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More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres

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Pith's one-line read The paper proves that replacing the pretzel knot P(−2,3,7) by any Montesinos knot K(2,3,|6s+1|) in a roll-spun construction still yields branched double covers that are homotopy 4-spheres, producing a strictly larger family of exotically…

desk verdict Genuine extension of Miyazawa's construction to all K(2,3,|6s+1|), with clean group theory; the only real soft spot is the load-bearing, under-proved geometric identification in Lemma 4.1. read the letter →

arxiv 2507.03798 v1 pith:4OJ6SD23 submitted 2025-07-04 math.GT

Judson Kuhrman This is my paper · ORCID
classification math.GT MSC 57K1057K40
keywords exoticRP2-knotshomotopy4-spherestwist-rollspunknotsMontesinosbrancheddoublecoversBrieskornspherestrianglegroupsrealSeiberg-Witteninvariant
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that a known infinite family of exotically knotted projective planes in the 4-sphere can be enlarged by swapping the single pretzel knot used as input for any Montesinos knot of the form K(2,3,|6s+1|). The central result is that the branched double cover of the roll-spun version of each such knot is a homotopy 4-sphere. If true, this yields topologically unknotted $RP^{2}$-knots whose real Seiberg-Witten invariant realizes every odd positive integer, and homotopy 4-spheres and homotopy $CP^{2}$s carrying involutions that are topologically standard but not smoothly standard. The paper also shows that adding two twists to a twist-roll-spun knot changes the branched double cover by a Gluck twist, so the earlier homotopy sphere is a Gluck twist on $S^{4}$.

What carries the argument

The machinery is the correspondence between twist-roll-spun 2-knots and their branched double covers. For an even number of twists, the fundamental group of the branched double cover is the quotient of π_1(Σ_2($S^{3}$,K)) by the normal subgroup generated by the homotopy class of the fixed knot (Corollary 3.4), so the cover is a homotopy sphere exactly when that class is a weight element, meaning an element that normally generates the whole group. For K=K(2,3,|6s+1|), the cover is the Brieskorn sphere Σ(2,3,|6s+1|), whose fundamental group is a central extension of the hyperbolic triangle group Δ(2,3,|6s+1|), the orientation-preserving symmetry group of a hyperbolic tiling by triangles. The paper identifies the fixed knot's image in Δ with the explicit word $c^{{(r+1)/2}}$ $b^{{(q+1)/2}}$ a $b^{{-(q+1)/2}}$ $c^{{-(r+1)/2}}$ a from equation (3), and shows this word normally generates Δ when q=3 and r=|6s+1|.

What would settle it

Take the case s=1 and compute π_1(Σ_2($S^{4}$, ρK(2,3,7))) from a surgery description, or check in Δ(2,3,7) whether the word $c^{4}$ $b^{2}$ a $b^{{-2}}$ $c^{{-4}}$ a normally generates the group; if either fails, the identification in Lemma 4.1 or the normal-generation result in Proposition 4.2 is wrong.

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Extended reading notes

Core claim

The paper claims that the construction of exotically unknotted $RP^{2}$-knots does not require the pretzel knot P(−2,3,7): for every integer s, the 2-knot obtained by roll-spinning the Montesinos knot K(2,3,|6s+1|) satisfies Σ_2($S^{4}$, ρK(2,3,|6s+1|)) ≃ $S^{4}$, i.e. its branched double cover is a homotopy 4-sphere. This is enough to force the connect-sum of the roll-spun knot with an unknotted $RP^{2}$ to be topologically unknotted. The same roll-spun knots yield homotopy $CP^{2}$s after connect-sum with ±$CP^{2}$, and the real Seiberg-Witten invariant of these surfaces takes all odd positive integer values. The mechanism is group-theoretic: an element represented by the fixed knot in the Brieskorn sphere Σ(2,3,|6s+1|) normally generates its fundamental group, and by the central-extension relation this makes the branched double cover simply connected.

Load-bearing premise

The proof stands on Lemma 4.1's identification of the fixed knot with an explicit word in a triangle group, an identification supported only by a short geometric description; if the word is wrong, the normal-generation step does not apply to the fixed knot.

Editorial extensions

If this is right

  • Every odd positive integer appears as the real Seiberg-Witten invariant of a topologically unknotted RP^2-knot in S^4.
  • There are homotopy 4-spheres whose branching involutions do not preserve any positive scalar curvature metric and are not smoothly conjugate to standard involutions.
  • If any of the new homotopy spheres is shown to be diffeomorphic to S^4, the construction produces an infinite family of inequivalent smooth involutions on S^4.
  • Adding 2k twists to a twist-roll-spun knot changes its branched double cover by a k-fold Gluck twist, and the earlier pretzel-input homotopy sphere is a Gluck twist on S^4.
  • A specific torus T_1(T(2,3),g) in S^4 admits a nontrivial torus surgery to S^4 while being neither topologically unknotted nor a turned torus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the real invariant takes the same value for two different parameters s, connect-sums of different roll-spun knots with an unknotted RP^2 produce examples with equal invariants; distinguishing them smoothly would require a new invariant.
  • The same normal-generation pattern might hold for other triples (2,q,r) beyond (2,3,|6s+1|), which would characterize a larger class of Montesinos inputs without changing the proof.
  • A direct computational check of the triangle-group word for small s would test the theorem's main geometric identification before deeper 4-manifold invariants are needed.
  • The new homotopy CP^2s are not covered by the earlier argument for standardness, since their input knots generally do not admit lens-space surgeries; whether they are diffeomorphic to CP^2 is a concrete open question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 7 minor

Summary. The paper aims to enlarge Miyazawa's infinite family of exotically unknotted embeddings of RP^2 in S^4 by replacing the single pretzel knot P(-2,3,7) with the infinite family of Montesinos knots K(2,3,|6s+1|). The main theorem states that for every integer s, the branched double cover of the roll-spun knot rho K(2,3,|6s+1|) is a homotopy 4-sphere. From this the author derives that connect-sums of these roll-spun knots with an unknotted RP^2 produce topologically unknotted but smoothly exotic RP^2-knots whose real Seiberg-Witten invariant takes all odd positive values, and corresponding homotopy 4-spheres and homotopy CP^2s with interesting involutions. A secondary theorem interprets certain branched double covers as Gluck twists and shows Miyazawa's original homotopy sphere is a Gluck twist on S^4. The proof strategy is group-theoretic: identify the homotopy class of the fixed knot in the Brieskorn sphere Sigma(2,3,r) with an explicit element of the triangle group Delta(2,3,r), prove that this element normally generates, and then lift this to the fundamental group of the branched double cover.

Significance. If the main theorem and its supporting identification are correct, the paper makes a substantial contribution to the study of exotic knotted surfaces in dimension four. It replaces a single example with an infinite family, shows that the real Seiberg-Witten invariant achieves all odd positive values, and produces new homotopy spheres and homotopy CP^2s with involutions that are topologically but not smoothly standard. The group-theoretic core, especially the normal-generation computation in Proposition 4.2, is explicit, checkable, and appears to be correct. The paper also provides a clean reinterpretation of branched double covers of twist-roll-spun knots in terms of Gluck twists, and it clearly identifies open questions and limitations. The main weakness is that the bridge from the three-dimensional geometry of the fixed knot to the explicit algebraic word in the triangle group is justified by only a short geometric sketch; this is the load-bearing point for the main theorem.

major comments (1)
  1. [Lemma 4.1, Eq. (3)] Lemma 4.1 identifies, up to a central power, the homotopy class of the fixed knot in the branched double cover Sigma(2,q,r) with the explicit element c^{(r+1)/2} b^{(q+1)/2} a b^{-(q+1)/2} c^{-(r+1)/2} a of Delta(2,q,r). This identification is the sole bridge between the geometry of the fixed knot and the normal-generation computation of Proposition 4.2, which in turn is what makes Theorem 1 follow. The proof, however, is a two-paragraph geometric sketch: it asserts that the fixed knot projects to the hyperbolic isometry whose axis is the reflection line, that the translation length is 2(A+B+C), and that the word in (3) is read off from the rotation sequence in Figures 1 and 2. The correspondence between the fixed knot and that particular isometry, the placement of the vertices, and the order and orientation of the rotations are not verified by an independent computation or by a derivation from the Seifert-fibered description of Sigma(2,q,r). A wrong rotation sequence, a different conjugacy representative, an inverse word, or an incorrect power of the central element would invalidate Proposition 4.2 as applied and collapse Theorems 1-3. I am not claiming the lemma is false; the algebra of Proposition 4.2 checks out and the final word (3) is internally consistent. But the proof as written does not close the gap, and because this is the load-bearing step, the manuscript should supply a complete derivation or an independent verification (for example, from the presentation in Section 4.1 together with the known action of the branching involution, or a direct computer check using a presentation of Sigma(2,3,r)).
minor comments (7)
  1. [Section 1] There is a typo in the section heading 'I ntroduction', and the abstract has a formatting issue with the embedding arrow 'RP2 ,→ S4'.
  2. [Section 4.1] In the sentence defining the fundamental group of the Brieskorn sphere, 'isomoprhic' should be 'isomorphic'.
  3. [Theorem 4] The statement has a grammatical error: 'For any knot K ⊆ S3 and m, n, k ∈ Z, The branched double covers' should use 'the' instead of 'The' after the comma.
  4. [Lemma 4.1 proof] The composition of rotations in the proof is described verbally; the argument would be much easier to check if the order of composition in the right-action convention were stated explicitly and a labeled diagram or a short calculation of the rotation angles were included.
  5. [Corollary 4.8] The assertion that Sigma(2,3,|6s+1|) is Dehn surgery on a twist knot for all s is used without a reference or proof; a citation or a one-line justification would be appropriate.
  6. [Section 5] In the last paragraph of Section 5, 'we our method is insufficient' should be 'our method is insufficient'.
  7. [Proposition 4.9] The check that a certain group is non-abelian is delegated to Sage; to make the paper self-contained, the specific computation or a short argument should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 follows from independent geometric and group-theoretic lemmas, not from its own conclusion.

full rationale

The central claim (Theorem 1) is derived through a chain whose ingredients are independent of the target result: Corollary 3.4 reduces homotopy-sphere to normal generation by the fixed knot; Lemma 4.1 identifies the fixed-knot class geometrically using the branching involution and hyperbolic tiling; Proposition 4.2 is a self-contained computation in the triangle group showing that the identified element normally generates; and the proof of Theorem 1 lifts this to the central extension. None of these steps assumes Theorem 1, and none is defined in terms of it. The proof of Lemma 4.1 is admittedly sketched, but a sketched geometric argument is a rigor concern, not circularity: the word in equation (3) is derived from the tiling geometry, not fitted to make Proposition 4.2 succeed. External results, such as Miyazawa's real Seiberg-Witten invariant, the Kang-Park-Taniguchi computation, and theorem of Hughes-Kim-Miller, are used as inputs and are not replaced by the paper's own claims. There are no load-bearing self-citations, and the author explicitly leaves open in Section 5 the questions the method cannot answer, such as whether the homotopy spheres are diffeomorphic to S4 or are Gluck twists. The paper therefore does not exhibit definitional, fitted, or self-citation circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard triangle-group facts and on five structural theorems from the literature ([26], [20], [14], [6], [16], [32]), all of which are cited and none of which assume the target result. The only potentially fragile input is the geometric identification in Lemma 4.1, which is a lemma with a sketch proof rather than an axiom.

assumptions (7)
  • domain assumption Plotnick's presentation for pi1 of branched double covers of twist-roll-spun knots (Lemmas 3.3, 3.6).
    Cited to [26]. This is the bridge from 4-manifold fundamental groups to the normal-generation question for the fixed knot class.
  • domain assumption Litherland's result that rho T(p,q) is isotopic to tau_{-pq} T(p,q) for torus knots.
    Cited to [20]. Used in Proposition 3.5 to handle the exceptional s=0 and s=-1 cases of Theorem 1.
  • domain assumption Hughes-Kim-Miller theorem that Sigma2(S4, tau_m rho_n K) depends on m mod 4.
    Cited to [14]. Used in Corollary 2.8 and Proposition 3.5; also reproved here via Theorem 4.
  • domain assumption Conway-Orson-Powell theorem: S#P is topologically unknotted iff its knot group is Z/2Z.
    Cited to [6]. Converts the homotopy-sphere condition into topological unknottedness of the RP2-knot, producing Theorems 2 and 3.
  • domain assumption Kang-Park-Taniguchi computation of |deg(K(2,3,|6s+1|))| = 4j +/- 1.
    Cited to [16]. Supplies the full range of odd Seiberg-Witten values; not verified in this paper.
  • standard math Classical facts about triangle groups Delta(2,3,r) and Brieskorn sphere fundamental groups (trivial abelianization, generation by any two generators, central extension structure).
    Used in Section 4 without proof; standard in low-dimensional topology.
  • domain assumption Zeeman's theorem that the +/-1-twist-spin of any knot is an unknotted 2-knot.
    Cited to [32]. Used in Lemma 2.6 and Proposition 3.5 to identify some branched double covers as S4.

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Cite this review

Pith. "Pith review of More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres." pith.science (2026). https://pith.science/paper/4OJ6SD23

@misc{pith2026250703798,
  author       = {Pith},
  title        = {Pith review of: More Exotic $\mathbbRP^2$-knots and Homotopy Spheres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OJ6SD23}},
  note         = {Machine review of arXiv:2507.03798}
}
abstract

We extend the infinite family of exotic embeddings $\mathbb{RP}^2 \hookrightarrow S^4$ constructed by Miyazawa to a strictly larger family of exotic embeddings, by showing that in place of the pretzel knot $P(-2, 3, 7)$, an infinite family of knots may be used as input to the construction. To this end, we prove that for any Montesinos knot of the form $K(2,3,|6s+1|)$, the branched double cover of the corresponding roll-spun knot is a homotopy sphere. This in turn produces a larger family of homotopy spheres and homotopy $\mathbb{CP}^2$s with potentially interesting involutions. We also observe that Miyazawa's homotopy sphere can be obtained from $S^4$ by a Gluck twist.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

32 extracted references · 26 canonical work pages

  1. [16]

    Cables of the figure-eight knot via real Frøyshov invariants

    Sungkyung Kang, JungHwan Park, and Masaki Taniguchi. Cables of the figure-eight knot via real Frøyshov invariants. preprint (2024), arxiv:2405.09295

    arXiv 2024

  2. [1]

    Jessica E. Banks. The Birman exact sequence for 3-manifolds. Bull. Lond. Math. Soc., 49(4):604–629, 2017

    work page 2017

  3. [2]

    Simon's conjecture for 2-bridge knots

    Michel Boileau, Steve Boyer, Alan W. Reid, and Shicheng Wang. Simon’s conjecture for 2-bridge knots, 2009. preprint (2009), arXiv:0903.2898

    work page Pith review arXiv 2009

  4. [3]

    The turned torus knot in S4

    Jeffrey Boyle. The turned torus knot in S4. J. Knot Theory Ramifications, 2(3):239–249, 1993

    work page 1993

  5. [4]

    A family of embedding spaces

    Ryan Budney. A family of embedding spaces. In Groups, homotopy and configuration spaces , volume 13 of Geom. Topol. Monogr., pages 41–83. Geom. Topol. Publ., Coventry, 2008

    work page 2008

  6. [5]

    Knots, volume 5 of De Gruyter Studies in Mathematics

    Gerhard Burde and Heiner Zieschang. Knots, volume 5 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition, 2003

    work page 2003

  7. [6]

    Unknotting nonorientable surfaces

    Anthony Conway, Patrick Orson, and Mark Powell. Unknotting nonorientable surfaces. preprint (2024), arxiv:2306.12305

    arXiv 2024

  8. [7]

    Embedded surfaces with infinite cyclic knot group

    Anthony Conway and Mark Powell. Embedded surfaces with infinite cyclic knot group. Geom. Topol., 27(2):739– 821, 2023

    work page 2023

Show all 32 references
  1. [8]

    S. M. Finashin, M. Kreck, and O. Ya. Viro. Nondiffeomorphic but homeomorphic knottings of surfaces in the 4- sphere. In Topology and geometry—Rohlin Seminar, volume 1346 of Lecture Notes in Math., pages 157–198. Springer, Berlin, 1988

    1988

  2. [9]

    Exotic embeddings of #6 RP2 in the 4-sphere

    Sergey Finashin. Exotic embeddings of #6 RP2 in the 4-sphere. In Proceedings of G¨ okova Geometry-Topology Confer- ence 2008, pages 151–169. G ¨okova Geometry/Topology Conference (GGT), G ¨okova, 2009

    2008

  3. [10]

    Gompf and Andr ´as I

    Robert E. Gompf and Andr ´as I. Stipsicz. 4 -manifolds and Kirby calculus , volume 20 of Graduate Studies in Mathe- matics. American Mathematical Society, Providence, RI, 1999

    1999

  4. [11]

    Allen E. Hatcher. A proof of the Smale conjecture, Diff (S3) ≃ O(4). Ann. of Math. (2), 117(3):553–607, 1983

    1983

  5. [12]

    Andrew J. Havens. Equivariant Surgeries and Irreducible Embeddings of Surfaces in Dimension Four. PhD thesis, ”Uni- versity of Massachusetts”, 2021

    2021

  6. [13]

    Groups killed by centralizing one ele- ment

    Geoff Robinson (https://mathoverflow.net/users/14450/geoff robinson). Groups killed by centralizing one ele- ment. MathOverflow. URL:https://mathoverflow.net/q/476505 (version: 2024-08-07)

    2024

  7. [14]

    Branched covers of twist-roll spun knots

    Mark Hughes, Seungwon Kim, and Maggie Miller. Branched covers of twist-roll spun knots. preprint (2024), arxiv:2402.11706

    2024 arXiv

  8. [15]

    Cyclic and finite surgeries on Montesinos knots

    Kazuhiro Ichihara and In Dae Jong. Cyclic and finite surgeries on Montesinos knots. Algebr. Geom. Topol., 9(2):731– 742, 2009

    2009

  9. [17]

    Gluck twist and unknotting of satellite 2-knots, 2020

    Seungwon Kim. Gluck twist and unknotting of satellite 2-knots, 2020. preprint (2020), arXiv:2009.07353

    2020 arXiv

  10. [18]

    Surgery on tori in the 4-sphere

    Kyle Larson. Surgery on tori in the 4-sphere. Math. Proc. Cambridge Philos. Soc., 164(1):109–124, 2018

    2018

  11. [19]

    New constructions and invariants of closed exotic 4- manifolds

    Adam Simon Levine, Tye Lidman, and Lisa Piccirillo. New constructions and invariants of closed exotic 4- manifolds. preprint (2023), arXiv:2307.08130

    2023 arXiv

  12. [20]

    R. A. Litherland. Deforming twist-spun knots. Trans. Amer. Math. Soc., 250:311–331, 1979

    1979

  13. [21]

    On 2-knots and connected sums with projective planes

    Vincent Longo. On 2-knots and connected sums with projective planes. preprint (2019), arXiv:1901.10972, 2019

    2019 arXiv

  14. [22]

    A. Malcev. On isomorphic matrix representations of infinite groups. Rec. Math. [Mat. Sbornik] N.S., 8/50:405–422, 1940

    1940

  15. [23]

    Stipsicz, and Urz ´ua Giancarlo

    Gordana Mati ´c, Ferit ¨Ozt ¨urk, Javier Reyes, Andr ´as I. Stipsicz, and Urz ´ua Giancarlo. An exotic 5 RP2 in the 4-sphere. preprint (2023), arXiv:2312.03617

    2023 arXiv

  16. [24]

    A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots

    Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots. preprint (2023), arXiv:2312.02041

    2023 arXiv

  17. [25]

    Montesinos

    Jos ´e M. Montesinos. On twins in the four-sphere. I. Quart. J. Math. Oxford Ser. (2), 34(134):171–199, 1983

    1983

  18. [26]

    Plotnick

    Steven P . Plotnick. Fibered knots inS4—twisting, spinning, rolling, surgery, and branching. InFour-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., pages 437–459. Amer. Math. Soc., Providence, RI, 1984

    1982

  19. [27]

    Plotnick

    Steven P . Plotnick. Equivariant intersection forms, knots in S4, and rotations in 2-spheres. Trans. Amer. Math. Soc., 296(2):543–575, 1986

    1986

  20. [28]

    Floer homology of Brieskorn homology spheres

    Nikolai Saveliev. Floer homology of Brieskorn homology spheres. J. Differential Geom., 53(1):15–87, 1999

    1999

  21. [29]

    Twist-roll spun knots

    Masakazu Teragaito. Twist-roll spun knots. Proc. Amer. Math. Soc., 122(2):597–599, 1994

    1994

  22. [30]

    SageMath, the Sage Mathematics Software System (Version 10.3) , 2025

    The Sage Developers. SageMath, the Sage Mathematics Software System (Version 10.3) , 2025. https://www.sagemath.org

    2025

  23. [31]

    On irreducible 3-manifolds which are sufficiently large

    Friedhelm Waldhausen. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2), 87:56–88, 1968

    1968

  24. [32]

    E. C. Zeeman. Twisting spun knots. Trans. Amer. Math. Soc., 115:471–495, 1965

    1965

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