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REVIEW 2 major objections 4 minor 15 references

Smoothly slice knots with Alexander polynomial 1 and high unknotting number

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that a smoothly slice knot with Alexander polynomial 1 and unknotting number exactly 5 exists, and the knot may be chosen smoothly doubly slice and amphicheiral.

desk verdict Genuinely new lower bound, but the theorem overclaims: only u(L) ≥ 5 is proven, not u(L) = 5. read the letter →

arxiv 2507.15592 v1 pith:7QHQK44N submitted 2025-07-21 math.GT

Lukas Lewark This is my paper · ORCID
classification math.GT MSC 57K1057K18
keywords unknottingnumberAlexanderpolynomial1smoothlysliceknotdoublyamphicheiralFloerhomologyU-torsionorderManolescu-Marengonknots
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

This paper proves that there exists a smoothly slice knot with Alexander polynomial 1 whose unknotting number is exactly 5, and that the knot may be chosen smoothly doubly slice and amphicheiral. The significance is that usual lower bounds for unknotting number—such as the tau-invariant or branched-cover homology—are also lower bounds for the smooth slice genus, so they cannot certify that a smoothly slice knot needs many crossing changes to untie. The proof exploits a torsion-order bound from knot Floer homology, which is one of the few lower bounds that can separate unknotting number from slice genus in this setting, and builds the example as the connected sum of a knot with its mirror.

What carries the argument

The central object is t(K), the maximal order of U-torsion in the minus version of knot Floer homology of K. It controls a Gordian-distance lower bound: any two knots joined by d crossing changes have |t difference| at most d, so t(K) is at most u(K). The paper computes t(ell) at least ell for the first six knots in the Knight Move family, uses standard identities for connected sums and mirrors to pass to J' # -J', and leans on a companion-paper proposition: a null-homologous twist can be replaced by one crossing change to an S-equivalent knot, preserving Alexander polynomial 1. The mirror sum is the construction that makes the final knot smoothly doubly slice and amphicheiral.

What would settle it

Following the constructive proof promised in Remark 7, write down the explicit knots J' and L and compute u(L) directly; Theorem 1 stands only if the result is exactly 5. A cheaper check targets the companion-paper proposition: enumerate knots within one crossing change of the sixth Knight Move knot and test whether any has Alexander polynomial 1; finding none would disprove the construction's key step.

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Extended reading notes

Core claim

In the paper's own terms, the discovery is: there is a smoothly doubly slice, amphicheiral knot L with Alexander polynomial 1 and unknotting number 5. The proof starts from the sixth member of the infinite knot family used to disprove the Knight Move conjecture, verifies by computer computation that its maximal U-torsion order t in minus knot Floer homology is at least 6, and then uses a companion-paper construction to replace the null-homologous twist that relates that knot to the unknot by a single crossing change to a knot J' that is S-equivalent to the unknot. The distance bound for t gives t(J') at least 5, and the mirror sum L = J' # -J' inherits t(L) at least 5, hence u(L) at least 5; the theorem states this lower bound is exact.

Load-bearing premise

The construction stands on the companion-paper proposition that a null-homologous twist can be traded for a single crossing change to an S-equivalent knot, which is only sketched here, and the exact value 5 also depends on an upper-bound argument that the printed text does not write out; if either premise fails, the claimed L with u(L) = 5 is not established.

Editorial extensions

If this is right

  • The torsion-order lower bound supplies concrete examples where a smoothly slice Alexander-polynomial-1 knot requires more than two crossing changes to untie.
  • The same recipe can be re-run for any knot satisfying the two hypotheses of Lemma 6, yielding smoothly doubly slice, amphicheiral Alexander-polynomial-1 knots with unknotting number at least n-1.
  • Because the final knot is amphicheiral and doubly slice, the high unknotting number is not caused by chirality or by a non-symmetric slice disk.
  • Proposition 5's constructivity, noted in Remark 7, means the theorem's existence claim is in principle accompanied by an explicit knot diagram.

Reading between the lines

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

  • If the companion-paper proposition extends to all members of the Knight Move family, then Conjecture 2 would turn the same lemma into a machine producing smoothly slice Alexander-polynomial-1 knots with arbitrarily large unknotting number, not just 5.
  • The lower-bound method used here isolates unknotting number from smooth slice genus: the constructed L has slice genus 0 because it is smoothly slice, while t(L) at least 5 forces many crossing changes, so t gives information that tau and other slice-genus bounds cannot.
  • In the present text, Lemma 6 is stated and proved only as a lower bound; the theorem's equality statement requires an additional upper-bound argument, which the paper does not write out but which the promised explicitness of the construction would make a finite check.
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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

2 major / 4 minor

Summary. The paper claims to prove the existence of a smoothly doubly slice, amphicheiral knot with Alexander polynomial 1 and unknotting number exactly 5. The construction starts with the Manolescu–Marengon knots κ_ℓ, uses the maximal order t(K) of U-torsion in knot Floer homology to get lower bounds on Gordian distance, invokes an unpublished proposition of Feller–Lewark to turn a null-homologous twist into a single crossing change, and forms the connected sum J′ # −J′ to obtain a smoothly doubly slice amphicheiral knot with Alexander polynomial 1. Lemma 6 proves only that this knot has unknotting number at least n−1; applying it with n=6 gives u(L) ≥ 5. The paper concludes Theorem 1 with u(L)=5.

Significance. If the advertised equality u(L)=5 were established, the result would be a notable advance: it would provide the first smoothly slice Alexander polynomial 1 knot with unknotting number greater than 2, with the extra doubly slice and amphicheiral properties. The use of U-torsion orders from knot Floer homology as a lower bound is interesting and appears to be a genuinely new mechanism for this class of knots. The computational verification for κ_ℓ with ℓ≤6 is carefully documented, with tables and an accompanying repository [Lew25], which is a strength. However, the central theorem as stated is not derived: the proof supplies only a lower bound, and the key topological step is outsourced to an unpublished companion paper. As it stands, the manuscript proves at most the existence of such a knot with unknotting number at least 5, not equal to 5.

major comments (2)
  1. [Section 2, Lemma 6 and Theorem 1] Lemma 6 concludes only that u(L) is at least n−1. Applying the lemma with n=6 gives u(L) ≥ 5, but Theorem 1 asserts u(L)=5. No upper bound u(L) ≤ 5 is proved anywhere in the manuscript: no diagram of L or explicit unknotting sequence is given, and the generic connected-sum bound u(J′ # −J′) ≤ 2u(J′) ≤ 2(u(κ_6)+1) ≤ 14 is far too weak. The sentence 'Theorem 1 is now inferred by applying the following lemma to K=κ_6' is therefore not valid; the proof establishes only a lower bound. To prove the stated theorem, the authors must either exhibit an unknotting sequence of length 5 for L or otherwise prove u(L) ≤ 5, or the statement of Theorem 1 and the abstract must be weakened to 'at least 5.'
  2. [Section 2, Proposition 5] Proposition 5 is the load-bearing step that converts the null-homologous twist relating K and J into a single crossing change producing a knot J′ that is S-equivalent to J, and hence has Alexander polynomial 1. The proposition is cited to the unpublished paper [FL] and only a two-sentence proof sketch is included. The sketch asserts the existence of a disk D′ with [Σ ⋔ D] = [Σ ⋔ D′] and a single proper arc intersection, but it does not actually justify the S-equivalence conclusion, which is essential for the rest of the argument. Since Lemma 6 depends entirely on this step, the main construction is conditional on an external, currently unavailable proof. A full proof of Proposition 5, or a publicly posted preprint of [FL], should be supplied before the result can be fully assessed.
minor comments (4)
  1. [Section 1, paragraph 1] The phrase 'there are no true lower bounds for the unknotting number' is overstated and could confuse readers: the invariants mentioned are genuine lower bounds, but they are not sensitive to Alexander polynomial 1. Rephrasing as 'no known lower bounds that are not also lower bounds for other geometric invariants' would be more accurate.
  2. [Section 2, proof of Lemma 6] The term 'S-equivalence' is used without a definition or reference. A brief definition, or a pointer to a standard source, would make the argument self-contained.
  3. [Tables 1–3] The color coding (green and red cells) used to explain the pairing argument is helpful in the digital version, but it is not accessible in monochrome printing. Adding a textual description of which cells are empty and which class is forced to pair would improve clarity.
  4. [References, [Lew25]] The companion computer calculations are cited to a GitHub repository; for reproducibility, it would be advisable to archive a version with a DOI or otherwise provide a permanent snapshot.

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing self-citation to unpublished [FL] proposition; Floer lower bound is independent, but Theorem 1's equality also omits the upper bound.

  1. self citation load bearing [Section 2, Proposition 5 (page 4)]
    "Proposition 5 ([FL]). Suppose two given knots K and J are related by a null-homologous twist. Then there exists a knot J ′ that is S-equivalent to J and related to K by a crossing change. A detailed proof of Proposition 5 will appear in the upcoming paper [ FL]."

    Lemma 6 uses Proposition 5 as the only bridge turning K_6's null-homologous twist into a single crossing change; this produces J' and hence L. The proposition is not proved here (only a sketch) and is attributed to [FL], the upcoming paper by the same author with P. Feller, with no code, formalization, or external verification supplied. The central existence step therefore rests on an unverified self-citation rather than on an independently checkable result.

full rationale

The derivation of the lower bound is not circular: Lemma 3 uses Szabó's calculator and the author's public data [Lew25] to establish t(K_ℓ)≥ℓ for ℓ≤6, and Theorem 4 is an external result of Alishahi–Eftekhary. The construction of L, however, depends on Proposition 5, whose proof is deferred to [FL], an upcoming paper coauthored by the present author; the sketch in this paper is too brief to be machine-checked and no independent confirmation is cited. That makes Proposition 5 a load-bearing self-citation under pattern 3. Separately, the paper's final inference is logically incomplete: applying Lemma 6 with n=6 gives only u(L)≥5, whereas Theorem 1 asserts u(L)=5. No upper bound u(L)≤5, unknotting sequence, or diagram for L is given, so the equality is not established by the text. This is an omitted-proof/correctness gap rather than a reduction-by-construction, so it does not itself increase the circularity score. Overall score 4: the independent Floer-theoretic content keeps the paper from being circular, but the pivotal proposition is an unverified self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof has no fitted constants. It relies on standard Floer theorems, a computer-assisted computation, and two unverified inputs: the unpublished Proposition 5 and the unstated upper bound for u(L) in Theorem 1. There are no invented entities.

assumptions (5)
  • standard math Knot Floer homology torsion bounds: |t(K)-t(J)| is at most the Gordian distance, t(K#J)=max(t(K),t(J)), and t(-K)=t(K).
    Used in Lemma 6 to convert t(J') >= n-1 into u(L) >= n-1; cited to AE20 and JMZ20.
  • domain assumption Proposition 5: a null-homologous twist can be replaced by a single crossing change to an S-equivalent knot.
    This is the central step producing J' with Alexander polynomial 1. The preprint gives only a proof sketch and cites the upcoming unpublished paper [FL] by Feller and Lewark.
  • domain assumption Szabo's HFK calculator outputs correct hat knot Floer homology for gamma_1 through gamma_6.
    Lemma 3 relies on tables produced by [Sza17]; no independent verification or formal certificate is provided.
  • domain assumption The Manolescu-Marengon knots gamma_l have the stated diagrams and satisfy u(gamma_l) <= l.
    Used to know u(gamma_6) <= 6 and that gamma_6 is related by a null-homologous twist; Figure 1 is not fully reproduced in the text.
  • standard math Zeeman's theorem that K#-K is smoothly doubly slice.
    Used in Lemma 6 to conclude L is smoothly doubly slice and amphicheiral; cited to [Zee65].

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Pith. "Pith review of Smoothly slice knots with Alexander polynomial 1 and high unknotting number." pith.science (2026). https://pith.science/paper/7QHQK44N

@misc{pith2026250715592,
  author       = {Pith},
  title        = {Pith review of: Smoothly slice knots with Alexander polynomial 1 and high unknotting number},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7QHQK44N}},
  note         = {Machine review of arXiv:2507.15592}
}
read the original abstract

We prove the existence of a smoothly doubly slice, amphicheiral knot with Alexander polynomial 1 and unknotting number 5.

Figures

Figures reproduced from arXiv: 2507.15592 by the authors.

Figure 1
Figure 1. The Manolescu–Marengon knots 1, 2, 6. The gray box for each of the ℓ contains a right-handed full twist on 2ℓ strands. For a diagram of 3, see [MM20]. 1 is the figure-eight knot. 2 is 17nh76 in the tables [BBP+25] [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Reference graph

Works this paper leans on

15 extracted references · 14 canonical work pages

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