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The knot quandles of Suciu's ribbon $n$-knots and automorphisms on the free group of rank two

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Pith's one-line read The knot quandles of the ribbon n-knots R_k are mutually non-isomorphic, and this paper proves it by showing the defining monodromies f_k are mutually non-conjugate automorphisms of the rank-two free group.

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arxiv 2509.07395 v2 pith:GKBX67TH submitted 2025-09-09 math.GT math.GR

Takuya Sakasai , Kokoro Tanaka This is my paper · ORCID
classification math.GTmath.GR MSC 57K1220F3457K4520F12
keywords knotquandlefiberedribbonn-knotsfreegroupofranktwogeneralizedAlexanderautomorphismconjugacytype
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 ribbon n-knots R_k all share the same knot group, so ordinary group data cannot tell them apart. This paper gives another proof that their knot quandles can: the quandles Q(R_k) are mutually non-isomorphic, and each has infinite type. The proof translates each knot quandle into a generalized Alexander quandle built from the rank-two free group and the monodromy automorphism f_k, then shows these automorphisms lie in distinct conjugacy classes. The core computation detects f_k^6 through a single integer, 3k^2 - 3k + 1, in the center of the 2-step nilpotent quotient of the free group. Because these integers are distinct, the monodromies cannot be conjugate, and the classification of connected generalized Alexander quandles turns that into non-isomorphism of the knot quandles.

What carries the argument

The central object is the family of automorphisms f_k of the rank-two free group F = ⟨a,b⟩ defined by f_k(a)=a^k b a^{-k} and f_k(b)=a^{k-1} b a^{-k}. The load-bearing identity is Lemma 2.2's computation f_k^6 = I(x_k) with x_k = a^k b^k (a^{-1}b)^{k-1} a^{-k} b^{-k} (a b^{-1})^{k-1}, and the subsequent reduction x_k ≡ [a,b]^{3k^2-3k+1} modulo [[F,F],F], an Aut(F)-equivariant invariant taking values in the center of the 2-step nilpotent quotient. The integer 3k^2-3k+1 distinguishes the conjugacy classes of the f_k and, through generalized Alexander quandles, the isomorphism classes of the knot quandles.

What would settle it

Look for a counterexample to the classification lemma: two connected generalized Alexander quandles GAlex(G,ψ) and GAlex(G,φ) that are isomorphic even though ψ and φ are not conjugate in Aut(G). Finding such a pair, even in a small group, would break the step from Proposition 2.3 to Theorem 3.1. Alternatively, an explicit quandle isomorphism between Q(R_k) and Q(R_l) for k≠l would directly refute the paper's claim.

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Extended reading notes

Core claim

The disagreement among these knots is visible already in elementary rank-two free group automorphisms. For each positive integer k, the monodromy f_k of the fibered ribbon n-knot R_k is the automorphism of F = ⟨a,b⟩ given by f_k(a)=a^k b a^{-k}, f_k(b)=a^{k-1} b a^{-k}. The paper proves f_k and f_l are never conjugate in Aut(F) for k≠l. It does so by computing the sixth power: f_k^6 is the inner automorphism I(x_k), and the element x_k, though a complicated commutator word, represents [a,b]^{3k^2-3k+1} in the center of [F,F]/[[F,F],F]. Since 3k^2-3k+1 is strictly increasing in k, the Aut(F)-orbits of the x_k are distinct. Using Inoue's description of knot quandles of fibered knots and the cl

Load-bearing premise

The bridge from distinct monodromies to distinct knot quandles rests on the lemma that connected generalized Alexander quandles on the same group are isomorphic exactly when their defining automorphisms are conjugate; if that lemma does not hold for this family, Proposition 2.3 alone does not imply quandle non-isomorphism.

Editorial extensions

If this is right

  • The ribbon n-knots R_k are distinguished by their knot quandles even though their knot groups are isomorphic.
  • Each knot quandle Q(R_k) has infinite type, so type alone does not separate them; the non-isomorphism is a finer algebraic distinction.
  • There exists an infinite family of n-knots for every n>1 with mutually isomorphic knot groups, infinite-type knot quandles, and mutually non-isomorphic knot quandles.
  • The proof offers a concrete algebraic route for showing non-conjugacy of free-group automorphisms induced by fibered knots: take a suitable power, read the resulting inner automorphism in the 2-step nilpotent quotient, and compare integers.

Reading between the lines

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

  • The integer 3k^2-3k+1 is an invariant carried by the sixth power of the monodromy and detected in a 2-step nilpotent quotient; it could plausibly be recast as a quandle cocycle or a characteristic class of the fiber bundle, giving the family a numerical signature independent of the classification lemma.
  • The same recipe likely works for other fibered n-knots: pass the monodromy to the inner automorphism group by taking a power, project the resulting element to the center of [F,F]/[[F,F],F], and use the integer one obtains to separate quandles without computing core groups or double branched covers.
  • Because the proof treats the classification lemma as a black box, a direct construction of the integer invariant at the level of quandles could bypass that lemma and yield a proof for knots whose monodromies are not so easily described.
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Referee Report

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Summary. The paper gives a new proof that the knot quandles of Suciu's ribbon n-knots R_k (n>1) are mutually non-isomorphic, even though their knot groups are isomorphic. The proof is purely group-theoretic: the monodromy automorphisms f_k of the rank-two free group F are shown to be pairwise non-conjugate in Aut(F). The key step (Proposition 2.3) computes f_k^6 as the inner automorphism I(x_k), then projects the elements x_k to the center of the 2-step nilpotent quotient [F,F]/[[F,F],F] ≅ Z, where they represent distinct integers 3k^2-3k+1. An equivariance argument shows the Aut(F)-orbits are distinct. Theorem 3.1 then applies a classification of connected generalized Alexander quandles to conclude that the quandles GAlex(F, f_k) ≅ Q(R_k) are non-isomorphic. The paper also proves that the type of each Q(R_k) is infinite by showing the order of f_k is infinite.

Significance. The result itself is not new—non-isomorphism of these knot quandles was proved by Jablonowski and later by Yasuda—but the method is genuinely different and conceptually transparent. The proof reduces a geometric/outer-automorphism problem to an explicit, checkable computation in the lower central series of F_2, and the authors are careful to give the intermediate expressions for f^6 and the projection of x_k. The paper also contributes a short proof of infinite type. If the classification lemma from [2, Lemma B.1] and [3] is accepted, the argument is sound and self-contained. The exposition is clear and the computational core is verifiable by hand, which makes this a useful alternative perspective on a known family of examples.

minor comments (4)
  1. [Section 2, definition of f_k] The definition f_k(a)=a^k b a^{-k}, f_k(b)=a^{k-1} b a^{-k} is initially surprising because the second word is not obviously part of an automorphism. The authors clarify later that f_k = I(a^k) ∘ f for f(a)=b, f(b)=a^{-1}b. It would help the reader to state this factorization immediately after the definition, or to remark that it proves f_k ∈ Aut(F).
  2. [Section 3, Theorem 3.1] The proof of Theorem 3.1 relies on the 'if and only if' classification of connected generalized Alexander quandles, cited to [2, Lemma B.1] and [3]. Since [3] is a preprint, it would be useful to quote the exact statement being used (including the hypotheses, e.g., connectedness and the same underlying group G) and to note explicitly that F is finitely generated and Q(R_k) is connected. This is a presentation issue; the cited result appears to apply.
  3. [Lemma 2.2] The final simplification of a^k f(a)^k ... f^5(a)^k (b^{-1}aba^{-1}) to the displayed x_k is labeled 'straightforward computation.' Given that this product is the starting point for Proposition 2.3, one or two intermediate lines would make the verification easier for the reader.
  4. [Proposition 2.3] The notation 'GL(2,Z) →det→ {±1}' is slightly informal. Consider replacing it with 'the determinant homomorphism GL(2,Z) → {±1}' for clarity.

Circularity Check

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No significant circularity identified.

full rationale

The central derivation is not circular. Proposition 2.3 is a self-contained computation in the automorphism group of the free group of rank two: the automorphisms f_k are shown to be mutually non-conjugate by evaluating their sixth powers I(x_k) in the Aut(F)-equivariant quotient [F,F]/[[F,F],F], where the images are [a,b]^{3k^2-3k+1} with distinct positive exponents. This argument does not use the target quandle theorem. Theorem 3.1 then combines this group-theoretic fact with two external results: Inoue's theorem identifying the knot quandle of a fibered n-knot with a generalized Alexander quandle GAlex(F,f_k), and the classification lemma [2, Lemma B.1]/[3] saying connected generalized Alexander quandles over the same group are isomorphic iff the defining automorphisms are conjugate. Neither of these results is derived from the present claim, and neither is authored by the present authors. The only self-citation is [14] (Tanaka–Taniguchi), used in Theorem 3.2 to identify the type of a generalized Alexander quandle with the order of its defining automorphism; that is a separate, independent result supporting a secondary statement, not the load-bearing non-isomorphism theorem. There is no fitted parameter renamed as a prediction, no uniqueness assertion imported from the authors' own prior work, and no ansatz smuggled in via self-citation. The applicability of the external classification lemma is a normal mathematical dependency, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no new postulated entities. The central proof rests on standard free-group theory and on several established quandle theorems cited from the literature. The only self-cited ingredient is [14] for the type of a generalized Alexander quandle, used in the secondary Theorem 3.2.

assumptions (6)
  • domain assumption Inoue's theorem: for a fibered oriented n-knot (n>1) with fiber group G and monodromy phi, the knot quandle is isomorphic to GAlex(G, phi).
    Invoked in Section 3 to identify Q(R_k) with GAlex(F, f_k), transferring the problem to automorphisms of the free group.
  • domain assumption Classification lemma: connected generalized Alexander quandles GAlex(G, psi) and GAlex(G, phi) are isomorphic iff psi and phi are conjugate in Aut(G).
    Used in the proof of Theorem 3.1, citing [2, Lemma B.1] and [3]; this is the bridge that makes Proposition 2.3 imply quandle non-isomorphism.
  • domain assumption The knot quandle Q(R_k) is connected.
    Used in the proof of Theorem 3.1, citing [12, Lemma 2.27], to apply the classification lemma.
  • domain assumption The type of a generalized Alexander quandle equals the order of its defining automorphism.
    Used in the proof of Theorem 3.2, citing [14, Proposition 2.1] (a self-citation by the second author); converts the infinite order of f_k into infinite quandle type.
  • standard math Periodic automorphisms of F_2 have order 2, 3, or 4.
    Used in Proposition 2.1, citing [11], together with the GL(2,Z) image of order 6 to conclude f_k has infinite order.
  • standard math The 2-nilpotent quotient [F,F]/[[F,F],F] is Aut(F)-equivariantly isomorphic to wedge^2 Z^2, isomorphic to Z, with action factoring through the determinant.
    Used in Proposition 2.3, with reference [9], to separate the Aut(F)-orbits of the elements x_k.

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Cite this review

Pith. "Pith review of The knot quandles of Suciu's ribbon $n$-knots and automorphisms on the free group of rank two." pith.science (2026). https://pith.science/paper/GKBX67TH

@misc{pith2026250907395,
  author       = {Pith},
  title        = {Pith review of: The knot quandles of Suciu's ribbon $n$-knots and automorphisms on the free group of rank two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKBX67TH}},
  note         = {Machine review of arXiv:2509.07395}
}
abstract

Jab{\l}onowski proved that the knot quandles of Suciu's $n$-knots, which share isomorphic knot groups, are mutually non-isomorphic, and Yasuda later gave a different proof. In this paper, we present yet another proof of this result by analyzing the conjugacy classes of certain automorphisms of the free group of rank two.

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Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [2]

    Edwin Clark, Larry A

    W. Edwin Clark, Larry A. Dunning, and Masahico Saito,Computations of quandle 2-cocycle knot invariants without explicit 2-cocycles, J. Knot Theory Ramifications26(2017), no. 7, 1750035, 22. MR 3660090

    work page 2017

  2. [3]

    Akihiro Higashitani, Seiichi Kamada, Jin Kosaka, and Hirotake Kurihara,Classification of generalized Alexander quandles, Preprint, arXiv:2406.01074 [math.GT] (2024), 2024

    work page Pith review arXiv 2024

  3. [1]

    Emil Artin,Zur Isotopie zweidimensionaler Fl¨ achen imR 4, Abh. Math. Sem. Univ. Hamburg 4(1925), no. 1, 174–177

    work page 1925

  4. [4]

    Ayumu Inoue,On the knot quandle of a fibered knot, finiteness and equivalence of knot quandles, Topology Appl.265(2019), 106811, 8

    work page 2019

  5. [5]

    Micha l Jab lonowski,On biquandle-based invariant of immersed surface-links, Yoshikawa ori- ented fifth move, and ribbon 2-knots, Preprint, arXiv:2505.14724 [math.GT] (2025), 2025

    work page Pith review arXiv 2025

  6. [6]

    Pure Appl

    David Joyce,A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra23 (1982), no. 1, 37–65

    work page 1982

  7. [7]

    Knot Theory Ramifications29(2020), no

    Taizo Kanenobu and Toshio Sumi,Suciu’s ribbon 2-knots with isomorphic group, J. Knot Theory Ramifications29(2020), no. 7, 2050053, 9. MR 4143003 THE KNOT QUANDLES OF SUCIU’S RIBBONn-KNOTS 5

    work page 2020

  8. [8]

    Lyndon and Paul E

    Roger C. Lyndon and Paul E. Schupp,Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1977 edition. MR 1812024

    work page 2001

Show all 16 references
  1. [9]

    MR 207802

    Wilhelm Magnus, Abraham Karrass, and Donald Solitar,Combinatorial group theory: Pre- sentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 207802

    1966

  2. [10]

    Sb.161(1982), no

    Sergei Vladimirovich Matveev,Distributive groupoids in knot theory, Mat. Sb.161(1982), no. 1, 78–88

    1982

  3. [11]

    Univ., Canberra, 1973), Lecture Notes in Math., vol

    Stephen Meskin,Periodic automorphisms of the two-generator free group, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973), Lecture Notes in Math., vol. Vol. 372, Springer, Berlin-New York, 1974, pp. 494–498. MR 369551

    1973

  4. [12]

    Takefumi Nosaka,Quandles and topological pairs: Symmetry, knots, and cohomology, Springer, 2017

    2017

  5. [13]

    Suciu,Infinitely many ribbon knots with the same fundamental group, Math

    Alexander I. Suciu,Infinitely many ribbon knots with the same fundamental group, Math. Proc. Cambridge Philos. Soc.98(1985), no. 3, 481–492. MR 803607

    1985

  6. [14]

    Kokoro Tanaka and Yuta Taniguchi, 2-knots with the same knot group but different knot quandles, J. Math. Soc. Japan (to appear), arXiv:2308.07782 [math.GT]

    arXiv

  7. [15]

    thesis, University of Illinois at Chicago, 1984

    Steven Karl Winker,Quandles, knot invariants, and the n-fold branched cover, Ph.D. thesis, University of Illinois at Chicago, 1984

    1984

  8. [16]

    Jumpei Yasuda,Knot quandles distinguish Suciu’s ribbon knots, Preprint, arXiv:2508.15129 [math.GT] (2025), 2025. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan Email address:sakasai@ms.u-tokyo.ac.jp Department...

    2025 arXiv

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