Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots

Michal Jablonowski

Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots

T0 review · 0 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The fundamental quandle, added to the knot group and second homotopy module, still does not determine the homotopy type of an oriented 2-knot exterior.

desk verdict Clean negative result: for the Plotnick–Suciu 2-knots, the fundamental quandle agrees while the first Postnikov invariant does not, so (π1, π2, Q) still doesn't determine the algebraic 2-type; the proof is a focused, correct observation. read the letter →

arxiv 2608.03818 v1 pith:J6BD727X submitted 2026-08-04 math.GT

Michal Jablonowski This is my paper · ORCID

classification math.GT MSC 57K45

keywords 2-knotsfundamentalquandlefirstPostnikovinvariantalgebraic2-typesecondhomotopymodulePlotnick–SuciuconstructionperipheralcosetBrieskornhomologyspheres

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's central claim is that the fundamental quandle does not fill the gap left by the knot group and the second homotopy module in determining the homotopy type of an oriented 2-knot exterior. For the Brieskorn Plotnick–Suciu pair, and also for larger lens-space families, the paper constructs 2-knots whose exteriors have isomorphic fundamental groups, second homotopy modules that match under a suitable group isomorphism, and isomorphic fundamental quandles, but whose first Postnikov invariants lie in different orbits under every compatible group and module isomorphism. Since the first Postnikov invariant is part of the algebraic 2-type, the exteriors are not homotopy equivalent. A sympathetic reader should care because the fundamental quandle is a strong meridional invariant, and this result says that even adding it to the usual pi_1/pi_2 package cannot recover the missing k-invariant. The paper also sharpens the Tanaka–Taniguchi examples: their second homotopy modules already separate the three knots, so those examples do not realize the new phenomenon.

What carries the argument

The load-bearing object is the peripheral coset quandle Cos(G,⟨m⟩,m), whose underlying set is the cosets of the meridian subgroup and whose operation is Pg ∗ Ph = P(gh⁻¹mh). Proposition 2.14 identifies the fundamental quandle of an oriented 2-knot with this coset quandle, so the quandle is completely determined by the group together with the conjugacy class and orientation of the meridian. The main proof works by showing, through a basepoint-explicit van Kampen computation (Lemma 4.3), that both exteriors contain the same boundary piece W = S¹_t × B³ whose distinguished loop μ_t is sent to the same literal element t in a common group Π for both exteriors. This makes the two oriented peripher

What would settle it

Compute explicit 3-dimensional chain models for the two exteriors in the smallest admissible case, say n₁=7 and n₂=13, resolve the two first Postnikov classes in H³(Π;π₂), and enumerate all compatible pairs (α,β) of group/module automorphisms of (Π,π₂); exhibiting such a pair that carries one class to the other would refute Theorem 4.6, while reproducing the claimed orbit separation in this concrete case would settle it positively.

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Extended reading notes

Core claim

Theorem 4.6 states that, for every pair of distinct integers n1,n2 > 5 each coprime to 6, the Brieskorn Plotnick–Suciu construction yields oriented 2-knots bK and bK' whose exteriors satisfy the following: their fundamental groups are isomorphic; there is a group isomorphism α0 and an α0-semilinear isomorphism β0 between the second homotopy modules; their fundamental quandles are isomorphic; yet no compatible pair (α,β) of a group isomorphism and an α-semilinear module isomorphism carries the first Postnikov invariant of bK to that of bK'. Hence the exteriors are not homotopy equivalent. The proof identifies both positive meridians with the same literal element t in a common group Π, so both

Load-bearing premise

The load-bearing premise is geometric: after the interior surgery, the two van Kampen identifications can be chosen so that the distinguished boundary meridian from the common S¹×B³ piece represents literally the same group element t in the common group Π for both exteriors; if that simultaneous identification failed, the two oriented peripheral triples would not coincide and the quandle isomorphism could break down.

Editorial extensions

If this is right

The compatible pair (π₁,π₂) together with the fundamental quandle does not determine the homotopy type of an oriented 2-knot exterior: Theorem 4.6 exhibits exteriors with all three isomorphic and distinct homotopy types.
For every N ≥ 2 there exist N oriented 2-knots whose exteriors are pairwise non-homotopy-equivalent while sharing fundamental group, second homotopy module, and fundamental quandle (Corollary 4.7).
The Alexander module and Alexander polynomial of all these examples agree (Λ/(t−2) and t−2, up to the usual unit), so Alexander-type invariants cannot separate the exteriors either.
The Tanaka–Taniguchi triples are excluded from the phenomenon: their π₂-modules are pairwise inequivalent under any isomorphism of the knot groups, so those examples are already distinguished by the compatible pair (π₁,π₂).
The quandle isomorphism can be chosen to induce the identity on the common group Π under the based identifications, showing that the quandle agreement is genuinely a peripheral phenomenon rather than an accident of the construction.

Reading between the lines

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

The paper leaves implicit that any invariant depending only on the group together with the meridian's conjugacy class and orientation—including counts of representations into finite groups with the meridian sent to a prescribed conjugacy class—cannot distinguish bX from bX', because the fundamental quandle is exactly that datum.
The mechanism is insensitive to internal orientation flips in the fiber summand: any two exteriors built from the same W#Y piece by surgery on the same word will have isomorphic quandles as long as the shared meridian can be based-identified to the same element, so quandle invariants are blind to the orientation change that moves the k-invariant.
This suggests that a homotopy classification of 2-knot exteriors will have to use the full algebraic 2-type including the k-invariant, and that the peripheral coset description of the quandle cannot be upgraded to a complete invariant by adding only module-level data.
A testable extension is to apply the same basepoint-explicit peripheral comparison to the other families discussed in the paper, such as the ribbon family with distinct quandles, to determine whether any pair there can have isomorphic quandles while retaining different modules; the methods of the paper provide the template.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper delivers exactly what the title says: for the Plotnick–Suciu 2-knots, the fundamental quandle is the same while the first Postnikov invariants are inequivalent, so the triple (π1, π2, Q) doesn't determine the algebraic 2-type. That is the one thing to know. The new content is Lemma 4.3, which identifies the distinguished meridian as the same literal element t in both exteriors through a careful van Kampen basepoint argument, and then uses the peripheral coset description to get isomorphic quandles. That is clean and convincing. The Tanaka–Taniguchi module non-isomorphism (Theorem 3.6) is also new and correctly proved via the kernel of the action, and it usefully separates those examples from the Plotnick–Suciu phenomenon.

What the paper does well: it is honest about what is imported. The non-realizability of the k-invariants comes from Plotnick–Suciu, and the lens-space family from Suciu's thesis, and the author states exactly which results are being used. The orientation conventions are handled carefully—the distinction between reversing the fiber orientation and reversing the knot orientation is explicit. The Alexander module computation is a nice extra, and Proposition 2.16 (Alexander module determined by the quandle) is a clean formal statement that makes the examples' agreement even more pointed.

The soft spots are minor. The genuinely new geometric input is thin—essentially one lemma plus a coset identification—but that is not a flaw; the paper is deliberately a focused observation about known examples. The reliance on Suciu's unpublished thesis [Suc84] is a small barrier for readers who want to check the lens-space family, though the needed statements are quoted clearly. The equivalence between "no compatible (α, β)" and "no k-invariant orbit" is imported from the MacLane–Whitehead realization theorem; the paper explains this but doesn't reprove it, which is appropriate. I stress-tested Lemma 4.3 and found no gap: the shared W-piece, the common word r, and the basepoint identifications all line up.

Who should read this: anyone working on 2-knot classification, quandle invariants, or the algebraic 2-type. It doesn't introduce a new method, but it closes a natural question and corrects a plausible guess. It deserves a serious referee; the proofs are clear and, as far as I can tell, correct. My recommendation: send it out, accept after minor comments.

Referee Report

0 major / 4 minor

Summary. The paper addresses whether the fundamental quandle, combined with the compatible pair (π1, π2), determines the first Postnikov invariant (and hence the homotopy type) of an oriented 2-knot exterior. It first proves (Theorem 3.6) that the Tanaka–Taniguchi triples, although having isomorphic knot groups and abstractly isomorphic second homotopy groups, have pairwise non-isomorphic second homotopy modules; they therefore do not realize the phenomenon in question. The central result is Theorem 4.6: for the Brieskorn Plotnick–Suciu pair, after orienting both 2-knots so that the common distinguished boundary meridian μ_t is positive, the fundamental quandles are isomorphic (Theorem 4.5) via the peripheral coset description, while the first Postnikov invariants are inequivalent under every compatible group/module isomorphism. The argument is extended in Corollary 4.7 to arbitrarily large families using Suciu's lens-space construction, and the common Alexander module is computed explicitly as Λ/(t−2) in Lemma 4.9.

Significance. The main result gives a clean negative answer to a natural question: adding the fundamental quandle to the compatible pair (π1, π2) does not determine the Postnikov invariant of a 2-knot exterior. The key new geometric step is Lemma 4.3, which identifies the distinguished meridian in both constructions; I checked this point and it is sound. The paper is careful to separate new contributions from quoted results of Plotnick–Suciu and Suciu, and it supplies explicit module computations, including the Alexander-module calculation. The Tanaka–Taniguchi comparison is also a useful clarification, showing that those examples are distinguished already at the level of π2-modules.

minor comments (4)

[Section 2 (Definitions 2.13 and Proposition 2.14)] The coset quandle operation is written with a subscript in Definition 2.13 but without it in Proposition 2.14. A brief remark that the subscript is omitted would remove ambiguity.
[Remark 4.11] In the Fox calculus display, the second derivative should be written with an augmentation bar: it is the image of the unreduced derivative under t↦t, x↦1. As typeset, the equality ∂r/∂x = t−2 could be misread as an unreduced Fox derivative.
[Abstract and Section 4.1] The abstract refers to 'admissible pairs of Brieskorn parameters' but the body states the hypotheses directly in Theorem 4.6 (distinct n1,n2>5, each coprime to 6). Aligning the terminology would avoid confusion.
[Proposition 2.14 proof] The proof uses the fact that all positive based meridians in a 2-knot group are conjugate without spelling out the standard argument or giving a reference. This is a minor self-containedness issue.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the quandle isomorphism is a geometrically grounded peripheral argument, and the Postnikov inequivalence is imported from the external Plotnick–Suciu theorem.

full rationale

The central derivation chain is self-contained against external benchmarks. Theorem 4.6 combines the Plotnick–Suciu result [PS85] that the two exteriors have isomorphic π1 and semilinearly isomorphic π2 but inequivalent first Postnikov invariants with a new peripheral argument (Lemma 4.3, Theorem 4.5) showing that the fundamental quandles agree. Lemma 4.3 is not a fit or a renaming: it uses the explicit fact that both exteriors are obtained from the same W = S^1_t × B^3 by surgery on the same word r = txt^{-1}x^{-2}, so van Kampen identifies both boundary meridians with the same literal generator t of Π. Theorem 4.5 then applies the standard peripheral-coset description of the fundamental quandle (Proposition 2.14), which is proved from the noose definition. No parameter is fitted to the target statement, and no Postnikov conclusion is assumed from the quandle side. The only self-citation, [Jab25], is used in the introduction to note that Suciu's ribbon 2-knots have mutually non-isomorphic quandles; this is peripheral to the main theorem and corroborated by independent references [Yas25, ST26]. The paper also explicitly declines to rely on the sketched [PS85, Theorem 8.1] for the large-family result, using instead Suciu's complete thesis argument. Thus no circularity concern is supported by the text; the score reflects only the presence of a minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on prior results about the Plotnick-Suciu construction and twist spinning, plus standard theorems in algebraic topology. No free parameters are fitted, and no new entities are postulated. The paper's own contributions are the peripheral quandle argument and the module non-equivalence proof.

assumptions (4)

domain assumption Plotnick-Suciu theorem: the two exteriors bX and bX' have isomorphic fundamental groups and semilinearly isomorphic second homotopy modules, and no compatible pair of isomorphisms carries the first Postnikov invariant from one to the other, provided the closed fibers admit no orientation-reversing h
Used as the backbone of Theorem 4.6. The paper verifies the no orientation-reversing homotopy equivalence hypothesis for Brieskorn homology spheres in Section 4.1 using [NR78] and [Mil75], but does not reprove the theorem itself.
domain assumption Zeeman twist-spinning theorem: the exterior of the s-twist spin of a classical knot K fibers over S^1 with fiber the punctured s-fold cyclic branched cover of S^3 along K, with monodromy the canonical deck transformation.
Invoked in Section 3 and Section 4.1 to identify the fibers of the Tanaka-Taniguchi and Brieskorn Plotnick-Suciu examples as punctured Brieskorn homology spheres.
domain assumption Brieskorn homology spheres Σ(2,3,n) with n > 5 coprime to 6 are aspherical and admit no orientation-reversing homotopy equivalence, because their Seifert Euler number has absolute value 1/(6n), not zero.
Used in Section 4.1 to verify the hypotheses of the Plotnick-Suciu theorem. The asphericity fact is attributed to [Mil75]; the Euler-number obstruction is applied from [NR78, Theorem 8.2].
standard math MacLane-Whitehead realization theorem: for 3-complexes, the existence of a map inducing isomorphisms on π1 and π2 is equivalent to the existence of a compatible pair of isomorphisms carrying k-invariants.
Used in the proofs of Theorem 4.6 and Corollary 4.7 to translate the Plotnick-Suciu and Suciu results about maps into the statement about Postnikov invariant orbits.

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Pith. "Pith review of Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots." pith.science (2026). https://pith.science/paper/J6BD727X

@misc{pith2026260803818,
  author       = {Pith},
  title        = {Pith review of: Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6BD727X}},
  note         = {Machine review of arXiv:2608.03818}
}

abstract

For every admissible pair of Brieskorn parameters in the Plotnick--Suciu construction, we obtain a pair of oriented $2$-knots whose knot groups are isomorphic, whose second homotopy modules are semilinearly isomorphic under a suitable group isomorphism, and whose fundamental quandles are isomorphic, while no compatible group and module isomorphisms carry one first Postnikov invariant to the other. Consequently, their exteriors are not homotopy equivalent. Thus, the knot group, the second homotopy module up to semilinear equivalence, and the fundamental quandle do not determine the homotopy type of an oriented $2$-knot exterior. Using an alternative construction from Suciu's thesis based on punctured lens spaces, the same peripheral argument yields, for every $N\geq2$, a family of $N$ oriented $2$-knots with these properties. We additionally show that the Tanaka--Taniguchi examples with isomorphic knot groups and distinct fundamental quandles have pairwise inequivalent second homotopy modules: no isomorphism between two of the knot groups makes the corresponding second homotopy modules semilinearly isomorphic.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Di Prisa, D

Bais, V., A. Di Prisa, D. Hartman, C.-S. Hsueh, M. Kegel, A. Merz, M. Pencovitch, A. Ray, D. Santoro, P. Tru\"ol, and L. Wakelin, On the Detection of Knotted Spheres by Their Traces in High Dimensions, preprint, 2025, arXiv:2511.07251, https://arxiv.org/abs/2511.07251

work page arXiv 2025

[2] [2]

Carter, J. S., S. Kamada, and M. Saito, Surfaces in 4 -Space, Encyclopaedia of Mathematical Sciences, vol. 142, Springer-Verlag, Berlin, 2004, https://doi.org/10.1007/978-3-662-10162-9

doi:10.1007/978-3-662-10162-9 2004

[3] [3]

Kasprowski, 4 -Manifolds with a Given Boundary, preprint, 2025, arXiv:2510.18836, https://arxiv.org/abs/2510.18836

Conway, A., and D. Kasprowski, 4 -Manifolds with a Given Boundary, preprint, 2025, arXiv:2510.18836, https://arxiv.org/abs/2510.18836

work page arXiv 2025

[4] [4]

H., The Derived Module of a Homomorphism, Advances in Mathematics, vol

Crowell, R. H., The Derived Module of a Homomorphism, Advances in Mathematics, vol. 6, 1971, pp. 210--238, https://doi.org/10.1016/0001-8708(71)90020-0

work page doi:10.1016/0001-8708(71)90020-0 1971

[5] [5]

Rourke, Racks and Links in Codimension Two, Journal of Knot Theory and Its Ramifications, vol

Fenn, R., and C. Rourke, Racks and Links in Codimension Two, Journal of Knot Theory and Its Ramifications, vol. 1, no. 4, 1992, pp. 343--406, https://doi.org/10.1142/S0218216592000203

doi:10.1142/s0218216592000203 1992

[6] [6]

McA., Twist-Spun Torus Knots, Proceedings of the American Mathematical Society, vol

Gordon, C. McA., Twist-Spun Torus Knots, Proceedings of the American Mathematical Society, vol. 32, no. 1, 1972, pp. 319--322, https://doi.org/10.1090/S0002-9939-1972-0288752-8

work page doi:10.1090/s0002-9939-1972-0288752-8 1972

[7] [7]

Jab onowski, M., On Biquandle-Based Invariant of Immersed Surface-Links, Yoshikawa Oriented Fifth Move, and Ribbon 2 -Knots, preprint, 2025, arXiv:2505.14724, https://arxiv.org/abs/2505.14724

arXiv 2025

[8] [8]

Joyce, D., A Classifying Invariant of Knots, the Knot Quandle, Journal of Pure and Applied Algebra, vol. 23, no. 1, 1982, pp. 37--65, https://doi.org/10.1016/0022-4049(82)90077-9

doi:10.1016/0022-4049(82)90077-9 1982