{"id":"750b9c3e-2868-46e4-9fd7-3bb696709540","arxiv_id":"2608.06625","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new height invariant from immersed-curve knot Floer homology gives rigidity results for knots that ribbon-concord to cable knots.","lead":"This paper studies which knots can sit below a cable knot in the ribbon concordance order, a directed version of knot equivalence. It introduces a new obstruction from knot Floer homology and proves the proposed rigidity holds for several large classes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.3's h_min monotonicity rests on an unproved assertion that ribbon concordance gives a literal inclusion of immersed curves; this underpins Theorems 1.2, 1.3, and 1.10.","rationale":"I read the paper as a serious attempt to verify Conjecture 1.1 in special cases using a new numerical invariant h_min. The cabling formulas (Propositions 2.5 and 2.6) are clearly derived from the immersed-curve recipe of [HW23], and the topological arguments in the appendix are careful and give useful background. The central chain of reasoning is: monotonicity of h_min under ribbon concordance plus the cabling formula for h_min force rigidity for cables, torus knots, and genus-one knots. The weakest point in this chain is the monotonicity step: Proposition 2.3 asserts that a ribbon concordance induces a literal subset of immersed curves, which would immediately give the height inequality. However, the authors do not supply a proof or reference for this geometric assertion, and it is not an obvious consequence of Zemke's algebraic injectivity. Since Proposition 2.3 is used in Theorems 1.2, 1.3, and 1.10, any failure or gap here undermines the central claims. I agree with the reader's identification of this as the weakest assumption. A second gap is the F-summand claim for fibered knots cited to [BVV18], but that is used only in the primeness results and is less central. Given that the main engine rests on an unproved step, the appropriate verdict remains CONDITIONAL: the theorems are plausible and likely correct if the missing proof is supplied, but they are not yet established as written.","tokens_in":15854,"tokens_out":13966,"duration_ms":120047,"concrete_test":"Attempt to prove Proposition 2.3 directly from Zemke's injective filtered chain map [Zem19, Theorem 1.1] without passing through the immersed-curve subset assertion: define h_min algebraically as the minimum Alexander-grading gap of a homologically inessential U- or V-tower summand in CFK, and show that an injective grading-preserving map forces h_min(J) ≥ h_min(K). If such a derivation succeeds, the geometric language in the proof is a harmless shorthand; if it fails or produces a counterexample, the paper's central monotonicity result is unproved and the main theorems lose their engine.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the assertion in the proof of Proposition 2.3 that Zemke's injectivity theorem implies a literal inclusion of immersed curves: \"In terms of immersed curves, this means that γ(J) is a subset of γ(K).\" No proof or citation is given for this geometric translation, and it is not a formal consequence of the algebraic statement. Zemke's theorem gives an injective grading-preserving filtered map on CFK complexes; the immersed-curve invariant is a decorated-curve encoding that packages the same homology data, but an injection of complexes does not automatically imply containment of the associated curves. This assertion is used to prove the monotonicity inequality h_min(J) ≥ h_min(K) for J ≤ K, which is then invoked in Theorem 1.2 (swapped cable case), Theorem 1.3 (primeness contradiction), and Theorem 1.10 (genus-one case). If the subset claim is false or even merely unproved, the contradictions that drive these theorems do not follow, and the partial verifications of Conjecture 1.1 are unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies ribbon concordance to cable knots, proposing a conjecture (Conjecture 1.1) that a nontrivial ribbon-concordance predecessor of a (p,q)-cable must itself be a (p,q)-cable. It proves the conjecture under additional hypotheses: when the predecessor is a cable of the same companion (Theorem 1.2), a torus knot (Theorem 1.9), or a genus-one knot (Theorem 1.10), and it verifies the conjecture for a class of slice companions with zero simplicial volume (Corollaries 1.4 and 1.5). The main technical tool is a new numerical invariant h_min(K), the minimum height of a homologically inessential component in the immersed-curve reformulation of knot Floer homology, together with a claimed monotonicity under ribbon concordance (Proposition 2.3) and a cabling formula (Proposition 2.5). The paper also proves a primeness criterion for fibered knots (Theorem 1.6) and records concordance-classification results for cables in Appendix A.","tokens_in":15886,"tokens_out":20498,"duration_ms":168195,"significance":"If the central monotonicity statement can be established, the paper gives the first systematic evidence for a natural conjecture about the ribbon-concordance order, and h_min is a clean, parameter-free obstruction that may be useful in other problems. The paper is largely self-contained and transparent: the main proofs are explicit, and the authors properly credit the prior knot Floer results (Zemke's injectivity, the cabling formula for immersed curves, the F-summand input for fibered knots) on which they rely. The results include concrete computational examples (Corollary 1.5) and a careful treatment of concordance of cables in the appendix. However, the proof of Proposition 2.3 contains a load-bearing gap: the geometric translation of Zemke's algebraic injection to an inclusion of immersed curves is asserted without proof, and this step is used in Theorems 1.2, 1.3, 1.9, and 1.10. The paper is therefore not yet in publishable form.","major_comments":[{"comment":"The proof of monotonicity h_min(J) >= h_min(K) for J <= K rests on the sentence \"In terms of immersed curves, this means that γ(J) is a subset of γ(K).\" This is an unproved geometric translation of Zemke's injectivity theorem ([Zem19, Theorem 1.7]). Zemke's result is an injective grading-preserving map of filtered knot Floer complexes; it does not formally imply that the associated immersed curves are nested. Since h_min is defined directly from the immersed curves, the inequality requires the stronger statement that every homologically inessential component of γ(J) appears as a component of γ(K). Please provide a proof or a precise reference for this inclusion, or give an alternative argument for the monotonicity of h_min.","section":"Section 2, Proposition 2.3"},{"comment":"The primeness conclusion depends on the assertion \"γ(J) has a homologically inessential component, which implies that both γ(K_{p,q}) and γ(K) also have a homologically inessential component [Zem19, HW23].\" The first implication (from J to K_{p,q}) again uses the unproved subset claim from Proposition 2.3; the second (from K_{p,q} to K) is an implicit use of the contrapositive of the cabling formula in Proposition 2.5, which the paper never states. Once Proposition 2.3 is supplied, this argument should be written out explicitly, since it is needed to reach the contradiction h_min(J) >= h_min(K_{p,q}) >= 3.","section":"Section 4, Theorem 1.3"}],"minor_comments":[{"comment":"When applying Proposition A.4 in the proof of Theorem 1.9, the authors should state explicitly that the torus knot J = T_{r,s} is viewed as the (r,s)-cable of the unknot, which is slice. Without this clarification, the hypotheses of Proposition A.4 are not visibly satisfied.","section":"Section 5, Theorem 1.9 proof"},{"comment":"The convention h_min(K)=0 when γ(K)=γ_0(K) creates an ambiguity if an inessential component of height 0 could exist. Please clarify why height-0 inessential components do not occur, or modify the definition to distinguish \"no inessential components\" from \"minimum height 0.\"","section":"Section 2, Remark 2.1"},{"comment":"The proof of Lemma 2.2 is very terse; expanding it would help, since the relationship between the height of a curve component and the span of Alexander gradings of HFK is a central geometric fact used repeatedly in the paper.","section":"Section 2, Lemma 2.2"},{"comment":"The change-of-basis argument to split off the box {a,b,c,d} as a direct summand is sketched. I recommend spelling out the elimination of incoming arrows in more detail, in particular why the replacements do not introduce new arrows into the box.","section":"Section 3, Proposition 3.1"},{"comment":"The sentence \"implies that both γ(K_{p,q}) and γ(K) also have a homologically inessential component\" is misleading: the text should say that an inessential component of γ(J) forces one in γ(K_{p,q}) and hence, by the cabling formula, one in γ(K). The current phrasing invites confusion about the logical order.","section":"Section 4, Theorem 1.3"},{"comment":"The author name is typeset as \"JUNGHW AN PARK,\" which should be \"JUNGHWAN PARK.\"","section":"Title page"}],"recommendation":"major_revision","confidential_remarks":"The central gap is the unproved assertion that ribbon concordance induces an inclusion of the associated immersed curves. The rest of the paper's structure is credible, and the appendix computations are valuable, so I would be willing to look at a revision that supplies a proof or a citable reference for this geometric inclusion. If the inclusion statement is in fact known in the literature, the authors should cite it explicitly and reconcile it with the algebraic statement of Zemke's theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper introduces a genuinely new invariant, h_min, defined from the immersed-curve model of knot Floer homology, and uses it to verify a natural conjecture about ribbon concordance to cable knots in three substantial cases: same-companion cables, torus knots, and genus-one predecessors. The cabling formula for h_min (Proposition 2.5) and the monotonicity claim (Proposition 2.3) are the engine, and the applications are real, not routine repackaging. The paper is well organized, the definitions are precise, and the appendix on cable concordance via signature integrals is a useful service. The primeness criterion for fibered knots is also interesting, though it leans on an unproved assertion about the F-summand in HFK of fibered knots that is merely cited to BVV18.\n\nThe soft spot is exactly where the stress-test points: the proof of Proposition 2.3 asserts that Zemke's injectivity on knot Floer complexes 'in terms of immersed curves, means that γ(J) is a subset of γ(K).' That geometric translation is not proved and is not a formal consequence of the algebraic statement. It is load-bearing: every monotonicity argument, including the inheritance of inessential components in Theorem 1.3 and the contradictions in Theorems 1.2 and 1.10, depends on this inclusion. If the inclusion is false, the main rigidity theorems do not follow. This is a serious gap, but it is also the kind of thing that may be true and provable; the authors may simply have regarded it as obvious.\n\nThe secondary issues are minor. The F-summand input is cited without derivation, and the paper uses ChatGPT for part of a computation, but the authors say they verified it, which is fine. There is no circularity: the theorems do not assume the conjecture they test, and the citations to prior work are standard background.\n\nOverall, this is a serious paper from the right people, with a new invariant and real partial results. The curve-inclusion step needs to be supplied or replaced before the results are fully trustworthy. A referee who knows immersed curves should be asked to check that step carefully. I would send this to peer review without hesitation, and I would probably cite it once the inclusion is sorted out.","headline":"Promising new obstruction to ribbon concordance with a load-bearing unproved curve-inclusion step; worth refereeing.","tokens_in":16616,"tokens_out":1987,"would_cite":true,"duration_ms":20320,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57R58"],"pacs":[],"model":"deepseek-v4-flash","headline":"Cable knots are rigid: if a nontrivial knot admits a ribbon concordance into a (p,q)-cable, the paper proves it must itself be the (p,q)-cable in three broad cases.","keywords":["ribbon concordance","cable knots","knot Floer homology","immersed curves","minimum-height invariant","torus knots","satellite knots","genus bounds"],"falsifier":"Search for a ribbon concordance $J \\le K_{p,q}$ with $J$ not a $(p,q)$-cable, or for a nontrivial $K$ with $K_{q,p} \\le K_{p,q}$ and $p \\ne q$; finding either would contradict Theorems 1.2 and Conjecture 1.1.","tokens_in":15427,"feed_emoji":"🪢","tokens_out":8284,"duration_ms":67047,"temperature":0.7,"pith_summary":"This paper targets a structural conjecture about the directed version of knot concordance: if a nontrivial knot $J$ admits a ribbon concordance into a $(p,q)$-cable $K_{p,q}$ (with $p>1$), then $J$ must itself be the $(p,q)$-cable of some knot. The authors prove the conjecture in three settings: when $J$ is already a cable of the same companion $K$, when $J$ is a torus knot, and when $J$ has genus one. They also show that any nontrivial predecessor of a cable of a fibered knot is prime, and that for a broad family of slice companions the predecessor must be a cable. The engine is a new invariant, the minimum-height invariant $h_{\\min}$, extracted from the immersed-curve formulation of knot Floer homology, which is monotone under ribbon concordance and obeys the cabling formula $h_{\\min}(K_{p,q}) = p \\cdot h_{\\min}(K) - p + 1$. If the conjecture is right, the class of cables is closed under taking ribbon-concordance predecessors, a rigidity property that ordinary concordance cannot see.","feed_headline":"Cable knots are rigid under ribbon concordance","feed_subtitle":"A new height invariant from Floer homology shows predecessors of a (p,q)-cable must also be (p,q)-cables.","key_machinery":"The load-bearing object is $h_{\\min}(K)$, the minimum height of a homologically inessential component among the immersed curves $\\gamma(K)$ that knot Floer homology associates to a knot in a marked torus; height is the vertical spread of a curve between its highest and lowest intersection levels, and $h_{\\min}$ is set to $0$ when no such component exists. The paper proves that $h_{\\min}$ is monotone under ribbon concordance ($J \\le K$ forces $h_{\\min}(J) \\ge h_{\\min}(K)$ whenever the former is nonzero) and that cabling multiplies it almost linearly: $h_{\\min}(K_{p,q}) = p \\cdot h_{\\min}(K) - p + 1$. Together with the genus detection of knot Floer homology and a lemma showing that connected sums of fibered knots produce $h_{\\min} = 2$, these formulas turn a numerical count of curve height into a rigidity machine that rejects non-cable predecessors.","core_discovery":"The central claim is that ribbon concordance into a nontrivial cable is highly rigid: the predecessor inherits the cable structure. Formally, Conjecture 1.1 states that if $p>1$ and a nontrivial knot $J$ satisfies $J \\le K_{p,q}$, then $J$ is itself a $(p,q)$-cable, and the paper establishes this in the cases where $J$ is a cable of the same companion (then $J$ is exactly $K_{p,q}$), where $J$ is a torus knot (then $J = T_{p,q}$), and where $J$ has genus one (then $J = T_{p,q}$ and $\\{p, |q|\\} = \\{2,3\\}$). The rigidity is forced by two height measurements attached to a knot's immersed-curve invariant: the essential component's height $h_0$, which is invariant under ordinary concordance, and the new minimum inessential height $h_{\\min}$, which is monotone under ribbon concordance. The cabling formulas $h_0(K_{p,q}) = p \\cdot h_0(K) + (p-1)(|q|-1)$ and $h_{\\min}(K_{p,q}) = p \\cdot h_{\\min}(K) - p + 1$ turn these measurements into obstructions that rule out every alternative predecessor.","pith_inferences":["If the full conjecture holds, the ribbon-concordance partial order restricts to a partial order on cable parameters: compatibility of a predecessor cable with a target cable is determined by winding number and framing in a way that mirrors concordance but is strictly more rigid.","The monotonicity of $h_{\\min}$ is a promising general obstruction: one could compute $h_{\\min}$ for known ribbon-concordance pairs to test whether the asserted geometric inclusion of immersed curves holds in practice, independent of the algebraic proof.","The same height machinery may apply to iterated cables or general satellites, where the cabling formula would need a replacement for the $p$-scaling rule; a counterexample there would localize how much cable rigidity depends on the linear cabling formula."],"forward_implications":["If $J \\le K_{p,q}$ and $J$ is a cable of $K$, then $J$ is exactly $K_{p,q}$; in particular, a companion knot $K$ cannot ribbon-concord to any of its nontrivial cables.","A nontrivial ribbon-concordance predecessor of a cable of any fibered knot is prime, so composite knots cannot flow into such cables.","For a fibered companion with zero simplicial volume, every nontrivial predecessor must itself be a cable, verifying Conjecture 1.1 for that class.","A genus-one predecessor of a cable must be the trefoil $T_{2,3}$ or its mirror, so no other genus-one knot ribbon-concordes to a cable.","For any companion with $h_0(K) \\ne 0$, the genus of a predecessor $J$ satisfies $g(J) \\ge p + g(T_{p,q})$."],"supporting_citations":[{"why":"Supplies the ribbon-concordance injection of knot Floer complexes from which $h_{\\min}$ monotonicity is derived.","marker":"[Zem19]"},{"why":"Defines the immersed-curve invariant in a marked torus that carries the height measurements.","marker":"[HR W24]"},{"why":"Provides the cabling recipe for immersed curves, yielding both cabling formulas used as obstructions.","marker":"[HW23]"},{"why":"Shows fibered knots have an $\\mathbb{F}$-summand in $HFK^-$, used to force $h_{\\min}=2$ for composite fibered predecessors.","marker":"[BVV18]"},{"why":"Characterizes zero-simplicial-volume knots as the closure of the unknot under connected sum and cabling.","marker":"[Gor83]"},{"why":"Gives monotonicity of simplicial volume under ribbon concordance, used to conclude a predecessor is a cable.","marker":"[AR26]"},{"why":"Records that a ribbon-concordance predecessor of a fibered knot is fibered, needed in Theorem 1.3.","marker":"[Miy18]"},{"why":"Provides genus detection by knot Floer homology, used for the $h_{\\min}$ genus bounds and genus monotonicity.","marker":"[OS04a]"},{"why":"Gives the signature cabling formula that pins down cable parameters under ordinary concordance.","marker":"[Lit79]"},{"why":"Computes the average signature of torus knots, used to exclude swapped or degenerate cable parameters.","marker":"[BO10]"}],"fun_headline_variants":["Ribbon concordance to a cable forces cable predecessor","New height invariant proves cable rigidity","Cable predecessors must be cables, Floer heights say","Height invariant pins cable predecessors","Ribbon concordance exposes cable structure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes, rather than proves, that the ribbon-concordance injection on knot Floer complexes appears in immersed-curve form as a literal inclusion of the predecessor's curve in the target's curve; if that geometric translation fails, $h_{\\min}$ monotonicity, and with it the main rigidity theorems, does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Ribbon concordance to a cable forces cable predecessor","New height invariant proves cable rigidity","Cable predecessors must be cables, Floer heights say","Height invariant pins cable predecessors","Ribbon concordance exposes cable structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000178,"raw_usage":{"total_tokens":1302,"prompt_tokens":959,"completion_tokens":343,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":275}},"tokens_in":575,"tokens_out":343,"duration_ms":3572,"temperature":1.0,"reasoning_tokens":275,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T04:11:42.101220+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a ribbon concordance $J \\le K_{p,q}$ with $J$ not a $(p,q)$-cable, or for a nontrivial $K$ with $K_{q,p} \\le K_{p,q}$ and $p \\ne q$; finding either would contradict Theorems 1.2 and Conjecture 1.1.","supporting_citations":[],"review_version":1}