General slice-Bennequin characterization of tight contact 3-manifolds and the tight stratum with vanishing Ozsváth–Szabó contact invariant (Hedden–Raoux)

Hedden and Raoux (PNAS 2021) conjectured a 4-dimensional characterization of tightness for contact structures on closed 3-manifolds: $(Y,\xi)$ is tight if and only if the slice-Bennequin inequality $tb_\xi(\partial\Sigma)+|rot_{[\Sigma]}(\partial\Sigma)|\le -\chi(\Sigma)$ holds for every smoothly embedded surface $\Sigma\subset Y\times[0,1]$ whose boundary is a Legendrian link in $Y\times\{1\}$. They proved the inequality for all structures with non-vanishing Ozsváth–Szabó contact invariant, and the reverse implication follows from the classical Eliashberg–Bennequin inequality; the open core is the forward direction for the many known tight structures whose contact invariant vanishes. As of the authors' December 2025 reformulation the conjecture remains open, and no structural description is known for the tight-but-undetected stratum in which the open direction lives. Either a proof or a counterexample would give an intrinsic, effectively checkable criterion for the tight class on arbitrary closed 3-manifolds.

Background

A contact structure $\xi$ on an oriented 3-manifold $Y$ is a smooth field of tangent planes $\xi=\ker\alpha$ that is maximally non-integrable. It is called overtwisted if $Y$ contains an overtwisted disk — an embedded disk whose boundary is tangent to $\xi$ in such a way that the disk's framing agrees with the contact framing — and tight otherwise. Eliashberg (1989) proved that overtwisted structures satisfy an h-principle — a flexibility principle under which the existence and classification of geometric structures reduce to pure homotopy theory — so their classification reduces to the homotopy theory of plane fields; tightness is where rigidity lives, and there is no analogous classification of tight structures on a general closed 3-manifold.

A knot or link $L\subset(Y,\xi)$ everywhere tangent to $\xi$ is Legendrian. Its Thurston–Bennequin number $tb(L)$ is the linking number of $L$ with its push-off along the contact framing, and its rotation number $rot_{[\Sigma]}(L)$, relative to a surface $\Sigma$ with $\partial\Sigma=L$, is the relative Euler number of $\xi$ over $\Sigma$. Eliashberg and Bennequin proved the classical intrinsic characterization of tightness: $(Y,\xi)$ is tight if and only if $tb_\xi(\partial\Sigma)+|rot_{[\Sigma]}(\partial\Sigma)|\le -\chi(\Sigma)$ for every smoothly embedded surface $\Sigma\subset Y$ with Legendrian boundary, where $\chi$ is the Euler characteristic.

In the standard tight $S^3$, Rudolph strengthened this to the slice-Bennequin inequality, in which the Seifert-type surface $\Sigma\subset S^3$ may be replaced by any smooth surface properly embedded in the 4-ball (a slice surface); his proof used Kronheimer and Mrowka's gauge-theoretic solution of the Milnor conjecture (the unknotting number of the $(p,q)$ torus knot is $(p-1)(q-1)/2$), and conversely the slice-Bennequin inequality implies the Milnor conjecture — tying 3-dimensional contact rigidity to smooth 4-dimensional topology.

Heegaard Floer homology assigns to a closed $Y$ a vector-space invariant $\widehat{HF}(-Y)$, and Ozsváth and Szabó (2005) attached to a contact structure a contact class $c(\xi)\in\widehat{HF}(-Y)$. If $c(\xi)\neq 0$ then $(Y,\xi)$ is tight, and in many settings the class detects fillability by symplectic 4-manifolds (a symplectic filling of $(Y,\xi)$ is a compact symplectic 4-manifold whose convex boundary, with its induced plane field, is $(Y,\xi)$); $c(\xi)$ always vanishes for overtwisted structures. Non-vanishing is, however, strictly stronger than tightness: Ghiggini produced infinitely many universally tight (tight in every finite cover) contact manifolds with trivial contact class; Ghiggini–Honda–Van Horn-Morris showed that Giroux torsion (the presence of a standard contact model on an embedded $T^2\times[0,1]$) forces vanishing; Massot gave torsion-free examples; and Li–Zhou (2025) produced tight but algebraically overtwisted structures — structures for which all holomorphic-curve invariants, including $c(\xi)$, vanish — by contact $+1$-surgery along maximum-$tb$ positive torus knots in the standard contact sphere. No structural description of this tight-but-undetected stratum is known.

Hedden and Raoux (arXiv:2010.13162; PNAS 2021) conjectured that tightness is precisely a 4-dimensional phenomenon: the Bennequin bound should characterize tightness even when the tested surfaces are allowed to sit in the 4-manifold $Y\times[0,1]$ rather than in $Y$ itself. Using knot Floer homology (the refinement of Heegaard Floer theory to pairs $(Y,K)$, a knot $K\subset Y$) and its $\tau$-invariant — an integer homological bound on the slice genus — together with relative adjunction inequalities they proved the conjecture, along with companion statements about fiber surfaces of fibered links and subsurfaces of open book pages (an open book decomposition is a fibration of the complement of a link over $S^1$ whose pages are Seifert surfaces for the binding; by Giroux's correspondence every contact structure is supported by one), for all structures with $c(\xi)\neq 0$. What their method cannot reach is exactly the undetected stratum $c(\xi)=0$, where the problem stands open.

Problem Statement

Primary question (Hedden–Raoux Conjecture 1, PNAS 2021). For a closed oriented contact 3-manifold $(Y,\xi)$, is it true that $(Y,\xi)$ is tight if and only if

$tb_\xi(\partial\Sigma)+|rot_{[\Sigma]}(\partial\Sigma)|\le -\chi(\Sigma)$

holds for every smoothly embedded surface $\Sigma\hookrightarrow Y\times[0,1]$ whose boundary $\partial\Sigma\subset Y\times\{1\}$ is Legendrian, where $tb_\xi$ is the Thurston–Bennequin number, $rot_{[\Sigma]}$ the rotation number computed in the trivialization of $\xi$ over $\Sigma$, and $\chi$ the Euler characteristic? The implication from the inequality to tightness is elementary (surfaces lying in $Y\subset Y\times[0,1]$ recover the classical Eliashberg–Bennequin characterization), and Hedden and Raoux proved the inequality for all structures with non-vanishing Ozsváth–Szabó contact invariant. The open core is therefore: does the slice-Bennequin inequality hold in $Y\times[0,1]$ for every tight $(Y,\xi)$, in particular for the stratum with vanishing contact invariant? Equivalently, in the authors' December 2025 formulation: does every symplectic (pseudo-holomorphic) surface in a thickening of an arbitrary tight contact manifold minimize slice genus among smooth surfaces in the same relative homology class with the same boundary — that is, is tightness precisely a symplecto-geometric property?

The scientific objective is an intrinsic, effectively checkable characterization of the tight class on arbitrary closed 3-manifolds through the slice-Bennequin equivalence; the open direction concerns exactly the stratum of tight contact structures with vanishing Ozsváth–Szabó contact invariant, currently known only through scattered families of examples (Ghiggini; Ghiggini–Honda–Van Horn-Morris; Massot; Conway–Kaloti–Kulkarni; Li–Zhou) with no unifying structural description.

References

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