Positive braid closures and taut foliations

We study taut foliations on the complements of non-split positive braid closures in $S^3$. If $L$ is such a link with components $L_1,\ldots,L_n$ and at least one component is not the unknot, then the Dehn surgery along a multislope $(s_1,\ldots,s_n)\in\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for $i=1,2,\ldots, n$ yields a non-L-space that admits a co-oriented taut foliation.

💡 Research Summary

The paper addresses a central problem in 3‑manifold topology: the relationship among co‑oriented taut foliations, left‑orderability of the fundamental group, and Heegaard‑Floer L‑spaces, encapsulated in the L‑space conjecture. While the implication “taut foliation ⇒ non‑L‑space” is known, the converse directions remain largely open. The author focuses on a concrete class of links—non‑split closures of positive braids in S³—and proves that a wide family of Dehn surgeries on these links produce manifolds that are simultaneously non‑L‑spaces and admit co‑oriented taut foliations.

The main theorem states: let L be the closure of a non‑split positive braid with components L₁,…,L_n, and assume at least one component is not the unknot. For any multislope (s₁,…,s_n)∈ℚⁿ satisfying s_i < 2g(L_i)−1 for each i, the Dehn surgery on L along this multislope yields a 3‑manifold that is a non‑L‑space and carries a co‑oriented taut foliation. Consequently, for positive p‑1,1 L‑space knots (which are known to be positive braid closures), the two implications “p ⇒ c” (p‑surgery yields a non‑L‑space) and “p ⇒ b” (the resulting manifold has a taut foliation) hold automatically.

The proof proceeds by constructing a branched surface B embedded in the link complement ν(L). The construction is carried out in three stages. First, a minimal positive braid diagram is chosen, and a set of points on the diagram is selected so that the diagram decomposes into over‑arcs and under‑arcs. Second, a train track τ is built by locally smoothing each crossing into a short horizontal segment, preserving the over/under distinction. Third, τ is thickened, a loop S¹ is added, and a disk D₂ is attached, producing the branched surface B. The author shows that B admits a natural co‑orientation and that its boundary is a train track on ∂ν(L).

To obtain a laminar branched surface (in the sense of Li), the author introduces a family of simple closed curves γ₁,…,γ_k on the annulus S¹×I satisfying eight geometric conditions (G1)–(G8). These conditions guarantee that each curve intersects every vertical fiber exactly once, consists of an arc inside a bigon of τ and an arc on τ, and that the associated disks D_i are disjoint and transverse to the I‑fibers of the fibered neighborhood N(B). An explicit greedy algorithm (Algorithm 1) constructs the γ_i in linear time with respect to the number of braid crossings, a substantial improvement over previous exponential‑search methods.

Splitting B along the union of the disks D_i yields a new branched surface B₁. The author verifies that B₁ satisfies Li’s laminar criteria: the horizontal boundary is incompressible and B‑incompressible, the complement is irreducible, there are no Reeb components, and there are no sink or half‑sink disks. Hence B₁ carries an essential lamination, which can be thickened to a co‑oriented taut foliation on the surgered manifold.

The paper also discusses how this construction differs from earlier approaches that start from a Seifert surface and perform sutured manifold decompositions. By first “pinching” the Seifert surface to obtain a simpler branched surface, the author reduces the combinatorial complexity of the subsequent splittings. The greedy algorithm’s linear complexity makes the method amenable to computer implementation, potentially allowing systematic exploration of taut foliations for large families of braid closures.

Finally, the author situates the result within the broader landscape of the L‑space conjecture. Since every positive p‑1,1 L‑space knot is a positive braid closure, the main theorem immediately yields non‑L‑spaces with taut foliations for all such knots, confirming two of the three conjectured equivalences for this class. The remaining open direction—showing that every non‑L‑space obtained from these surgeries has left‑orderable π₁—remains unresolved, even for specific examples such as the (2,3,7) pretzel knot.

In summary, the paper provides a new, algorithmically efficient construction of laminar branched surfaces for complements of non‑split positive braid closures, and uses this to prove that a broad family of Dehn surgeries on these links produce non‑L‑spaces admitting co‑oriented taut foliations. This advances our understanding of the L‑space conjecture for an important and well‑studied class of knots and links.