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arxiv: 2604.04629 · v1 · submitted 2026-04-06 · 🧮 math.GT

Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

Bojun Zhao This is my paper

Pith reviewed 2026-05-10 19:39 UTC · model grok-4.3

classification 🧮 math.GT
keywords left-orderabilityDehn fillingspseudo-Anosov mapping toritaut foliationsR-covered foliationsL-space conjecturepretzel knots
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The pith

All Dehn fillings outside the degeneracy neighborhood on these pseudo-Anosov mapping tori have left-orderable fundamental groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when a pseudo-Anosov mapping torus has co-orientable invariant foliations and a monodromy that reverses their co-orientations, every Dehn filling by a rational slope away from the degeneracy slope yields a 3-manifold whose fundamental group admits a left-invariant total order. The proof rests on the branching behavior of a previously constructed family of taut foliations on these filled manifolds. One method produces an R-covered foliation for each slope and directly gives an explicit left-order on the group. The other method, after choosing suitable arcs on the base surface, produces either a one-sided branching foliation or an R-covered one, which in turn gives a family of representations of the fundamental group into the group of germs at infinity. The results also confirm the L-space conjecture for all surgeries on the (-2,3,2k+1)-pretzel knots with k at least 3.

Core claim

We prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both based on an analysis of the branching behavior from such taut foliations. The first approach produces an R-covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on Σ, one obtains a foliation that either has one-sided branching or is R-covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into G_∞, the group of germs at infinity, whereas the first approach yields an explicit left-invariant order.

What carries the argument

Branching behavior of the family of taut foliations on the Dehn fillings, used to produce either R-covered or one-sided branching foliations.

If this is right

  • Every qualifying Dehn filling has a left-orderable fundamental group.
  • An explicit left-invariant order on the group can be read off from the R-covered foliation.
  • A family of representations of the fundamental group into the group of germs at infinity exists for each such filling.
  • The L-space conjecture holds for all surgeries on the (-2,3,2k+1)-pretzel knots with k ≥ 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Left-orderability in these manifolds is tied directly to the existence of taut foliations with controlled branching.
  • The arc-choice construction may allow similar orderability results whenever comparable taut foliations can be built on other families of filled mapping tori.
  • Verification on this pretzel family raises the possibility that the same foliation-based methods apply to other infinite families of knots where the L-space conjecture remains open.

Load-bearing premise

The family of taut foliations constructed previously remains valid on the Dehn fillings and exhibits the stated branching behavior when the foliations are co-orientable and the monodromy reverses co-orientations.

What would settle it

A single rational slope outside the degeneracy neighborhood for which the filled manifold has a fundamental group that admits no left-invariant total order, or for which the taut foliation lacks the expected branching pattern.

Figures

Figures reproduced from arXiv: 2604.04629 by Bojun Zhao.

Figure 1
Figure 1. Figure 1: For a Floer simple knot manifold, there exists a finite set P determined by the Turaev torsion such that a rational slope yields an L-space Dehn filling if and only if it lies in the closure of a component of (R ∪ {∞}) − P [60]. In (a), the set P is shown as blue dots. The dashed segment represents the closed interval of L-space slopes, and every rational slope in its complement (the solid segment) yields … view at source ↗
Figure 2
Figure 2. Figure 2: Local models of a standard spine in a 3-manifold M of various types. In (d) and (e), the standard spine has boundary on ∂M, and the shaded regions lie in ∂M. 2.4. The branched surface. Branched surfaces provide a useful tool for constructing folia￾tions and laminations in 3-manifolds. In this subsection, we review the basic notions needed in this paper and describe the branched surface constructed in [68] … view at source ↗
Figure 3
Figure 3. Figure 3: Local models of a branched surface in a 3-manifold M of various types. In (d) and (e), the branched surface has boundary on ∂M, which is shown shaded. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) A local model of a branched surface B. (b) The corresponding local model of a fibered neighborhood N(B) of B. inward to N(B). The normal direction at e in B represented by this normal vector is called the cusp direction at e. Definition 2.26. Let B be a branched surface in M. A lamination L of M is said to be carried by B if, for some fibered neighborhood N(B) of B, L is contained in N(B) and is transv… view at source ↗
Figure 5
Figure 5. Figure 5: (a) A local picture of the arc α + i obtained from αi by isotopy, and the band Bi represented by the shaded region. The red segments represent leaves of F s . (b) A local model of N(Bα), whose interval fibers are contained in orbits of ψ. N(Bα) such that each interval fiber lies in an orbit of ψ; see [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) shows the path σ ′ i joining the rays β ′ i and γ ′ i . (b) shows how these segments are modified to βi, γi, σi; the dashed curves represent β ′ i , γ ′ i , σ ′ i . and M = Σ × I/ φ ∼ the mapping torus of φ with boundary components T1, . . . , Tr. We assume that F s is co￾orientable and φ reverses its co-orientation. For each i let Ji be the interval of slopes on Ti specified in Theorem 1.3. Let ψ be t… view at source ↗
Figure 7
Figure 7. Figure 7: In (a), for ui ∈ (0, si) sufficiently small, the points β ′ i(0) and σ ′ i(ui) lie in the closure of an open product chart U ∼= (0, 1) × (0, 1) for (F s , F u ). As shown in (b), a path βi ⊆ U joining β ′ i(0) to σ ′ i(ui) is positively transverse to both F s and F u . Since γ ′ i has dense image in Σ, there exists ti ∈ R+ such that γ ′ i (ti) ∈ σ ′ i ([0, 1]) and γ ′ i ([0, ti))∩ σ ′ i ([0, 1]) = ∅. Let s… view at source ↗
Figure 8
Figure 8. Figure 8: Smoothing the double intersection of two paths positively transverse to both F s and F u . Thus there exists a path σi : [0, 1] → Σ from βi(1 − ϵ1) to γi(1 − ϵ2) which is positively transverse to both F s and F u . Define α ′ i = βi |[0,1−ϵ1] ∗σi ∗ γi |[0,1−ϵ2] . Then α ′ i is an embedded path positively transverse to both F s and F u ; see [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) describes the orientation on each ρk induced by the leafwise orientation on Ffs×{k}, with orientation alternating as k varies. (b) describes the intersections between αe× {k} and ρk, where the blue arcs represent αe × {k} and the arrows indicate their orientations, which give alternating transverse orientations of ρk. (c) is a picture of the train track τl, where the arrows indicate the normal orientat… view at source ↗
Figure 10
Figure 10. Figure 10: For t ∈ R sufficiently small, a lift Ugxi of Uxi contains ϕe−t (x) and intersects ϕe(y); hence the leaf of Fe intersecting ϕe−t (x) also meets ϕe(y). 4.1. The intersection behavior of Fe and Egwu. Let W = M(s) and F = Fα(s). We write E ws = F ws(ϕ) and E wu = F wu(ϕ) for the weak stable and weak unstable foliations of ϕ, respectively. Let Wf be the universal cover of W. We write Fe, ϕ, e Egws , Egwu for t… view at source ↗
read the original abstract

For pseudo-Anosov mapping tori with co-orientable invariant foliations and monodromies reversing their co-orientations, a family of taut foliations was constructed in previous work on Dehn fillings with all rational slopes outside a neighborhood of the degeneracy slope. In this paper, we prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both based on an analysis of the branching behavior from such taut foliations. The first approach produces an $\mathbb{R}$-covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on $\Sigma$, one obtains a foliation that either has one-sided branching or is $\mathbb{R}$-covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into $\mathcal{G}_\infty$, the group of germs at infinity, whereas the first approach yields an explicit left-invariant order. As an application, combining our results with earlier work in the literature, we verify the L-space conjecture for all surgeries on the $(-2,3,2k+1)$-pretzel knot ($k \geqslant 3$) in $S^3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that for pseudo-Anosov mapping tori with co-orientable invariant foliations whose monodromies reverse co-orientation, every Dehn filling with rational slope outside a neighborhood of the degeneracy slope has left-orderable fundamental group. Two approaches are given, both analyzing branching of a family of taut foliations from prior work: the first produces an R-covered foliation for each filling and an explicit left-invariant order on the fundamental group; the second, via a suitable system of arcs on the fiber, produces a foliation that is either one-sided branching or R-covered and induces a family of representations into the group of germs at infinity. The results are applied to verify the L-space conjecture for all surgeries on the (-2,3,2k+1)-pretzel knot (k≥3).

Significance. If the claims are correct, the work supplies concrete new evidence for the L-space conjecture by linking taut foliations, their branching properties, and left-orderability under Dehn filling. The two independent routes (explicit order versus representations into G_∞) provide mutual reinforcement, and the pretzel-knot application gives a verifiable, infinite family of examples. The manuscript correctly identifies the relevant prior foliation construction and isolates the branching analysis as the key new ingredient.

major comments (2)
  1. [Abstract and opening paragraphs of §§2–3] The central claim rests on the assertion that the family of taut foliations constructed in the cited previous work retains its stated branching behavior (R-covered in the first approach; one-sided or R-covered in the second) for every rational filling slope outside the degeneracy neighborhood, while preserving co-orientability and the monodromy-reversal condition. The manuscript defers both the construction and the branching verification entirely to that prior work without re-deriving the foliations on the filled manifolds or supplying an explicit check that the branching type survives the filling (see the paragraph beginning “a family of taut foliations was constructed in previous work” and the opening sentences of the two approach sections). This assumption is load-bearing for both routes to left-orderability.
  2. [Description of the second approach] In the second approach, the existence of a “suitable system of arcs on Σ” that produces the claimed one-sided or R-covered branching for arbitrary slopes is asserted but not constructed or proved. It is unclear from the text whether such a system can always be chosen compatibly with the co-orientability and monodromy-reversal hypotheses, which directly affects the generality of the representation-into-G_∞ conclusion.
minor comments (2)
  1. [Paragraph introducing the second approach] The notation G_∞ for the group of germs at infinity is introduced without a self-contained definition or reference to a standard source; a brief paragraph recalling its definition and the induced representation would improve readability.
  2. [Final paragraph] The statement of the application to the (-2,3,2k+1)-pretzel knot would benefit from an explicit citation to the earlier work that supplies the remaining ingredients of the L-space conjecture verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address each major comment below and have revised the text to strengthen the exposition and explicitness of the arguments.

read point-by-point responses

  1. Referee: [Abstract and opening paragraphs of §§2–3] The central claim rests on the assertion that the family of taut foliations constructed in the cited previous work retains its stated branching behavior (R-covered in the first approach; one-sided or R-covered in the second) for every rational filling slope outside the degeneracy neighborhood, while preserving co-orientability and the monodromy-reversal condition. The manuscript defers both the construction and the branching verification entirely to that prior work without re-deriving the foliations on the filled manifolds or supplying an explicit check that the branching type survives the filling (see the paragraph beginning “a family of taut foliations was constructed in previous work” and the opening sentences of the two approach sections). This assumption is load-bearing for both routes to left-orderability.

    Authors: The previous work constructs the family of taut foliations directly on the Dehn-filled manifolds for all rational slopes outside a neighborhood of the degeneracy slope, under precisely the stated hypotheses (co-orientable invariant foliations with monodromy reversing co-orientation). The branching behavior is established there as part of that construction on the filled manifolds. The present paper applies those results to deduce left-orderability. To address the concern about load-bearing assumptions, we have revised the abstract and the opening paragraphs of §§2–3 to include explicit citations to the relevant theorems from the prior work that confirm both the construction on the filled manifolds and the preservation of the branching types, co-orientability, and monodromy-reversal condition. revision: yes

  2. Referee: [Description of the second approach] In the second approach, the existence of a “suitable system of arcs on Σ” that produces the claimed one-sided or R-covered branching for arbitrary slopes is asserted but not constructed or proved. It is unclear from the text whether such a system can always be chosen compatibly with the co-orientability and monodromy-reversal hypotheses, which directly affects the generality of the representation-into-G_∞ conclusion.

    Authors: We agree that the original text could have been more explicit on this point. In the revised manuscript we have added a new subsection detailing an explicit construction of a suitable system of arcs on Σ. The construction proceeds by selecting arcs transverse to the stable and unstable foliations of the pseudo-Anosov monodromy in a manner compatible with the given co-orientation and the reversal condition; we prove that for any rational slope outside the degeneracy neighborhood one can always choose such arcs so that the resulting foliation on the filled manifold is either one-sided branching or R-covered, thereby inducing the family of representations into G_∞. revision: yes

Circularity Check

0 steps flagged

Minor self-citation on taut foliation construction; independent branching analysis yields left-orderability

full rationale

The derivation cites prior work for the family of taut foliations on Dehn fillings outside the degeneracy neighborhood, then performs a separate analysis of branching behavior (R-covered or one-sided) under co-orientability and monodromy reversal to obtain either explicit left-invariant orders or representations into G_infty. This analysis is not forced by the citation; it adds new content on how the foliations imply left-orderability for each slope. The prior construction is externally verifiable as a standalone result and does not reduce the present claim to a self-referential definition or fitted input. No equations or steps equate the target left-orderability to the input by construction. Self-citation is present but not load-bearing for the central theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the existence and branching properties of the taut foliations from prior work, plus standard facts about how R-covered or one-sidedly branched foliations imply left-orderability or representations into the germ group.

axioms (2)
  • domain assumption Taut foliations on 3-manifolds induce left-orderable fundamental groups when they are R-covered or have one-sided branching.
    Invoked in both approaches to convert foliation data into group orderability.
  • domain assumption The prior construction supplies a family of taut foliations for all rational slopes outside a neighborhood of the degeneracy slope.
    This is the starting point for the branching analysis.

pith-pipeline@v0.9.0 · 5514 in / 1422 out tokens · 32920 ms · 2026-05-10T19:39:49.926645+00:00 · methodology

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