$ is toroidal and carries infinitely many tight structures (the torus-bundle cases were classified by Honda), while $M(\pm1),M(\pm2),M(\pm3)$ are small Seifert fibered spaces with known classifications, e.g. $M(1)\cong\Sigma(2,3,7)$ has exactly two tight structures (Mark and Tosun) and $M(-1)$ has exactly one (Tosun). Conway and Min completed the remaining slopes as follows. For $r=\frac{p}{q}\in(0,1]\cap\mathbb{Q}$ write the negative continued fraction $-\frac{q}{p}=[r_0,\dots,r_n]=r_0-\frac{1}{r_1-\dots-\frac{1}{r_n}}$ with $r_0\leq-1$ and $r_i\leq-2$ for $i\geq1$, and set $\Phi(r)=|r_0(r_1+1)\cdots(r_n+1)|$, extending $\Phi$ to all of $\mathbb{Q}$ by $\Phi(r+1)=\Phi(r)$; set $\Psi(r)=0$ for $r\geq-3$ and $\Psi(r)=\Phi\!\left(-\frac{1}{r+3}\right)$ for $r<-3$. Their Theorem 1.1 states that for $r\in\mathcal{R}=((-\infty,-4)\cup[-3,0)\cup[1,4)\cup[5,\infty))\cap\mathbb{Q}$ the manifold $M(r)$ supports exactly $2\Phi(r)$ tight contact structures up to isotopy for $r\in[1,4)\cup[5,\infty)$, and exactly $\Phi(r)+\Psi(r)$ for $r\in(-\infty,-4)\cup[-3,0)$, distinguished by their Heegaard Floer contact classes (an invariant in Ozsváth and Szabó's Heegaard Floer homology whose nonvanishing certifies tightness). A Legendrian knot is a knot everywhere tangent to $\xi$; contact surgery is the contact-geometric analogue of Dehn surgery performed along a Legendrian knot. The structures counted by $\Psi$ arise by negative contact surgery on Legendrian figure-eight knots in the standard tight $(S^3,\xi_{std})$ (studied earlier by Conway), while those counted by $\Phi$ arise by negative contact surgery on Legendrian figure-eight knots in the overtwisted sphere $(S^3,\xi^1_{OT})$, the unique overtwisted contact structure on $S^3$ with (normalized) 3-dimensional homotopy invariant $d_3=1$ ($d_3$ being the classical invariant of the underlying oriented 2-plane field); such knots are called non-loose, meaning their complements are tight even though the ambient sphere is overtwisted. Their Theorem 1.4 shows every tight structure on $M(r)$, $r\in\mathcal{R}$, is strongly symplectically fillable, that all $\Psi$-counted ones are Stein fillable, and that all tight structures on $M(r)$ are Stein fillable whenever $r\geq-9$. The upper bound is obtained through a convex-surface decomposition of the genus-one fiber of the figure-eight complement (a convex surface is one admitting a transverse vector field whose flow preserves $\xi$, so that the contact data reduce to its dividing curves, the curves where the flow direction is tangent to $\xi$), a technique introduced for the figure-eight and torus knots by Etnyre and Honda; this shelling analysis is exactly what fails on the three slope intervals left open.

Problem Statement

Let $K\subset S^3$ be the figure-eight knot, and for rational $r$ let $M(r)$ be the closed 3-manifold obtained by Dehn surgery on $K$ with coefficient $r$ (remove a tubular neighborhood of $K$ and reglue it so that the meridian of the knot maps to the slope-$r$ curve on the boundary torus). Define the counting functions $\Phi,\Psi:\mathbb{Q}\to\mathbb{Z}$ of Conway and Min as follows: for $r=\frac{p}{q}\in(0,1]\cap\mathbb{Q}$ write the negative continued fraction $-\frac{q}{p}=[r_0,\dots,r_n]=r_0-\frac{1}{r_1-\dots-\frac{1}{r_n}}$ with $r_0\leq-1$ and $r_i\leq-2$ for $i\geq1$, set $\Phi(r)=|r_0(r_1+1)\cdots(r_n+1)|$, extend $\Phi$ to all of $\mathbb{Q}$ by $\Phi(r+1)=\Phi(r)$, and set $\Psi(r)=0$ for $r\geq-3$ and $\Psi(r)=\Phi\!\left(-\frac{1}{r+3}\right)$ for $r<-3$. Complete the classification program of Conway and Min for tight contact structures on this family, namely: (1) For every rational $r\in(-4,-3)\cup(0,1)\cup(4,5)$, determine the exact number of isotopy classes of tight contact structures on $M(r)$ and classify them. The known lower bounds are $2\Phi(r)$ structures for $r>0$ and $\Phi(r)+\Psi(r)$ for $r<0$, obtained by contact surgery; the source states these are believed not to be sharp, so the answer must either match the lower bound or identify the additional structures. (2) Within the already classified range $\mathcal{R}=((-\infty,-4)\cup[-3,0)\cup[1,4)\cup[5,\infty))\cap\mathbb{Q}$, determine for every rational $r<-9$ whether each of the $\Phi(r)$-counted tight contact structures on $M(r)$ is Stein fillable, closing the gap between Theorem 1.4's strong fillability and its partial Stein fillability statement; and extend the same fillability determination to whatever structures part (1) produces on the remaining intervals. Parts (1) and (2) jointly complete one scientific objective: a full enumeration of the tight contact structures on every Dehn surgery on the figure-eight knot, together with their fillability status. The source poses a further, logically independent question (its Question 1.8): for a general fibered knot $K\subset S^3$ and rational $r\in[0,1)$, whether the number of isotopy classes of tight contact structures on $S^3_{n+r}(K)$ is independent of $n$ for all sufficiently large $n$; that asymptotic question about arbitrary fibered knots is recorded here as adjacent open context and is deliberately not part of this problem's completion boundary.

References

1. James Conway and Hyunki Min, "Classification of tight contact structures on surgeries on the figure-eight knot", Geometry & Topology 24(3):1457-1517, 2020. DOI: 10.2140/gt.2020.24.1457; arXiv:1901.06066. https://arxiv.org/abs/1901.06066 (source of the open questions posed here)
2. John B. Etnyre and Ko Honda, "Knots and Contact Geometry I: Torus Knots and the Figure Eight Knot", Journal of Symplectic Geometry 1(1):63-120, 2001. DOI: 10.4310/jsg.2001.v1.n1.a3. https://doi.org/10.4310/jsg.2001.v1.n1.a3
3. James Conway, "Contact surgeries on Legendrian figure-eight knots", Journal of Symplectic Geometry 17(4):1061-1078, 2019. DOI: 10.4310/jsg.2019.v17.n4.a4; arXiv:1610.03943. https://arxiv.org/abs/1610.03943
4. Ko Honda, "On the classification of tight contact structures II", Journal of Differential Geometry 55(1):83-143, 2000. DOI: 10.4310/jdg/1090340567. https://doi.org/10.4310/jdg/1090340567
5. Thomas E. Mark and Bülent Tosun, "Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres", Advances in Mathematics 335:878-895, 2018. DOI: 10.1016/j.aim.2018.07.023; arXiv:1603.07710. https://arxiv.org/abs/1603.07710 (classification of the two tight structures on $M(1)\cong\Sigma(2,3,7)$, cited in the source's introduction)
6. Bülent Tosun, "Tight small Seifert fibered manifolds with $e_0=-2$", Algebraic & Geometric Topology 20:1-27, 2020. DOI: 10.2140/agt.2020.20.1; arXiv:1510.06948. https://arxiv.org/abs/1510.06948 (classification on $M(-1)$ and the torus-knot surgery classifications motivating the source's Question 1.8)
7. Vincent Colin, Emmanuel Giroux, and Ko Honda, "Finitude homotopique et isotopique des structures de contact tendues", Publications Mathématiques de l'IHÉS 109:245-293, 2009. DOI: 10.1007/s10240-009-0022-y. https://doi.org/10.1007/s10240-009-0022-y (toroidality criterion for infinitely many tight structures)
8. John Etnyre and Nur Saglam, "Surgeries on the trefoil and symplectic fillings", 2023. arXiv:2308.00068. https://arxiv.org/abs/2308.00068 (trefoil analogue)
9. Hyunki Min and Isacco Nonino, "Tight contact structures on a family of hyperbolic L-spaces", 2023. arXiv:2311.14103. https://arxiv.org/abs/2311.14103 (Whitehead-link successor classification)
10. Mahan Mj and Balarka Sen, "Tight contact structures on hyperbolic homology 3-spheres", 2024. arXiv:2404.14698. https://arxiv.org/abs/2404.14698
11. Hyunki Min and Konstantinos Varvarezos, "Bordered contact invariants and half Giroux torsion", 2025. arXiv:2506.14050. https://arxiv.org/abs/2506.14050 (adjacent tool development)
12. John B. Etnyre, Hyunki Min, and Anubhav Mukherjee, "Non-loose torus knots", 2022. arXiv:2206.14848. https://arxiv.org/abs/2206.14848 (develops the non-loose-knot surgery theory underlying the $\Phi$-counted structures, for torus rather than figure-eight knots; adjacent tool development)
13. Isacco Nonino, "An infinite family of hyperbolic 3-manifolds without tight projectively Anosov flows", 2024. arXiv:2411.08716. https://arxiv.org/abs/2411.08716 (consumes the source classification)
14. John B. Etnyre, "Tight and overtwisted contact structures", 2025. arXiv:2504.13096. https://arxiv.org/abs/2504.13096 (survey)