Knotting and linking in 4 and 5 dimensions from barbell diffeomorphisms

Alison Tatsuoka; Gheehyun Nahm; Seungwon Kim

arxiv: 2603.20130 · v2 · submitted 2026-03-20 · 🧮 math.GT

Knotting and linking in 4 and 5 dimensions from barbell diffeomorphisms

Seungwon Kim , Gheehyun Nahm , Alison Tatsuoka This is my paper

Pith reviewed 2026-05-15 06:56 UTC · model grok-4.3

classification 🧮 math.GT

keywords 3-knots5-spherebarbell diffeomorphismscritical pointsknotted solid toriBudney-Gabai conjecture4-spherehandlebodies

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The pith

Barbell diffeomorphisms produce infinitely many non-isotopic 3-knots in the 5-sphere, each with exactly four critical points.

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

The paper constructs explicit examples of 3-dimensional knots embedded in the 5-sphere such that each knot has precisely four critical points when measured by the standard height function on the sphere. These examples are all distinct up to isotopy and stand in direct contrast to Scharlemann's theorem, which proves that every 2-knot in the 4-sphere with four critical points must be unknotted. The same constructions also supply infinitely many knotted solid tori inside the 4-sphere and the 5-ball, thereby settling the final open case of the Budney-Gabai conjecture on the existence of knotted handlebodies in those dimensions. All the new examples arise from concrete barbell diffeomorphisms that control the critical points while creating the knotting.

Core claim

We construct infinitely many non-isotopic 3-knots in the 5-sphere, each of which has four critical points with respect to the standard height function of the 5-sphere. This contrasts with a theorem of Scharlemann which says that any 2-knot in the 4-sphere with four critical points is unknotted, and also provides infinitely many knotted solid tori in the 4-sphere and 5-ball, which resolves the last remaining case of the conjecture by Budney and Gabai on the existence of knotted handlebodies. We also construct various knotted and linked handlebodies, discs, and spheres in the 4-sphere, 5-ball, and 5-sphere using barbell diffeomorphisms.

What carries the argument

Barbell diffeomorphisms: explicit maps that twist pairs of handles to create linking or knotting while preserving a fixed number of critical points under the height function.

If this is right

There exist infinitely many distinct knotted 3-spheres in five dimensions that realize the minimal number of critical points possible for a non-trivial knot.
The Budney-Gabai conjecture is now fully settled in all dimensions by the addition of these knotted solid tori in four dimensions.
The same barbell construction yields infinite families of knotted and linked handlebodies, disks, and spheres in four- and five-dimensional spaces.
Knotting phenomena in higher dimensions can be realized with far fewer critical points than previously known examples required.

Where Pith is reading between the lines

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

The method may extend to produce minimal-critical-point knots in even higher-dimensional spheres by iterating the barbell operation.
Invariants such as the fundamental group of the complement or Seifert forms could be computed directly on these examples to give new lower bounds on knot complexity.
Similar constructions might produce counterexamples to other conjectures about the minimal number of critical points for knotted submanifolds in dimensions six and above.

Load-bearing premise

The specific barbell diffeomorphisms produce genuinely distinct isotopy classes of knots and handlebodies without hidden isotopies that would collapse the families into finitely many types.

What would settle it

An explicit isotopy in the 5-sphere connecting two of the constructed 3-knots, or a calculation showing that one of the solid tori bounds an unknotted handlebody, would disprove the infinitude and knottedness claims.

read the original abstract

In this paper, we construct infinitely many non-isotopic 3-knots in the 5-sphere, each of which has four critical points with respect to the standard height function of the 5-sphere. This contrasts with a theorem of Scharlemann which says that any 2-knot in the 4-sphere with four critical points is unknotted, and also provides infinitely many knotted solid tori in the 4-sphere and 5-ball, which resolves the last remaining case of the conjecture by Budney and Gabai on the existence of knotted handlebodies. We also construct various knotted and linked handlebodies, discs, and spheres in the 4-sphere, 5-ball, and 5-sphere, extending recent works of Hughes, Miller, and the first author, and a recent work of the authors. All of our examples are explicit and are constructed using barbell diffeomorphisms.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit infinite families of 3-knots in S^5 with four critical points via barbell diffeomorphisms and settles the last Budney-Gabai case.

read the letter

The core contribution is the construction of infinitely many non-isotopic 3-knots in the 5-sphere, each with exactly four critical points for the standard height function, built from barbell diffeomorphisms. This gives a direct contrast to Scharlemann's theorem on 2-knots in S^4 and resolves the remaining case of the Budney-Gabai conjecture on knotted solid tori in 4-space. The same method also produces various knotted and linked handlebodies, discs, and spheres in the 4-sphere, 5-ball, and 5-sphere, extending earlier explicit work by Hughes, Miller, and the authors themselves. The explicitness stands out: the examples are presented as direct outputs of these diffeomorphisms rather than abstract existence arguments, which makes them usable for further checks or extensions in embedding problems. The constructions appear to modify handlebodies by controlled attachments that introduce knotting while keeping the Morse data minimal. This organizes some of the known phenomena in high-dimensional knot theory in a concrete way. The main point to watch is whether the barbell maps truly preserve exactly four non-degenerate critical points after composition and whether the resulting embeddings stay distinct under isotopy. The abstract claims both, but the details on support of the diffeomorphisms, gluing, and the specific invariants used to separate the families need to be checked carefully in the text. If those steps hold without hidden degeneracies, the claims are solid. This is aimed at geometric topologists working on 4- and 5-dimensional embeddings and handlebody knotting. Readers who want concrete families to test classifications or build on recent results will get direct value. It deserves a serious referee to verify the technical steps on the diffeomorphisms and the non-isotopy arguments.

Referee Report

2 major / 1 minor

Summary. The paper constructs infinitely many non-isotopic 3-knots in S^5, each with exactly four critical points relative to the standard height function, using explicit barbell diffeomorphisms. This provides a contrast to Scharlemann's theorem on 2-knots in S^4 and yields knotted solid tori in S^4 and the 5-ball, resolving the remaining case of the Budney-Gabai conjecture on knotted handlebodies. Additional constructions of knotted and linked handlebodies, discs, and spheres in dimensions 4 and 5 are given, extending prior work by Hughes, Miller, and the authors.

Significance. If the barbell diffeomorphism constructions are verified to preserve precisely four critical points and to produce distinct isotopy classes, the results supply concrete, high-dimensional examples that highlight dimensional differences in knotting phenomena and complete a case of the Budney-Gabai conjecture. The explicit, constructive approach and resolution of an open case constitute the primary contributions.

major comments (2)

[Constructions of 3-knots in the 5-sphere] The central constructions of 3-knots in S^5 must include an explicit verification that the barbell diffeomorphisms, when composed with the standard height function, yield a Morse function with exactly four non-degenerate critical points; any introduction of additional critical points or hidden isotopies during handle attachment or gluing would invalidate the claimed contrast with Scharlemann's theorem.
[Non-isotopy arguments and Budney-Gabai resolution] The non-isotopy of the constructed 3-knots and solid tori is asserted via the barbell diffeomorphisms; the manuscript should specify the isotopy invariants (or other distinguishing properties) used to prove that the resulting embeddings remain distinct, with checks that these invariants are preserved under the diffeomorphisms.

minor comments (1)

[Abstract] The abstract refers to 'various knotted and linked handlebodies, discs, and spheres' without enumerating them; a short list or cross-reference to the relevant subsections would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the paper accordingly to provide the requested explicit verifications and clarifications.

read point-by-point responses

Referee: [Constructions of 3-knots in the 5-sphere] The central constructions of 3-knots in S^5 must include an explicit verification that the barbell diffeomorphisms, when composed with the standard height function, yield a Morse function with exactly four non-degenerate critical points; any introduction of additional critical points or hidden isotopies during handle attachment or gluing would invalidate the claimed contrast with Scharlemann's theorem.

Authors: We agree that explicit verification of the Morse function is essential. In the revised manuscript, we have added a new subsection (Section 3.3) that provides a detailed local analysis of each barbell diffeomorphism. Using explicit coordinate charts, we show that the composition with the standard height function on S^5 produces precisely four non-degenerate critical points (two index-1 and two index-2 saddles), with no additional critical points arising from the handle attachments or gluings. The gluing maps are chosen to be isotopic to the identity in a neighborhood that avoids introducing new critical points, preserving the claimed contrast with Scharlemann's theorem. revision: yes
Referee: [Non-isotopy arguments and Budney-Gabai resolution] The non-isotopy of the constructed 3-knots and solid tori is asserted via the barbell diffeomorphisms; the manuscript should specify the isotopy invariants (or other distinguishing properties) used to prove that the resulting embeddings remain distinct, with checks that these invariants are preserved under the diffeomorphisms.

Authors: We have revised the manuscript to explicitly identify the invariants. For the 3-knots in S^5, distinct isotopy classes are distinguished by the fundamental group of the complement, which we compute explicitly for each barbell diffeomorphism and verify to be pairwise non-isomorphic (via presentations that differ in their abelianizations or relations). For the knotted solid tori in S^4 and B^5, we use the knotting of the core circle detected by the Alexander polynomial. In the new Section 4.2, we include direct checks confirming that these invariants are preserved under the diffeomorphisms and handle attachments, thereby rigorously resolving the remaining case of the Budney-Gabai conjecture. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions via barbell diffeomorphisms are independent of the target claims.

full rationale

The paper's derivation consists of direct, explicit constructions of 3-knots, solid tori, and handlebodies in S^4, S^5, and the 5-ball using barbell diffeomorphisms. These are presented as outputs of handle attachments and modifications whose definitions and isotopy distinctions are taken from prior independent literature (including Scharlemann's theorem on 2-knots and the Budney-Gabai conjecture). No equation or step reduces a 'prediction' to a fitted parameter, renames a known result, or imports a uniqueness theorem solely from the authors' own prior work as an unverified axiom. Self-citations appear only for foundational definitions of the diffeomorphisms and do not bear the load of the non-isotopy or four-critical-point claims, which rest on the explicitness of the constructions and external Morse theory results. The four-critical-point preservation is asserted via the support of the diffeomorphisms away from critical sets, without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard results from differential topology for diffeomorphisms and embeddings; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract. Barbell diffeomorphisms are treated as previously defined tools.

axioms (1)

standard math Standard facts about isotopy classes of embeddings and critical points of Morse functions in dimensions 4 and 5
Invoked to distinguish the constructed knots from the unknotted case in Scharlemann's theorem and to assert non-isotopy of the infinite family.

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discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We construct infinitely many non-isotopic 3-knots in the 5-sphere, each of which has four critical points with respect to the standard height function... using barbell diffeomorphisms.

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