{"id":"a4aa261b-ac3d-48fe-85a7-4d3d706926f0","arxiv_id":"2507.15592","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A new construction claims the first smoothly slice Alexander polynomial 1 knot with unknotting number 5, using Manolescu-Marengon knots and Floer torsion bounds.","lead":"The paper claims to construct a smoothly slice knot with Alexander polynomial 1 that needs exactly five crossing changes to unknot. Such knots with unknotting number above 2 had not previously been proven to exist.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Even assuming the unpublished Proposition 5, Lemma 6 yields only u(L) ≥ 5; no upper bound u(L) ≤ 5 is proven, so Theorem 1's equality is not established.","rationale":"The reader's REJECT is justified, but the most load-bearing problem is not only the yet-unpublished Proposition 5. The more decisive gap is internal: the proof does not connect the lower bound to equality. Lemma 6 is explicitly stated as 'at least n−1', and applying it with n=6 yields only u(L) ≥ 5. Nothing in the text gives u(L) ≤ 5. This is not a matter of external verification or consensus; it is a missing logical step. Even accepting the computer-assisted knot Floer homology computations (Tables 1–3), the Gordian distance bound Theorem 4, and Proposition 5, the conclusion 'unknotting number equals 5' does not follow. The lower-bound machinery is sound and the paper's contribution might be salvageable as a '≥5' result, but as written the main theorem is unproven. The reader's rationale also notes this gap, so our agreement is partial: we identify a different primary weakest assumption, but the verdict is unchanged.","tokens_in":5098,"tokens_out":8217,"duration_ms":83846,"concrete_test":"Examine the proof of Lemma 6 and of Theorem 1 for any argument bounding u(L) above by n−1. If none exists, the theorem as stated is unsupported. Once [FL] is available, explicitly construct J' from 6 via Proposition 5, form L = J'#−J', and compute u(L) (e.g., by exhaustive crossing-change search if the diagram is small enough). If u(L) ≠ 5, the claim fails; if u(L) = 5, supply the missing upper-bound proof. In the meantime, confirm that the paper nowhere establishes u(L) ≤ 5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 6 concludes 'unknotting number at least n − 1'. Applying it with n = 6 (K = 6) gives a knot L with u(L) ≥ 5. Theorem 1 claims u(L) = 5. The word 'equals' requires both inequalities, but the argument provides no u(L) ≤ 5. The proof of Lemma 6 uses Theorem 4 to get u(L) ≥ t(L) ≥ n − 1, a lower bound only. There is no construction of an unknotting sequence for L, no diagram of L, and no general bound on u(J' # −J') in terms of u(J') that would cap it at 5. Indeed L = J' # −J' with J' obtained from 6 by one crossing change; the available bound is u(L) ≤ 2u(J') ≤ 2(u(6)+1) ≤ 14. Thus, even if Proposition 5 is true, the stated theorem does not follow logically. The paper proves a weaker statement: existence of a smoothly doubly slice, Alexander polynomial 1, amphicheiral knot with unknotting number at least 5. That is not the abstract's claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5336,"tokens_out":9005,"duration_ms":98849,"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":[{"comment":"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.'","section":"Section 2, Lemma 6 and Theorem 1"},{"comment":"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.","section":"Section 2, Proposition 5"}],"minor_comments":[{"comment":"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.","section":"Section 1, paragraph 1"},{"comment":"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.","section":"Section 2, proof of Lemma 6"},{"comment":"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.","section":"Tables 1–3"},{"comment":"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.","section":"References, [Lew25]"}],"recommendation":"reject","confidential_remarks":"The manuscript has a sound-looking computational lower-bound mechanism, but the main theorem as stated is not proved: Lemma 6 gives only u(L) ≥ 5, and the missing upper bound is indispensable. In addition, Proposition 5, which is central to the whole construction, is not proved in the manuscript and rests on an unpublished companion paper. These are not presentational issues; they concern the validity of the advertised result. I would not rule out a publishable paper built on the same ideas if the authors can supply the missing upper-bound argument and make Proposition 5 fully available, but the current submission does not support its abstract."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This short note proves the existence of a smoothly doubly slice, amphicheiral knot with Alexander polynomial 1 and unknotting number at least 5. That is a new and interesting result: previous to this, no smoothly slice Alexander polynomial 1 knot with u > 2 was known. The method is also appealingly direct: take the Manolescu–Marengon knot λ_6, use a null-homologous twist to get a knot J' of Alexander polynomial 1, and then L = J' # -J'. The lower bound comes from the maximal order of U-torsion in knot Floer homology, via a clean trick that extracts t(K) from the hat homology tables. The exposition is clear, the computational data is supplied with a GitHub link, and the arguments up to the final step are standard.\n\nThe problem is the theorem statement. Lemma 6, which is the engine, concludes only u(L) ≥ n−1. Applied to λ_6 with n=6, that gives u(L) ≥ 5. Theorem 1 then claims \"unknotting number equals 5.\" The equal sign requires an upper bound, and none is provided. The obvious bound u(J'#-J') ≤ 2u(J') ≤ 14 is far from 5. So the abstract and Theorem 1 overclaim. This is not a minor wording issue; the entire point of the title and abstract is that the unknotting number is exactly 5. What is actually proven is the existence of a knot with unknotting number at least 5, which would still be the first such example with Alexander polynomial 1, and in that form the paper would be sound.\n\nA second concern is that Proposition 5, the step that converts a null-homologous twist into a crossing change, is cited to an upcoming paper by the author with P. Feller and only sketched here. The sketch is plausible and the statement is believable, but a referee would need to see the full proof. That is a normal thing to ask for, but it does mean the paper as written is not self-contained.\n\nOn balance: the core idea is good and the lower-bound result is probably correct. But the equality claim is unproven, and that is a load-bearing flaw. I would send it to a serious referee, with instructions to require either a proof of the upper bound or a revised theorem stating \"at least 5.\" The paper deserves attention, but it needs revision before it is publishable.","headline":"Genuinely new lower bound, but the theorem overclaims: only u(L) ≥ 5 is proven, not u(L) = 5.","tokens_in":5849,"tokens_out":3173,"would_cite":false,"duration_ms":31668,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57K18"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["unknotting number","Alexander polynomial 1","smoothly slice knot","doubly slice","amphicheiral knot","knot Floer homology","U-torsion order","Manolescu-Marengon knots"],"falsifier":"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.","tokens_in":4897,"feed_emoji":"🪢","tokens_out":24530,"duration_ms":264062,"temperature":0.7,"pith_summary":"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.","feed_headline":"A slice knot requiring 5 unknotting moves is proved to exist","feed_subtitle":"First proven smoothly slice Alexander-polynomial-1 knot that needs more than 2 crossing changes to untie.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the infinite knot family whose sixth member is the starting knot K.","marker":"[MM20]"},{"why":"Proves the torsion-order distance bound |t(K)-t(J)| <= d(K,J) used to lower-bound unknotting number.","marker":"[AE20]"},{"why":"The computer program whose output tables establish t(ell) >= ell for ell = 1 through 6.","marker":"[Sza17]"},{"why":"The companion paper containing the skeleton-proof proposition that converts a null-homologous twist into a crossing change to an S-equivalent knot.","marker":"[FL]"},{"why":"Gives the identities t(K#J) = max(t(K), t(J)) and t(-K) = t(K) used to compute t(L).","marker":"[JMZ20]"},{"why":"Shows that K#-K is smoothly doubly slice, giving L its doubly slice and amphicheiral properties.","marker":"[Zee65]"}],"fun_headline_variants":["Slice knot with Alexander polynomial 1 has unknotting number 5","First slice knot with Alexander polynomial 1 to need 5 unknotting moves","Doubly slice knot with Alexander polynomial 1 has unknotting number 5","Amphicheiral slice knot with Alexander polynomial 1 and unknotting number 5"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Slice knot with Alexander polynomial 1 has unknotting number 5","First slice knot with Alexander polynomial 1 to need 5 unknotting moves","Doubly slice knot with Alexander polynomial 1 has unknotting number 5","Amphicheiral slice knot with Alexander polynomial 1 and unknotting number 5"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002324,"raw_usage":{"total_tokens":8852,"prompt_tokens":731,"completion_tokens":8121,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":347,"completion_tokens_details":{"reasoning_tokens":8030}},"tokens_in":347,"tokens_out":8121,"duration_ms":58169,"temperature":1.0,"reasoning_tokens":8030,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:30:03.579053+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}