{"id":"f7338fb2-efbc-42b4-a274-09a85595530d","arxiv_id":"1908.06447","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every p at least 2 there exist knots with arbitrarily large gaps between tu_{p-1} and tu_p, and torus knots with unbounded gaps between tu_1 and tu_2.","lead":"The paper proves that consecutive generalized unknotting numbers, called untwisting numbers, can differ by arbitrarily large amounts for every p at least 2, and that torus knots have arbitrarily large gaps between the first two untwisting numbers. It gives explicit constructions using cable knots and a quadratic upper bound for tu_2 of torus knots.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is sound, but the proof's use of Hom14 to infer ε(T_{2,2n+1})=1 should be checked: the cited corollary may be stated for Seifert genus, not slice genus.","rationale":"I read the paper's central claim as Theorem 1: for every p≥2 and m≥1 there is a knot K with tu_{p-1}(K)-tu_p(K)≥m. The construction is a (p,1)-cable K_p of K=T_{2,2n+1}, with n=m(p-1). The proof is genuinely elegant: a crossing change in K induces a null-homologous twist on 2p strands in K_p, so tu_p(K_p)≤u(K)=n; Hom's formula gives τ(K_p)=pn; the slice-genus bound gives pn≤g_4(K_p)≤p·tu_p(K_p)≤pn; hence tu_p(K_p)=n and g_4(K_p)=pn. The lower bound tu_{p-1}(K_p)≥pn/(p-1)=n+m then follows. I checked the satellite twisting lemma and the band-move proposition; both are standard and correctly applied. The one place where the proof could fail is the transition from τ(K)=g_4(K) to ε(K)=1. That implication is plausible for the chosen torus knot, but the text states it for any knot with τ=u, and Hom's corollary may be phrased using the Seifert genus. Because the chosen example satisfies the needed hypothesis, this is a fixable exposition gap, not a fatal flaw. The index error in the even-k sum in Theorem 2 is real but separate and does not affect Theorem 1. I therefore see no reason to move away from the reader's conditional verdict.","tokens_in":4755,"tokens_out":23533,"duration_ms":236596,"concrete_test":"Look up Hom14, Corollary 4 and Theorem 1, and check the exact hypotheses. If Corollary 4 states τ(K)=g_4(K) ⇒ ε(K)=1, the proof is valid as written. If it states τ(K)=g_3(K) ⇒ ε(K)=1, verify that T_{2,2n+1} satisfies τ=g_3=n (true, since torus knots are genus-minimizing), and edit the proof to use the torus knot example rather than 'any K with τ=u'. If the corollary fails even for T_{2,2n+1}, recompute τ of the (p,1)-cable directly from Hom's cable formula and compare with pn.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The equality chain proving Theorem 1 has three inputs: the satellite upper bound tu_p(K_p) ≤ u(K), the slice-genus lower bound tu_p ≥ g_4/p, and the value τ(K_p)=pτ(K) obtained from Hom's cable formula. The first two are routine, so the load-bearing step is the computation of τ(K_p). The proof argues that since τ(K)=u(K), one has τ(K)=g_4(K), and that [Hom14, Corollary 4] implies ε(K)=1; then [Hom14, Theorem 1] gives τ(K_p)=pτ(K)=pn. If Hom's corollary is actually stated for the Seifert genus g(K) rather than the smooth slice genus g_4(K), then the inference from τ(K)=g_4(K) to ε(K)=1 is not justified for an arbitrary knot with τ=u. The proof says 'take K to be any knot with τ(K)=u(K)=n', which would be an overgeneralization in that case. For the explicit choice K=T_{2,2n+1}, however, the Seifert genus equals the slice genus, so the stronger hypothesis is satisfied and the argument can be repaired by restricting to torus knots. This is a precision issue in the written proof, not a counterexample to the theorem, but it is exactly the point on which the main result depends.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5012,"tokens_out":9943,"duration_ms":97663,"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":[{"comment":"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.","section":"Section 2, proof of Theorem 1"}],"minor_comments":[{"comment":"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":"Section 3, proof of Theorem 2"},{"comment":"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.","section":"Section 1 and Section 3"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Propositions and lemmas"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhere 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.\n\nThe 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.\n\nThis 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.","headline":"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.","tokens_in":5572,"tokens_out":5403,"would_cite":true,"duration_ms":50427,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Gaps between consecutive untwisting numbers can be arbitrarily large.","keywords":["untwisting number","unknotting number","null-homologous twist","cable knot","torus knot","slice genus","tau-invariant"],"falsifier":"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.","tokens_in":1722,"feed_emoji":"🪢","tokens_out":5004,"duration_ms":142669,"temperature":0.7,"pith_summary":"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.","feed_headline":"Untwisting numbers can be arbitrarily far apart","feed_subtitle":"For any p, some knot's (p-1)th and pth untwisting numbers differ by a chosen amount.","key_machinery":"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}.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the bordered Heegaard Floer homology cable formula for τ and the corollary that τ=g_4 implies ε=1, which yields τ(K_p)=pτ(K).","marker":"[Hom14]"},{"why":"Introduces the pth untwisting number and proves the gap between tu_1 and tu_2 can be arbitrarily large, the phenomenon this paper extends.","marker":"[Inc16]"},{"why":"Defines the τ-invariant and its slice-genus lower bound, which the proof uses to compute τ(K_p) and to control untwisting numbers.","marker":"[OS03]"},{"why":"Provides the background that gaps between tu_p and the minimal untwisting number can be arbitrarily large, framing the consecutive-gap question.","marker":"[Inc17]"}],"fun_headline_variants":["Untwisting number gaps grow without bound","Arbitrarily large gaps between consecutive untwisting numbers","Torus knots show unbounded untwisting gaps","For any p, tu_{p-1} and tu_p differ arbitrarily","Untwisting numbers: consecutive gaps can be any size"],"cache_read_input_tokens":7680,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Untwisting number gaps grow without bound","Arbitrarily large gaps between consecutive untwisting numbers","Torus knots show unbounded untwisting gaps","For any p, tu_{p-1} and tu_p differ arbitrarily","Untwisting numbers: consecutive gaps can be any size"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1218,"prompt_tokens":846,"completion_tokens":372,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":290}},"tokens_in":462,"tokens_out":372,"duration_ms":3644,"temperature":1.0,"reasoning_tokens":290,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:47:28.029405+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}