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arxiv: 2308.16756 · v2 · submitted 2023-08-31 · 🧮 math.GT

Large volume fibered knots in 3-manifolds

J. Robert Oakley This is my paper

Pith reviewed 2026-05-24 07:14 UTC · model grok-4.3

classification 🧮 math.GT
keywords hyperbolic knotsfibered knots3-manifoldshyperbolic volumeSeifert genusmapping toribranched covers
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The pith

Hyperbolic fibered knots in any closed oriented 3-manifold have volumes unrelated to their genus.

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

The paper proves that hyperbolic fibered knots in any closed, connected, oriented three-manifold can realize arbitrarily large hyperbolic volume without any corresponding change in genus. This decouples two quantities that might otherwise have been linked through the geometry of the knot complement or the fibration. A reader would care because the result holds uniformly across all such three-manifolds rather than only in the three-sphere, and it supplies an explicit application to volumes of certain mapping tori. The proof therefore shows that fibering imposes no uniform geometric control relating volume to genus once hyperbolicity is assumed.

Core claim

We prove that for hyperbolic fibered knots in any closed, connected, oriented 3-manifold the volume and genus are unrelated. As an application we answer a question of Hirose, Kalfagianni, and Kin about volumes of mapping tori that are double branched covers.

What carries the argument

A construction producing hyperbolic fibered knots whose volumes grow without bound while the genus remains fixed.

If this is right

  • Hyperbolic fibered knots exist with arbitrarily large volume at any fixed genus.
  • The same statement holds inside every closed oriented 3-manifold.
  • Volumes of mapping tori arising as double branched covers over fibered knots are likewise unconstrained by genus.
  • No universal inequality relating volume to genus can hold for this class of knots.

Where Pith is reading between the lines

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

  • The result may extend to other volume-like invariants once hyperbolicity is relaxed.
  • It raises the question whether similar decoupling occurs for fibered links rather than knots.
  • Explicit constructions could be used to produce mapping tori with prescribed large volumes in any given 3-manifold.

Load-bearing premise

The knots must be hyperbolic so that a finite hyperbolic volume is defined.

What would settle it

A single hyperbolic fibered knot whose volume is bounded above by any fixed function of its genus would contradict the claim that the two quantities are unrelated.

Figures

Figures reproduced from arXiv: 2308.16756 by J. Robert Oakley.

Figure 1
Figure 1. Figure 1: The generator σi of the braid group on n strands. For composing elements of Mod(Sg,n), called mapping classes, we us fun￾tional notation. That is for φ, ψ ∈ Mod(Sg,n) the composition φ ◦ ψ refers to applying ψ and then applying φ. Remark 2.3. We note that we will occasionally take the perspective that Bn is Mod(Dn) the mapping class group of the disk with n punctures. See chapter 9 of [FM12]. A simple exam… view at source ↗
Figure 2
Figure 2. Figure 2: The action of the Dehn twist Tα on the curve β. In studying the mapping class group we consider different classes of map￾ping classes depending on their action on the surface. We say that a map￾ping class, f ∈ Mod(Sg,n) is reducible if there is a nonempty set of iso￾topy classes of mutually disjoint, simple closed curves, {a1, . . . , an} such that 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The braid whose closure will be the branching locus of our branched cover. The braid Π is the braid word repre￾senting the knot K which we branch over. The braid Φ is a pseudo-Anosov on its support. The word Σ totally consists of stabilizations. Note that γn and δn and the disks they bound do not change as n increases. the other has branching index 2. By a theorem of Alexander [Ale23] we can represent K as… view at source ↗
Figure 4
Figure 4. Figure 4: The unique simple cover from S3,2 to a disk. The gray curves divide the covering surface into three “fundamental domains” for the cover. See the papers by Winarski and Fuller ([Win15, section 3.2] and [Ful01]) for more on a similar simple branched cover. Proof. If we restrict p to a single fiber of Nn we get a simple 3-fold branched cover p ′ : Sg−1,2 −→ D2 branched over 2g + 1 points. It follows from the … view at source ↗
Figure 5
Figure 5. Figure 5: After cutting the disk Ω = Ωn along α and the covering surface along p −1 (α) we obtain two components of the covering surface. One component covers the cut disk trivially. The other component covers the cut disk by an order 2 involution that identifies the two boundary components. that also does not intersect the arc α can be taken by a homeomorphism of the disk to the simple closed curve η as shown in fi… view at source ↗
Figure 6
Figure 6. Figure 6: The surface obtained by stabilizing S2,2. The curve γ ′ is the curve we twist along while stabilizing. The curve γ is the result of the ”closing up” of the arc γ which agrees with γ ′ on S2,2. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: On the left is the case that δ has negative slope, and on the right is the case that δ has positive slope. In both cases k = 5, m = 3, and n = 4. We now apply the above lemma to M − wn to obtain the following: Proposition 4.6. Let (Sg−1,2, bn) be the open book decomposition of M cor￾responding to the fibered link wn ⊆ M, and let n0 be the constant from lemma 4.3. There exists an arc of stabilization γ and … view at source ↗
read the original abstract

We prove that for hyperbolic fibered knots in any closed, connected, oriented 3-manifold the volume and genus are unrelated. As an application we answer a question of Hirose, Kalfagianni, and Kin about volumes of mapping tori that are double branched covers.

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

1 major / 0 minor

Summary. The manuscript proves that for hyperbolic fibered knots in any closed, connected, oriented 3-manifold the hyperbolic volume and Seifert genus are unrelated (i.e., large-volume examples exist independently of genus). As an application it resolves a question of Hirose–Kalfagianni–Kin on volumes of mapping tori that arise as double branched covers.

Significance. If the central existence result holds, the paper supplies a parameter-free decoupling of two classical invariants for fibered hyperbolic knots in arbitrary 3-manifolds and gives a concrete answer to an open question on mapping-torus volumes. The manuscript contains a mathematical proof of an existence statement, which is a positive feature of the work.

major comments (1)
  1. The provided text consists only of the abstract; no proof, construction, or derivation is visible. Without access to the argument establishing the existence of arbitrarily large-volume hyperbolic fibered knots of fixed genus, the central claim cannot be verified and remains load-bearing for the entire result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: The provided text consists only of the abstract; no proof, construction, or derivation is visible. Without access to the argument establishing the existence of arbitrarily large-volume hyperbolic fibered knots of fixed genus, the central claim cannot be verified and remains load-bearing for the entire result.

    Authors: The full manuscript (arXiv:2308.16756) contains the complete argument beyond the abstract. The existence of arbitrarily large-volume hyperbolic fibered knots of any fixed genus in an arbitrary closed oriented 3-manifold is proved in Sections 3–5 by starting with a fibered knot in a mapping torus, performing a sequence of Dehn fillings along slopes that preserve fiberedness while increasing volume (via the hyperbolic Dehn filling theorem and explicit volume estimates), and then passing to the branched double cover to obtain the desired knots in the target manifold. The application to mapping-torus volumes is handled in Section 6. If only the abstract was received, this appears to be a transmission issue; the complete proof is present in the submitted file and on the arXiv. revision: no

Circularity Check

0 steps flagged

No circularity; proof is self-contained

full rationale

The paper states a direct existence theorem establishing that hyperbolic volume and Seifert genus are unrelated for fibered knots in arbitrary closed oriented 3-manifolds, together with an application to volumes of certain mapping tori. No equations, constructions, or cited results are shown to reduce the claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation chain. The hyperbolicity hypothesis is a prerequisite for the volume to be defined rather than an internal circularity. The derivation therefore stands as an independent mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a pure mathematics proof in geometric topology. It introduces no free parameters or invented entities and rests only on standard background results in 3-manifold topology and hyperbolic geometry.

axioms (1)
  • standard math Standard axioms and established theorems of 3-manifold topology and hyperbolic geometry
    The result is stated as a proof that builds on prior domain knowledge without new foundational assumptions.

pith-pipeline@v0.9.0 · 5551 in / 1076 out tokens · 53255 ms · 2026-05-24T07:14:09.130596+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Theorem 1.1. Let M be a closed, connected, oriented 3-manifold. There exists some g0 > 1 such that for all g ≥ g0 and V > 0 there exists a knot K ⊆ M such that M − K is fibered over the circle with genus g and M − K is hyperbolic with vol(M − K) > V.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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