(1,1) non-L-space knots are persistently foliar

We prove that (1,1) non-L-space knots in $S^3$ and lens spaces are persistently foliar. This provides positive evidence for the L-space conjecture.

💡 Research Summary

The paper addresses the knot‑level version of the L‑space conjecture, which predicts that a non‑L‑space rational homology sphere admits a co‑oriented taut foliation. Delman and Roberts formulated a “persistently foliar” property for knots: a knot is persistently foliar if, except for a single meridional slope, every boundary slope of the knot complement is strongly realized by a co‑oriented taut foliation intersecting the boundary torus in a linear foliation of that slope.

The author focuses on (1,1) knots—knots admitting a genus‑one Heegaard splitting with two base points—lying in S³ or a lens space. Such knots can be described by reduced (1,1) diagrams parametrized by a 4‑tuple (p,q,r,s). Prior work (Greene‑Lewallen‑Vafaee, 2018) characterizes (1,1) L‑space knots precisely as those whose diagrams are coherent: all rainbow arcs around a base point point in the same direction. Consequently, a non‑L‑space (1,1) knot must be incoherent, i.e., it contains at least one pair of rainbow arcs with opposite directions.

The core of the paper is a construction of branched surfaces inside the knot complement that carry essential laminations, which can be thickened to taut foliations realizing all boundary slopes. The construction proceeds in several stages:

1. Heegaard Branched Surface: Starting from a reduced (1,1) diagram, the author builds a Heegaard branched surface by attaching disks to the torus Σ on opposite sides of the α‑ and β‑curves, orienting the branch direction to the left of both curves. Small disks are removed from the two bigons containing the base points, yielding a branched surface B.

2. Sink/Source Tubes: Using the hyperelliptic involution symmetry, the diagram contains exactly one sink bigon and one source bigon, together with a unique sink α‑arc and source α‑arc among the boundaries of hexagons or octagons. The β‑sink (resp. β‑source) sectors connect these bigons to the corresponding α‑arcs, forming a sink tube (resp. source tube). Lemma 2.4 guarantees that these tubes contain all β‑sink (resp. β‑source) sectors.

3. Reversing a Quadrilateral Sector: Within the Heegaard branched surface there is a distinguished quadrilateral sector S₀ that is simultaneously an α‑source and a β‑source. By reversing the co‑orientation of S₀ (the “reversing” operation described in Construction 2.10), the branch direction on its boundary flips outward, effectively eliminating a potential sink disk.

4. Laminarity Checks: A branched surface is laminar if (i) its horizontal boundary is incompressible and non‑spherical, (ii) there are no monogons, (iii) no Reeb components, and (iv) no sink disks. The author verifies each condition for the modified branched surface B_T:

   • The horizontal boundary consists of incompressible surfaces inherited from the Heegaard torus.
   • The construction of sink/source tubes and the removal of the bigon interiors prevent monogons.
   • No torus carried by B_T bounds a solid torus, so Reeb components are absent.
   • The reversing step eliminates any sink disk; any remaining disks are shown to be non‑sink by examining branch directions.

5. Full Carrying of Laminations: Lemma 2.8 (derived from Li’s laminar surface theory) states that a sink‑disk‑free, torus‑free branched surface with non‑disk horizontal components fully carries a lamination. Applying this lemma, the author concludes that B_T fully carries an essential lamination.

6. From Laminations to Foliations: Standard techniques (e.g., Gabai’s sutured manifold hierarchy) allow the lamination to be thickened into a co‑oriented taut foliation. Because the construction works for any boundary slope other than the meridian, each such slope is strongly realized.

The main theorem (Theorem 1.3) asserts that every (1,1) non‑L‑space knot in S³ or a lens space is persistently foliar. As a corollary (Corollary 1.4), these knots admit no reducible surgeries, aligning with classical cabling conjectures.

Methodologically, the paper blends several recent advances:

• Rasmussen’s Heegaard foliation idea (2020) for constructing foliations from Heegaard splittings.
• Tao Li’s branched‑surface laminarity criteria (2002, 2024) to guarantee essential laminations.
• The “reversing” operation, also used by Delman‑Roberts, to manipulate co‑orientations and eliminate sink disks.
• Slope‑detection techniques from Boyer‑Clay and Boyer‑Gordon‑Hu, though the paper’s primary contribution lies in using these detections to guide the branched‑surface construction rather than to prove new detection results.

Overall, the work provides a complete verification of the knot‑level L‑space conjecture for the entire class of (1,1) knots that are not L‑space knots. It demonstrates that the combinatorial data of a (1,1) diagram (specifically its incoherence) directly yields a laminar branched surface, which in turn produces the desired family of taut foliations. This not only strengthens evidence for the L‑space conjecture but also offers a concrete, diagrammatic method that may be adaptable to broader families of knots and 3‑manifolds.