Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy

This paper proves that Dehn fillings of mapping tori derived from orientation-preserving pseudo-Anosov homeomorphisms with co-orientation-reversing monodromy admit co-orientable taut foliations for a specific range of multislopes, a result that characterizes all non-L-space fillings for certain hyperbolic fibered knot manifolds, including $(-2,3,2q+1)$-pretzel knots.

The Gist

Imagine you have a piece of stretchy fabric (a surface) with some holes cut out of it. Now, imagine you twist and stretch this fabric in a very specific, chaotic way (a "pseudo-Anosov" twist) and then glue the top edge to the bottom edge. The result is a 3D shape, like a twisted tube or a donut with holes. Mathematicians call this a mapping torus.

The paper by Bojun Zhao is about what happens when you plug up the holes in this twisted tube with solid dough (a process called Dehn filling).

Here is the breakdown of the paper's story, using simple analogies:

1. The Big Mystery: The "L-Space" Riddle

Mathematicians are trying to solve a giant puzzle called the L-space conjecture. It connects three different ways of looking at a 3D shape:

• The "Boring" Shape (L-space): Some shapes are so simple and rigid that they don't have much "wiggle room" mathematically. They are like a perfectly stiff plastic ball.
• The "Orderly" Group: The shape's internal symmetry group can be arranged in a strict line (left-to-right) without any contradictions.
• The "Flowing" Foliations: The shape can be sliced into layers (like a loaf of bread) that flow smoothly everywhere without getting stuck or tangled.

The conjecture says: If a shape is NOT "boring" (not an L-space), then it must have "flowing layers" (a taut foliation).

Zhao's paper is about proving that for a specific type of twisted tube, if you plug the holes in a certain way, you guarantee that the resulting shape will have these smooth, flowing layers.

2. The Twist: The "Co-Orientation Reversing" Problem

Usually, when you twist a fabric, the "front" and "back" of the material stay consistent. But in this paper, Zhao looks at a special kind of twist where the fabric flips inside out as it goes around.

• Analogy: Imagine walking on a Möbius strip. If you walk around it once, your left hand becomes your right hand.
• In math terms, this is called co-orientation-reversing.
• The Problem: When the fabric flips inside out, the usual rules for finding those smooth "flowing layers" break down. Many ways of plugging the holes result in a "boring" shape (an L-space) that cannot have smooth layers.

3. The Solution: Finding the "Safe Zones"

Zhao's main achievement is finding the Safe Zones.

Imagine the edge of your twisted tube has a "danger zone" of angles where plugging the hole creates a boring, rigid shape.

• The Degeneracy Slope: This is the specific angle where the chaotic flow of the twist hits the wall and gets stuck. It's the "forbidden" angle.
• The Safe Intervals: Zhao calculated exactly which angles (slopes) are safe to plug the holes with. He found that if you plug the holes with angles outside a specific small range near the forbidden angle, the resulting shape will always have those smooth, flowing layers (co-orientable taut foliations).

The Metaphor:
Think of the twisted tube as a river with a whirlpool (the chaotic twist).

• If you throw a boat (plug the hole) into the whirlpool at the exact wrong angle, it gets sucked in and destroyed (becomes an L-space).
• Zhao figured out the exact compass directions where you can throw the boat, and it will glide smoothly over the water, creating a beautiful, continuous path (a taut foliation).

4. Why This Matters: The "Pretzel" Knots

The paper doesn't just stay in theory; it applies to real, famous knots, like the Pretzel Knot (specifically the $(-2, 3, 2q+1)$ variety).

• For these specific knots, Zhao showed that almost every way you can plug the hole (except for a tiny, specific range) creates a shape that is not an L-space.
• This confirms the L-space conjecture for these knots: "If it's not boring, it has smooth layers."

5. The Toolkit: Branched Surfaces

How did he prove this? He used a tool called a Branched Surface.

• Analogy: Imagine a subway map. The tracks (leaves of the foliation) merge and split at stations (branch points).
• Zhao built a specific "subway map" inside the twisted tube. He proved that this map is sturdy enough to guide the "trains" (the flow) all the way through the tube and out the other side, no matter how you plug the holes (as long as you stay in his "Safe Zones").
• Because the map is sturdy and covers the whole shape, it proves that the smooth layers exist.

Summary

Bojun Zhao took a complex, twisted 3D shape where the fabric flips inside out. He figured out exactly which angles you can use to seal the holes so that the shape remains "fluid" and full of smooth, flowing layers. This helps mathematicians solve a major puzzle about the nature of 3D shapes, proving that for these specific knots, "non-boring" shapes always have "smooth layers."

In one sentence: He found the "safe directions" to plug the holes in a twisted, inside-out tube to guarantee that the resulting shape flows smoothly rather than becoming rigid.

Technical Summary

1. Problem Statement

The paper addresses a central problem in low-dimensional topology related to the L-space conjecture, which posits the equivalence of three properties for a closed, orientable, irreducible 3-manifold $M$:

1. $M$ is not an L-space (a rational homology sphere with minimal Heegaard Floer homology rank).
2. The fundamental group $\pi_1(M)$ is left-orderable.
3. $M$ admits a co-orientable taut foliation (CTF).

While the implication (3) $\implies$ (1) is established, the converse remains open. The specific focus of this paper is pseudo-Anosov mapping tori $M = \Sigma \times I / \sim$ where the monodromy $\phi: \Sigma \to \Sigma$ is co-orientable but co-orientation-reversing.

• Context: For co-orientation-preserving monodromies, the set of Dehn filling slopes yielding CTFs is well-understood (often excluding only the degeneracy slope).
• The Gap: For co-orientation-reversing monodromies, the situation is more complex. Many such manifolds are known to be L-spaces (and thus admit no CTFs) for certain slopes, but a general characterization of the slopes that do yield CTFs was lacking, particularly for manifolds with multiple boundary components.

Goal: To explicitly determine the set of multislopes on the boundary $\partial M$ such that the resulting Dehn filling admits a co-orientable taut foliation, specifically for the co-orientation-reversing case.

2. Methodology

The author employs a geometric-topological approach centered on branched surfaces and train tracks, following the framework established by Gabai, Oertel, and Li.

A. Construction of a Laminar Branched Surface

1. Setup: Let $\Sigma$ be a compact orientable surface with boundary, and $\phi$ a pseudo-Anosov homeomorphism. The stable foliation $F_s$ is co-orientable.
2. Arc Selection: The author constructs a union of oriented, properly embedded arcs $\alpha \subset \Sigma$ that are positively transverse to the stable foliation $F_s$. These arcs connect specific "stable segments" on the boundary of $\Sigma$ defined by the singularities of $F_s$.
3. Branched Surface $B(\alpha)$: A branched surface $B(\alpha)$ is constructed in the mapping torus $M$ by taking the fiber $\Sigma \times \{0\}$ and attaching product disks $\alpha \times I$.
   • The "branch locus" is formed by the arcs $\alpha$ and their images under the monodromy.
   • The construction ensures $B(\alpha)$ is co-orientable.

B. Verification of Laminarity

To ensure $B(\alpha)$ carries an essential lamination (which can be perturbed into a taut foliation), the author verifies the criteria for a laminar branched surface (Li's definition):

• No Sink/Half-Sink Disks: Proven by analyzing the orientation of the arcs relative to the foliation. The co-orientation-reversing nature of $\phi$ is crucial here, ensuring that any potential disk bounded by the branch locus would contradict the transverse intersection properties of the foliation.
• No Trivial Bubbles: Proven by showing that the complement $M \setminus \text{Int}(N(B(\alpha)))$ contains no 3-balls, relying on the fact that $\Sigma \setminus \alpha$ contains no disk components.
• Essentiality: The surface satisfies the incompressibility and irreducibility conditions required by Gabai-Oertel.

C. Realization of Slopes via Boundary Train Tracks

The intersection of the branched surface with the boundary tori, $\tau = B(\alpha) \cap \partial M$, forms a train track.

• The author analyzes the measures on this train track.
• By constructing specific simple closed curves carried by $\tau$ and analyzing their algebraic intersection numbers with the meridian and longitude, the author determines exactly which rational slopes are "realized" (fully carried) by the train track.
• Key Calculation: The slopes realized depend on the degeneracy locus $(p_i; q_i)$ of the suspension flow and the order $c_i$ of the boundary components under $\phi$.

D. Application of Li's Theorem

Using Theorem 2.15 (Li), the author concludes that since $B(\alpha)$ is a laminar branched surface carrying no torus, and the boundary train tracks realize a specific set of multislopes, the Dehn filling along these slopes admits an essential lamination. This lamination is then extended to a co-orientable taut foliation.

3. Key Contributions and Results

A. Main Theorem (Theorem 1.5)

The paper provides an explicit description of the set of CTF filling multislopes for co-orientation-reversing pseudo-Anosov mapping tori.

• Let $T_i$ be boundary components with degeneracy locus $(p_i; q_i)$ and boundary component order $c_i$.
• The set of valid slopes $J_i$ on each $T_i$ is an open interval (or union of intervals) in $\mathbb{R} \cup \{\infty\}$ defined by the endpoints:
  $$\frac{p_i}{q_i + c_i} \quad \text{and} \quad \frac{p_i}{q_i - c_i}$$
• Specifically, $J_i$ is the interval (or union of intervals) not containing the degeneracy slope $\delta_{T_i} = p_i/q_i$.
• Result: Any Dehn filling with a multislope in the product of these intervals $J_1 \times \dots \times J_k$ admits a co-orientable taut foliation.

B. Corollary for Fibered Knots (Corollary 1.6)

For fibered knots in $S^3$ or lens spaces where the monodromy is co-orientation-reversing and preserves each boundary component ($c_i=1$), the CTF slopes are explicitly determined.

• If the degeneracy slope is $\delta = p/q$, the CTF slopes are those in the interval(s) between $p/(q+1)$ and $p/(q-1)$ that do not contain $\delta$.

C. Applications to Specific Knots

The paper applies these results to:

1. (-2, 3, 2q+1)-pretzel knots in $S^3$:
   • Confirms that for $q \ge 3$, these knots have co-orientation-reversing monodromy.
   • Proves that all rational slopes in $(-\infty, 2g(K)-1)$ are CTF surgery slopes.
   • Shows that the $n$-fold cyclic branched covers of these knots are non-L-spaces for all $n \ge 2$.
2. L-space knots in Lens Spaces:
   • Provides a census of hyperbolic fibered L-space knots in lens spaces (using the SnapPy census).
   • Identifies specific examples (Tables 1–4) where the derived CTF intervals exactly match the known maximal Non-L-space (NLS) filling intervals.
   • Demonstrates that for many such knots, the set of CTF slopes is precisely the set of NLS slopes, supporting the L-space conjecture in these cases.

4. Significance

• Resolution of a Specific Case: The paper solves the problem of finding CTF slopes for the co-orientation-reversing case, which was previously less understood than the preserving case.
• Support for the L-space Conjecture: By showing that for a wide class of hyperbolic fibered L-space knots (including pretzel knots and many in lens spaces), the set of slopes yielding CTFs coincides with the set of slopes yielding non-L-spaces, the paper provides strong evidence for the equivalence of (1) and (3) in the L-space conjecture.
• Methodological Advancement: The construction of the branched surface $B(\alpha)$ tailored to co-orientation-reversing monodromies offers a new tool for analyzing Dehn fillings. The explicit calculation of the realized slopes via the "phi-triple" $(c, p, q)$ provides a computable criterion for future research.
• New Examples: The paper generates a large list of explicit examples (via the census) where the L-space conjecture is verified, including cases where the knot is in a lens space rather than $S^3$.

In summary, Zhao establishes a precise geometric criterion for the existence of co-orientable taut foliations in Dehn fillings of a significant class of 3-manifolds, bridging the gap between foliation theory and Heegaard Floer homology in the context of the L-space conjecture.