# Verification report — KP-1.19 (K3 Problem 1.19: are twists about connected-sum spheres detected by knots?)

Verification date: 2026-10-03. Paper: "Asymmetric Hyperbolic Knots Detect Mapping Classes of 3-Manifolds: A Written
Proof for a Problem of the K3 List" (10 pages).

**Verdict.** The answer to the problem is yes, for closed, connected, oriented 3-manifolds. More is true: if K is a
knot in a closed, connected, orientable 3-manifold Y whose complement is hyperbolic with trivial isometry group,
then every diffeomorphism f of Y with f(K) isotopic to K is isotopic to the identity (Theorem A). Such knots exist
in every Y by a theorem of Kawauchi (1992). So every diffeomorphism that is not isotopic to the identity changes the
isotopy class of a knot. This is the conjecture of Aceto, Bregman, Davis, Park and Ray for closed manifolds, in the
smooth category. In addition, a twist about a connected-sum sphere that is not isotopic to the identity changes the
isotopy class of every hyperbolic knot (Theorem B). The proofs combine known theorems. Etnyre and Margalit have a
proof of the general statement (recorded since 2020); it has not appeared, and no written proof was found; the note
may coincide with their argument and claims no priority. The note is unrefereed. Two independent verification runs,
both AI-assisted, examined the argument; the second examined the final text line by line and found no mathematical
error.

## Statement checked
- **Primary source.** R. İ. Baykur, R. C. Kirby, D. Ruberman (eds.), *K3: A New Problem List in Low-Dimensional
  Topology*, Math. Surveys Monogr. 295, AMS 2026, doi:10.1090/surv/295, Problem 1.19 (p. 28 of the authors'
  preliminary version, which is the version that was read; the page was also read as a rendered image). In
  paraphrase:
  - Y = Y₁ # Y₂ is a connected sum of 3-manifolds, neither summand is S³, and Φ is a Dehn twist about the
    connected-sum sphere. Y₁ and Y₂ are assumed to be such that Φ is not isotopic to the identity. Is there a knot
    K ⊂ Y such that K and Φ(K) are not ambiently isotopic?
  - The remarks say that Aceto–Bregman–Davis–Park–Ray treated the analogous question for prime 3-manifolds (the
    Gluck twist of S¹ × S², and irreducible manifolds via free homotopy classes of loops), and that the remaining
    question is whether twists about connected-sum spheres are detected by knots.
- **History.** P. Aceto, C. Bregman, C. W. Davis, J. Park, A. Ray, *Isotopy and equivalence of knots in
  3-manifolds*, J. Lond. Math. Soc. (2) 113 (2026) e70600, arXiv:2007.05796.
  - Version 1 (July 2020): Theorem 1.1 for prime manifolds; Conjecture 1.8 states the same for all oriented
    3-manifolds; a note says that Etnyre and Margalit have a proof of the conjecture by different methods.
  - Versions 2 and 3 and the published version (2026): Theorem 1.1 is stated for each orientation-preserving
    diffeomorphism of a prime manifold; Corollary 4.4 treats connected sums of copies of S¹ × S²; Remark 1.6
    (Remark 4.5 in version 2) names twists along separating spheres as the main remaining case; a note says that
    Etnyre and Margalit communicated a different proof which extends to the non-prime case and has not yet
    appeared.
- **Which Problem 1.19.** The record is Problem 1.19 of the K3 list of 2026. Problem 1.19 of Kirby's list of 1997
  is a different problem (knots with the same 0-surgery).
- **Corpus record.** ulamai/UnsolvedMath, KP-1.19 (record 2678, version 1.6.0, upstream status `open`). The
  statement is a copy of the K3 text. The record is titled "Kirby Problem 1.19" in the set "Kirby's Problems in
  Low-Dimensional Topology".
- **Setting.** K3 says "3-manifolds" without further specification. Its remarks refer to the paper above, whose
  setting is smooth, closed, connected, oriented. The note treats this setting.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Problem 1.19 as posed, Y closed, connected, oriented: some knot K with Φ(K) not isotopic to K | yes | Corollary 1.2(c), from Theorem A |
| the same for knots as unoriented subsets (stronger than for oriented, parametrised knots) | yes | Theorem A; Section 2, conventions |
| every hyperbolic knot is moved by a non-trivial Φ | yes | Theorem B |
| every orientation-preserving diffeomorphism that acts trivially on π₁(Y) and is not isotopic to the identity moves every hyperbolic knot, if Y is not prime and not a connected sum of lens spaces and copies of S¹ × S² | yes | Theorem 4.4 |
| the same for connected sums of lens spaces and copies of S¹ × S² | not claimed; the method gives nothing there | Remark 4.5(2) |
| arbitrary diffeomorphism f not isotopic to the identity (also orientation-reversing), any closed orientable Y (also prime): some knot is moved | yes | Theorem A, Corollary 1.2(a) |
| equivalent knots are isotopic if and only if π₀Diff⁺(Y) is trivial (Conjecture 1.8 of version 1 of the paper above, for closed manifolds, in the smooth category) | yes | Corollary 1.2(b) |
| knots that are not hyperbolic | can be fixed by Φ: knots disjoint from the sphere or meeting it in two points | Remark 4.5(3) |
| 3-manifolds with boundary, non-orientable 3-manifolds | not treated | Scope paragraph |
| topological category (the setting of version 1 of the paper above) | not treated; the note is in the smooth category, as the published version of that paper | Scope paragraph |

## Results in the paper
- **Theorem 2.4 (Kawauchi, quoted).** For every knot L in a closed, connected, oriented 3-manifold Y there is an
  infinite family of knots L* ⊂ Y whose complements are hyperbolic of volume less than a constant C⁺, with trivial
  isometry group, and C⁺ is the supremum of the volumes. Read in the original (Osaka J. Math. 29 (1992), Main
  Theorem, parts (1) and (2), and the definitions on p. 299 and in Section 2); the proof was not read.
- **Section 2, standard facts.** For a hyperbolic knot K, with M = Y ∖ K, a cusp neighbourhood C and N = C ∪ K:
  N is a tubular neighbourhood of K; the exterior X is irreducible with incompressible boundary; the peripheral
  subgroup P has property (∗): if 1 ≠ a ∈ P and δaδ⁻¹ ∈ P then δ ∈ P.
- **Lemma 3.1.** A diffeomorphism of X that acts trivially on π₁(X) acts trivially on H₁(∂X) and preserves the
  orientation.
- **Lemma 3.2.** An isometry σ of M that preserves the meridian class up to sign extends from X to a
  diffeomorphism ĝ of Y of the same order; ĝ is the identity if σ is.
- **Proposition 3.3.** If f(K) is isotopic to K, then f is isotopic to such a ĝ. Steps: normal form (isotopy
  extension, uniqueness of tubular neighbourhoods); Mostow–Prasad rigidity; Waldhausen's theorem in the form of
  Hatcher–Wahl, Proposition 2.1 (orientation-preserving, pointed, smooth), after a change of base point;
  extension of the isotopy to Y through homeomorphisms; the solid torus (Cerf and Hatcher: Homeo(S¹ × D² rel ∂) is
  connected); passage from topological to smooth isotopy on the closed manifold Y (Cerf and Hatcher).
- **Theorem A and Corollary 1.2.** Consequences of Proposition 3.3 and Theorem 2.4.
- **Lemma 4.1.** A twist about a sphere induces the identity on π₁.
- **Lemma 4.2.** If every prime summand of Y₁ is a lens space or S¹ × S², the twist about the connected-sum sphere
  of Y₁ # Y₂ is isotopic to the identity (circle actions on lens spaces and S¹ × S²; rotation of a punctured ball).
- **Theorem 4.3 (Chen–Tshishiku, quoted).** On a closed oriented 3-manifold that is not prime and not a connected
  sum of lens spaces and copies of S¹ × S², an orientation-preserving diffeomorphism of finite order that acts
  trivially on π₁ is the identity. Quoted from the published version (Israel J. Math. 272 (2026), Lemma 6.1; see
  also Theorem 1.4).
- **Theorem 4.4 and Theorem B.** Consequences of Proposition 3.3, Theorem 4.3 and Lemmas 4.1 and 4.2.

## Checks (sanity checks of formulas; scripts and outputs in reproducibility/)
- **Author** (`lead/lead_checks.py`, standard library, under one second).
  - Lemma 3.2: on 14 pairs (lattice, linear part) the map τ is well defined on the solid torus V, is a product in
    the radial collar, and has the order of σ|T; 6,390 random tests.
  - Lemma 4.2(a): local form of the circle actions on L(p, q) for five pairs (p, q) and on S¹ × S²; 300 random
    tests.
  - Property (∗) in the figure-eight knot group, in exact arithmetic over Z[ω], for all 4,372 reduced words of
    length at most 7 in the two standard generators.
- **First independent verification run** (`independent_run/check_extension.py`): Lemma 3.2 on 10 pairs (lattice,
  linear part), 4,000 random tests; the rotation loop generates π₁(SO(3)).
- **Second independent verification run** (`independent_run_2/run2_checks.py`, standard library, about three
  seconds; written from the text of the note without reading the two programs above).
  - Lemma 3.2 in lattice coordinates, exact rational arithmetic: all symmetries A with Am = ±m of six lattices
    (square, rectangular, hexagonal, rhombic, centred rectangular, oblique) and primitive m with coordinates of
    absolute value at most 3; 1,792 configurations, 896 with each sign; 156,632 tests (equivariance, τ well
    defined, boundary and collar form, order of τ, freeness of the circle action, meridian curves).
  - Lemma 4.2(a): the circle action on L(p, q) for seven pairs (p, q), exact; the differential at the fixed point,
    by finite differences; the rotation loop lifts to a path from 1 to −1 in the unit quaternions; the isotopy on
    the shell.
  - Property (∗) in the figure-eight knot group, exact in Z[ω]: 118,097 reduced words of length at most 10; the
    31 elements among them that fix ∞ are parabolic translations and commute; (∗) holds for the meridian and for
    a second peripheral element.
  - The collar formula of Step 4 of the proof of Proposition 3.3, exact: 1,680 tests.
- The second run also extracted the source archive and ran the first two programs again; they reproduce their
  stored outputs.
- These computations test formulas. They are not part of any proof.

## Sources read
See `reproducibility/literature/sources_read.md` for the public copy, the size and the SHA-256 of each source and
for what was read in it.
- Read in the original: Kawauchi (1992), statement and definitions; Kawauchi (1993), Theorem 1.1; Chen–Tshishiku
  (published version); Hatcher (1983), theorem and appendix; Cerf (1959), Théorème 5 and Corollaire 1;
  Hatcher–Wahl (arXiv version 4), Section 2 with Proposition 2.1; Myers (1982), abstract, introduction and
  Theorem 6.1; Thurston's notes, Chapter 5; Hatcher's notes on 3-manifolds, Chapter 1;
  Aceto–Bregman–Davis–Park–Ray (arXiv versions 1, 2, 3 and the published version in an open repository copy); K3
  (authors' preliminary version).
- Not accessible or not consulted: Waldhausen (1968); Mostow (1973); Prasad (1973); Thurston (1982); Milnor
  (1962); Hirsch (1976). Waldhausen's theorem is used in the form of Hatcher–Wahl, Proposition 2.1. The
  formulation in Remark 3.4 (homotopy through maps of pairs) is the one commonly quoted and was not compared with
  the original.

## Independent verification runs
Both runs were AI-assisted. They are not peer review.

### First run (2026-10-03, on a working write-up)
It examined a working write-up of what is now Proposition 3.3 and Section 4, and of a variant proof of the
Chen–Tshishiku theorem that is no longer part of the note. Its verdicts:

| Item | Verdict |
|---|---|
| Proof correct and complete | yes; every step re-derived, no gap found |
| Answers the question as posed and as intended | yes (closed oriented Y, one knot, ambient isotopy) |
| Answered in the source itself | no |
| Published or posted answer | none found; Etnyre and Margalit have a proof of a more general statement, which has not appeared |
| Substance | slight: a consequence of published theorems and a classical argument; suitable for a short note |
| Presentation | fixes required |

The run also read Kawauchi's Main Theorem in the original and derived Theorem A from it; the working write-up had
only mentioned this route. Its required fixes (priority statement; Theorem A first; classical theorems checked
against the sources; citation of the published version of Chen–Tshishiku; small corrections; the record is
Problem 1.19 of the K3 list) were applied when the text was written. The run also asked that Etnyre and Margalit
and the scribe of the problem be contacted before posting. This has **not** been done; the note says that there
has been no contact.

### Second run (2026-10-03, on the final text, before the last revision)
The text written after the first run contained parts that no run had examined: Theorem A and its corollary, the
quotation of Kawauchi's theorem, Lemma 3.1, the argument that C ∪ K is a tubular neighbourhood, and a new
organisation of Steps 3 to 5 of the proof of Proposition 3.3. The second run examined the whole text line by line
and compared every quoted statement with the source.

| Area | Verdict |
|---|---|
| Statement fidelity (K3 Problem 1.19; all versions of the paper of Aceto et al.) | correct; the note answers the question as posed for closed, connected, orientable Y |
| Section 2, Lemma 3.1, Lemma 3.2, Proposition 3.3, Theorem A | correct |
| Quoted results (Hatcher–Wahl 2.1; Hatcher 1983, appendix (3) and (9); Cerf 1959; Kawauchi; Mostow–Prasad; Myers) | stated correctly; hypotheses match the use |
| Section 4, Theorem B | correct; one remark claimed too much |
| Computations | all pass (own program; rerun of the package) |
| Novelty and credit | no written proof found; two corrections of wording and credit |
| Presentation | small corrections |

Points examined in particular:
- *Hatcher–Wahl, Proposition 2.1.* The statement is for orientation-preserving diffeomorphisms that fix an
  interior base point and induce the identity of π₁ at that point; nothing is required on the boundary; the
  category is smooth. Step 3 arranges exactly these hypotheses (Lemma 3.1 gives the orientation; a point-push
  gives the base point).
- *Why "homotopic implies isotopic" holds here.* For Haken manifolds with boundary this needs a condition: the
  reflection of a product F × [0,1] is homotopic but not isotopic to the identity. It reverses the orientation.
  For the map u of Step 3, the condition of Waldhausen's theorem (a homotopy through maps of pairs) was verified
  directly, as in Remark 3.4; it can fail only for I-bundles over closed surfaces. So Step 3 is justified both by
  the published statement and by the classical form with its peripheral hypothesis.
- *Steps 4 and 5.* The collar formula defines a path of homeomorphisms of Y; the solid torus is handled by
  statements (3) (topological form) and (9) of Hatcher's appendix; the return to diffeomorphisms uses the theorem
  of Cerf and Hatcher on the closed manifold Y only.
- *Kawauchi's theorem.* The hypothesis "good (3,1)-manifold pair" holds for a knot in every closed, connected,
  oriented 3-manifold, S³ and reducible manifolds included; the exterior is hyperbolic of finite volume; the
  isometry group is the full one (it is identified with Out π₁ on p. 299).
- *Chen–Tshishiku.* Published version: Lemma 6.1, Theorem 1.4 (with the hypothesis on RP³ added in the published
  version), Theorem 2.4, Lemma 2.5, Remark 2.2, Theorem 5.1.

Required fixes of the second run and what was done:
1. Priority wording: the sources say that Etnyre and Margalit have a proof, communicated to the authors, not yet
   appeared; "announced" was replaced; the introduction now says that there has been no contact; "written and
   checked proof" became "written proof". Done.
2. Credit: the published version of the paper of Aceto et al. also treats connected sums of copies of S¹ × S²
   (Corollary 4.4). Added to the introduction.
3. Remark 4.5(2) claimed that every hyperbolic knot detects every non-trivial twist class in every Y. Restricted
   to the manifolds of Theorem 4.4, with the reason (Chen–Tshishiku, Theorem 5.1). Done.
4. The sentence of the introduction that announces Theorem 4.4 now states its hypothesis on Y. Done.
5. The appendix with a variant proof of the Chen–Tshishiku theorem was removed, with four references used only
   there. The run found no error in it, but the equivariant sphere theorem (Meeks–Yau, Dunwoody) and the results
   of Smith theory (Bredon) that it used could not be read in the originals. Theorem B rests on the published
   Lemma 6.1.
6. Theorem 2.2 and Remark 3.4: the role of the orientation hypothesis and of the peripheral condition is now
   stated; Step 3 says that Steps 4 and 5 use only a path of homeomorphisms. Done.
7. The existence of hyperbolic knots (Myers, Thurston) is now stated for closed manifolds only. Done.
8. Reference added for the fact that a compressible torus in an irreducible manifold bounds a solid torus or lies
   in a ball. Done.
9. Conventions of Chen–Tshishiku ("reducible"; the hypothesis on RP³) stated precisely. Done.
10. Verification paragraph of the note rewritten to describe both runs. Done.
11. Scope paragraph updated (sources seen and not seen; searches). Done.
12. First sentence of the introduction: smooth "unless we say otherwise". Done.
13. Bibliography: issue number of Chen–Tshishiku added; unused entries removed. Done.
14. Package rebuilt. Done.

Not applied: contact with Etnyre and Margalit and with the scribe of the problem. No contact is made at this
stage; the note says so.

The revision after the second run changed wording, credit, two remarks and the appendix only. No statement or
proof of Sections 2 to 4 was changed, apart from the added explanations listed in items 6 to 9 and one added
sentence on Kawauchi's imitations (the imitation of a knot is a knot). The revised text was read again in full
and every page of the PDF was inspected.

## Relation to the literature, novelty and scope
- **Searches (second run, 2026-10-03).** arXiv API: joint papers of Etnyre and Margalit (none); all arXiv papers
  of Etnyre (85) and of Margalit in the relevant categories (44); 35 keyword searches (twists about spheres;
  equivalent and isotopic knots; knots and mapping class groups of 3-manifolds; Gluck twists and knots; hyperbolic
  knots without symmetries; the K3 list; Problem 1.19; recent papers of the authors of the paper above and of the
  scribe). Papers citing the paper of Aceto et al. or that of Chen–Tshishiku (OpenAlex, OpenCitations, Semantic
  Scholar): three, unrelated. zbMATH, Crossref, the preprint page of Etnyre, one web search. Earlier stages also
  searched MathOverflow and the conference page behind the note of Aceto et al. No paper answering the problem and
  no statement of Theorem A was found. The requests are listed in `reproducibility/independent_run_2/queries_run2.log`
  and in `reproducibility/literature/`.
- **Not searched.** MathSciNet and Google Scholar. Citation data for papers of 2025–26 are incomplete.
- **Novelty.** The ingredients are known: Kawauchi's theorem (1992), the description of the symmetries of a
  hyperbolic knot by Mostow–Prasad and Waldhausen, Hatcher's theorem, and the theorem of Chen–Tshishiku. The
  note contributes the written proof that they settle the conjecture for closed manifolds and Problem 1.19, and
  Theorem B. Experts may regard Theorem A as an easy consequence of known results. Etnyre and Margalit have a
  proof since 2020 that has not appeared; whether it uses the same argument is not known to us. No priority is
  claimed. This negative search is not a proof of priority.
- **Scope.** Closed, connected, orientable 3-manifolds, smooth category. Not treated: manifolds with boundary,
  non-orientable manifolds, the topological category.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/