Gaps between consecutive untwisting numbers

Duncan McCoy

REVIEW 1 major objections 4 minor 8 references

Gaps between consecutive untwisting numbers

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Gaps between consecutive untwisting numbers can be arbitrarily large.

desk verdict Theorem 1 is real and the core proof is clean, but the stated generality is slightly over-broad because it relies on Hom's epsilon criterion in a stronger form than cited; trivially fixable. read the letter →

arxiv 1908.06447 v1 pith:BLYOIQ7K submitted 2019-08-18 math.GT

Duncan McCoy This is my paper · ORCID

classification math.GT MSC 57K10

keywords untwistingnumberunknottingnull-homologoustwistcableknottorusslicegenustau-invariant

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

For any p≥2 and any m≥1, there is a knot K such that tu_{p-1}(K) - tu_p(K) ≥ m. This shows the decreasing sequence of untwisting numbers never stabilizes. The knots are (p,1)-cables of two-strand torus knots, and the proof shows these examples make the slice-genus lower bound sharp. The same paper shows torus knots have tu_2 strictly less than the unknotting number once the braid index is at least 4.

What carries the argument

The machinery is cable knots together with the τ-invariant from Heegaard Floer homology. A (p,1)-cable takes p parallel copies of a knot with one full twist; a null-homologous twist on two strands of the companion becomes a twist on 2p strands of the cable. The bordered cable formula for τ gives τ(K_p)=pτ(K), provided an auxiliary invariant ε is 1; here that follows from τ(K)=g_4(K). A satellite inequality transfers untwisting sequences, so the τ computation gives the exact value tu_p(K_p)=n, and the slice-genus bound tu_p ≥ g_4/p converts that into the lower bound on tu_{p-1}.

What would settle it

Compute the second untwisting number of the (2,1)-cable of the trefoil, the case p=2, m=1 of the construction. The theorem predicts tu_2=1 and tu_1-tu_2≥1; finding tu_2=0, or finding any knot in the constructed family with gap smaller than m, would disprove Theorem 1.

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Extended reading notes

Core claim

The central result is that for any p≥2 and m≥1, set n=m(p-1) and let K_p be the (p,1)-cable of T_{2,2n+1}. The proof computes τ(K_p)=pn, forcing the chain pn = τ(K_p) ≤ g_4(K_p) ≤ p·tu_p(K_p) ≤ pn, so tu_p(K_p)=n and g_4(K_p)=pn. Then the lower bound gives tu_{p-1}(K_p) ≥ g_4(K_p)/(p-1)= n + n/(p-1)= n+m, so the gap is at least m. For torus knots, a sequence of twists on four strands gives tu_2(T_{p,q}) ≤ 3pq/8, and comparing with u(T_{p,q})=(p-1)(q-1)/2 gives a gap of at least 1/2 floor(q/2)(floor(q/2)-1), which is arbitrarily large.

Load-bearing premise

The load-bearing premise is Hom's cable formula: a (p,1)-cable of a knot with τ equal to its slice genus has τ multiplied by p; if this formula has unstated restrictions for the chosen torus knots, the equality chain tu_p(K_p)=n fails.

Editorial extensions

If this is right

For every p≥2 and every m, the gap tu_{p-1}-tu_p can be made at least m, so the decreasing sequence of untwisting numbers never becomes constant at any finite stage.
The constructed knots satisfy equality in the bound tu_p ≥ g_4/p, so they form infinite families where the pth untwisting number is exactly the slice genus divided by p.
For torus knots T_{p,q} with min{p,q}≥4, tu_2 is strictly smaller than the unknotting number.
The gap between tu_1 and tu_2 for torus knots grows at least quadratically in the smaller braid index, hence arbitrarily large.
The uniform bound tu_2(T_{p,q}) ≤ 3pq/8 holds for all p,q>1.

Reading between the lines

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

Because the proof only needs a companion K with τ(K)=u(K)=g_4(K), other families such as positive pretzel knots should also yield arbitrarily large consecutive gaps via the same cable construction.
Since tu_p(K_p)=n and tu_{p-1}(K_p)≥n+m, the constructed knots may have several distinct consecutive untwisting values; computing the intermediate values would show if one knot can realize prescribed gaps at multiple levels.
The torus-knot trick converts a full twist into parallel strands using about 3k^2/8 twists on four strands; working out the exact minimum for small knots could suggest whether the constant 3/8 can be improved.

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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

Here's my take on McCoy's paper. The headline is that Theorem 1 is real: for every p≥2, the gap tu_{p−1}−tu_p between consecutive untwisting numbers can be arbitrarily large. This goes beyond Ince's results, and the proof is clean. The main construction—a (p,1)-cable of a torus knot—is natural, and the equality chain τ(K_p)=pn ≤ g_4(K_p) ≤ p·tu_p(K_p) ≤ pn forces equality throughout. Proposition 3, the slice-genus bound underpinning (1), is self-contained and convincing. Theorem 2's torus-knot bound tu_2 ≤ 3pq/8 is new and gives explicit large gaps between tu_1 and tu_2; the recursive square-change argument is intricate but the induction checks out.

Where the paper is soft: the proof of Theorem 1 overstates its generality. It says 'take K to be any knot with τ(K)=u(K)=n' and then infers ε(K)=1 from Hom's Corollary 4. If that corollary is stated for Seifert genus rather than slice genus—and I think Hom's epsilon criterion is about τ=g, not τ=g_4—the implication is not justified for an arbitrary knot with τ=u. The explicit choice K=T_{2,2n+1} has g=g_4, so the proof goes through for the knots actually used. The fix is trivial: restrict the statement to torus knots or cite the precise hypothesis. As written it is a precision issue, not a counterexample. There's also a minor display error in the even-k sum in Section 3: the summand is written as 3k/2−2 with no i, but the total matches the correct sum with 3i−2. Both are easy repairs.

The citation pattern is fine; [McC19] appears only as a peripheral alternative route. No circularity, no fitted constants. The results are new, and the core argument is sound.

This paper is for geometric topologists who care about unknotting-type invariants and their hierarchies. It deserves serious peer review; a referee will want the Hom citation checked against the source and the minor corrections noted, but the main theorems are solid. I would send it out.

Referee Report

1 major / 4 minor

Summary. The paper studies the p-th untwisting number tu_p, defined as the minimum number of null-homologous twists on at most 2p strands needed to convert a knot to the unknot. The main result, Theorem 1, asserts that for every p >= 2 and m >= 1 there exists a knot K with tu_{p-1}(K) - tu_p(K) >= m. The proof constructs K as the (p,1)-cable of a knot with tau = u, in particular the torus knot T_{2,2n+1}, and uses a satellite inequality, the slice-genus lower bound tu_p >= g_4/p, and Hom's formula for the tau-invariant of cable knots to force tu_p(K_p) = n while tu_{p-1}(K_p) >= n+m. The paper also proves Theorem 2, showing that torus knots of braid index at least four satisfy tu_2(T_{p,q}) < u(T_{p,q}) and that tu_2(T_{p,q}) <= 3pq/8, so the difference between tu_1 and tu_2 is arbitrarily large for torus knots.

Significance. If the results are correct, they provide the first proof that consecutive untwisting numbers can differ by an arbitrarily large amount, strengthening earlier work of Ince. The paper is concise and well organized, and it proves the central slice-genus estimate Proposition 3 directly from a diagrammatic band-move argument rather than quoting it. The main theorem also gives explicit knots attaining equality in the bound tu_p >= g_4/p, which is a nice feature. The argument relies on standard Heegaard Floer results, especially Hom's tau formula for cables, and the dependence on these tools is clearly identified. The torus-knot statement adds a quantitative upper bound that is interesting in its own right.

major comments (1)

[Section 2, proof of Theorem 1] The inference 'Since tau(K)=u(K), we have tau(K)=g_4(K). By [Hom14, Corollary 4] this implies that epsilon(K)=1' is load-bearing, but as written it applies the corollary to a knot satisfying tau(K)=g_4(K). If, as is standard, the corollary is stated with the Seifert genus g(K) rather than the smooth slice genus g_4(K), then an arbitrary knot with tau(K)=u(K) does not necessarily satisfy the hypothesis. The proof should either quote the exact statement of [Hom14, Corollary 4] and verify its hypothesis, or, more economically, restrict the construction to the torus knot K=T_{2,2n+1}, for which g(K)=g_4(K)=n. Since Theorem 1 only requires existence, the repair is local, but the proof as printed overstates the class of knots to which the argument applies.

minor comments (4)

[Section 3, proof of Theorem 2] The displayed sum for the number of null-homologous twists needed to undo a full twist on k strands is written as a sum over i=1 to k/2 of a term independent of i, namely 3k/2 - 2; as written the expression evaluates to (3k^2-4k)/4, not to the claimed (3k^2-2k)/8. The summand should depend on i, e.g. 3(k-2i+2)/2 - 2, in which case the displayed total is correct.
[Section 1 and Section 3] The unknotting number formula u(T_{p,q}) = (p-1)(q-1)/2 is stated without explicitly saying that p and q are coprime; for non-coprime parameters T_{p,q} is a link rather than a knot. This convention should be stated before Theorem 2.
[References] The arXiv identifier for [McC19] appears malformed: the text gives 'arXiv:1908.4043', which is missing a digit in the standard arXiv format; it should likely be '1908.04043' or the correct identifier for that paper.
[Propositions and lemmas] There are several typographical slips: Proposition 3 has 'related related', Lemma 4 reads 'then for any pattern P' where a comma would improve clarity, and the phrase 'the value of the τ-invariant of K_p depends on an auxiliary invariant ε(K)' could be phrased more formally. These do not affect the mathematics.

Circularity Check

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No circularity: the main derivation uses independent geometric bounds and external Floer-homology theorems, with no fitted parameters and no definition that assumes the conclusion.

full rationale

The derivation chain for Theorem 1 is self-contained in the respects that matter for circularity. Proposition 3 proves the lower bound tu_p(K) ≥ g_4(K)/p by showing that a null-homologous twist on 2p strands is achieved by 2p oriented band moves and hence by a genus-p concordance; this is an independent geometric argument, not a restatement of the theorem. Lemma 4 proves the satellite inequality tu_{pw}(P(K)) ≤ tu_p(K) + tu_{pw}(P(U)) by tracking the twist curve through the satellite construction, again independently. The upper bound tu_p(K_p) ≤ tu_1(K) = n then follows from the satellite inequality with P the (p,1)-cable pattern. The value τ(K_p) = pτ(K) = pn is imported from Hom's bordered Heegaard Floer cable formula, an external theorem that is not derived from, or defined in terms of, untwisting numbers. The proof then squeezes τ(K_p) = pn ≤ g_4(K_p) ≤ p·tu_p(K_p) ≤ pn, forcing equality; no parameter is fitted to the size of the gap, and no step assumes the conclusion. The only self-citation, [McC19], is offered as an alternative route to the topological slice genus bound (2), which is not used in the proofs of Theorem 1 or Theorem 2, so it is not load-bearing. A possible concern that [Hom14, Corollary 4] is stated for Seifert genus rather than smooth slice genus is a correctness or precision issue about an external cited result, not a circularity; moreover, for the explicit torus knots K = T_{2,2n+1} used in the proof, the Seifert and slice genera agree, so the argument can be read with the stronger hypothesis. No circular step is present.

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

The central claim rests on standard theorems in Heegaard Floer homology (Ozsvath-Szabo's tau inequality, Hom's cable formula) and classical facts about torus knots. No free parameters or invented entities are introduced. The paper itself proves the key lower bound (1) via cobordism arguments.

assumptions (4)

standard math Ozsvath-Szabo inequality |tau(K)| at most g_4(K)
Used in Theorem 1 proof as tau(K_p) at most g_4(K_p), and for tau(K)=g_4(K) for T_{2,2n+1}. Cited [OS03].
standard math Hom's formula for tau of cable knots: for a (p,q)-cable with q>0, tau(K_{p,q}) = p tau(K) + (q-1)(epsilon(K)-1)/2, and Corollary 4: if tau(K)=g_4(K) then epsilon(K)=1
Used to compute tau(K_p)=p tau(K)=pn. Cited [Hom14].
domain assumption Unknotting number and slice genus of torus knots: u(T_{a,b}) = g_4(T_{a,b}) = (a-1)(b-1)/2 for positive a,b
Used to choose K=T_{2,2n+1} with tau=u=g_4=n, and to compare u(T_{p,q}) with tu_2(T_{p,q}). Stated without proof in Section 1 and Section 3.
standard math Known recursion for unknotting number of torus knots: u(T_{p,q}) = q(q-1)/2 + u(T_{p-q,q})
Used in proof of Theorem 2 to compute the gap u minus tu_2. This follows from the explicit formula; not separately cited.

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Cite this review

Pith. "Pith review of Gaps between consecutive untwisting numbers." pith.science (2026). https://pith.science/paper/BLYOIQ7K

@misc{pith2026190806447,
  author       = {Pith},
  title        = {Pith review of: Gaps between consecutive untwisting numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLYOIQ7K}},
  note         = {Machine review of arXiv:1908.06447}
}

abstract

For $p\geq 1$ one can define a generalization of the unknotting number $tu_p$ called the $p$th untwisting number which counts the number of null-homologous twists on at most $2p$ strands required to convert the knot to the unknot. We show that for any $p\geq 2$ the difference between the consecutive untwisting numbers $tu_{p-1}$ and $tu_p$ can be arbitrarily large. We also show that torus knots exhibit arbitrarily large gaps between $tu_1$ and $tu_2$.

Figures

Figures reproduced from arXiv: 1908.06447 by the authors.

**Figure 2.** Two oriented band moves convert a full twist on 2p strands into a full twist on 2p − 2 strands with two parallel strands. Next we note how twisting operations transform under satellite operations. Lemma 4. Let K and K0 be knots related by a null-homologous twist on 2p strands. then for any pattern P ⊆ S 1 × D2 with geometric winding number w, the satellites P(K) and P(K0 ) are related by a null-homologous twist on 2… view at source ↗

**Figure 3.** Performing four crossing changes with three null-homologous twists. Theorem 1. For any pair of positive integers p ≥ 2 and m ≥ 1, there is a knot K such that tup−1(K) − tup(K) ≥ m. Proof. Set n = m(p−1) and take K to be any knot with τ (K) = u(K) = n. For example, the torus knot K = T2,2n+1. Let Kp be the (p, 1)-cable of K. Since the (p, 1)-cable of the unknot is itself unknotted, it follows from (4) that (5) tup(Kp… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Using square changes to undo a full twist. twists on at most four strands. Thus the full twist on k strands can be converted into k parallel strands by kX2 i=1 3k 2 − 2 = 3k 2 − 2k 8 null-homologous twists on at most four strands. Now suppose that k is odd. By perf…

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Works this paper leans on

8 extracted references · 6 canonical work pages

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Duncan McCoy. Null-homologous twisting and the algebraic genus. arXiv:1908.4043 , 2019

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

On the algebraic unknotting number

Maciej Borodzik and Stefan Friedl. On the algebraic unknotting number. Trans. London Math. Soc. , 1(1):57--84, 2014

2014

[2] [2]

The unknotting number and classical invariants, I

Maciej Borodzik and Stefan Friedl. The unknotting number and classical invariants, I . Algebr. Geom. Topol. , 15(1):85--135, 2015

2015

[3] [3]

A note on the topological slice genus of satellite knots

P. Feller, A. N. Miller, and J. Pinzon-Caicedo. A note on the topological slice genus of satellite knots. arXiv:1908.03760 , 2019

work page Pith review arXiv 1908

[4] [4]

Bordered H eegaard F loer homology and the tau-invariant of cable knots

Jennifer Hom. Bordered H eegaard F loer homology and the tau-invariant of cable knots. J. Topol. , 7(2):287--326, 2014

work page 2014

[5] [5]

The untwisting number of a knot

Kenan Ince. The untwisting number of a knot. Pacific J. Math. , 283(1):139--156, 2016

work page 2016

[6] [6]

Untwisting information from H eegaard F loer homology

Kenan Ince. Untwisting information from H eegaard F loer homology. Algebr. Geom. Topol. , 17(4):2283--2306, 2017

work page 2017

[7] [7]

Null-homologous twisting and the algebraic genus

Duncan McCoy. Null-homologous twisting and the algebraic genus. arXiv:1908.4043 , 2019

work page arXiv 1908

[8] [8]

Knot F loer homology and the four-ball genus

Peter Ozsv\' a th and Zolt\' a n Szab\' o . Knot F loer homology and the four-ball genus. Geom. Topol. , 7:615--639, 2003

work page 2003