{"id":"ccd99fb6-c0b2-4576-9b89-bc6d166a14d8","arxiv_id":"1908.04397","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Cabling a knot transforms its immersed-curve Heegaard Floer invariant by the explicit plane map f_{p,q}; this recovers known cabling formulas and produces new independent concordance classes.","lead":"Heegaard Floer invariants of knot complements, drawn as immersed curves in a punctured torus, change under cabling by an explicit sliding and scaling map of the plane. This yields short geometric proofs of known cabling formulas and a new infinite family of topologically slice but smoothly independent knots.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's full generality rests on Proposition 12's unproved local-system extension of the merge operation; the phrase 'straightforward computation' is not a verification.","rationale":"Read in good faith, the paper gives a clear and plausible geometric picture: the cable operation is implemented by the explicit plane transformation f_{p,q}, and the examples involving the right-handed trefoil and Hom's τ and ε formulas are consistent. The central claim would be established if the merge operation from [6] is correctly imported and if its extension to local systems is valid. The most load-bearing gap is Proposition 12: it is a new assertion, not available in [6], and it is exactly what is needed to apply the formula to all components of HF(M), not only the distinguished component γ0. The paper itself flags that non-trivial local systems were not handled in [6], saying 'because non-trivial local systems are not handled there'. The proof of Proposition 12 occupies one sentence and refers to a computation that is not shown. This is a genuine correctness risk rather than a stylistic concern. If the local-system extension fails, Theorem 1 would be false as stated, even though its restriction to trivial local systems might still hold. A direct small computation with a non-diagonalizable local system would settle the issue. I therefore agree with the reader's conditional verdict; the paper should be accepted only once Proposition 12 is supplied, or the theorem is explicitly restricted to the class of curves for which the merge computation is verified.","tokens_in":16728,"tokens_out":7699,"duration_ms":82082,"concrete_test":"Perform the algebraic computation underlying Proposition 12 for a minimal nontrivial case: take ϑ to be a single type a_1 segment over F with a 2-dimensional local system Φ = [[1,1],[0,1]], and take γ to be a single c_1. Compute m(γ,ϑ) directly from the definition m(γ,ϑ) = CFDAA(P×S^1) ⊠ (γ,ϑ), using the trimodule of [3] and following the contraction through the toroidal grid of [6, Figures 10–12]. Verify that the resulting type D structure is the a_1 segment decorated by the same Φ, with no additional summands or change of basis. Repeat with a non-diagonalizable 3-dimensional Φ and with u = b_k to cover the other cases asserted in Proposition 12. If the resulting local system differs, Proposition 12 is false and Theorem 1 must be restricted to trivial local systems; if it matches, the claimed extension lands.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1 is stated for arbitrary immersed multicurves, including components decorated with non-trivial local systems. The proof applies the merge operation m(γ, ϑ) to ϑ = HF(M). Proposition 11, imported from [6], is proved only for trivial local systems. Proposition 12 extends it by asserting that for every decorated letter u, m(u, c_k) carries the same local system as u, with the proof deferred to 'a straightforward computation' in [6, Figures 10–12]. Since [6] explicitly does not handle local systems, Proposition 12 is not an imported theorem but a new claim. This is exactly the place where local-system data could interact with the gluing: the extra arrows encoding Φ pass through the c_k columns in the toroidal grid, and a basepoint, ordering, or contraction issue in the pairing with the bordered trimodule could change the holonomy on the image component. Without this check, the formula for γ_{p,q} is unverified for any K whose HF(M) has a non-trivial local system on a closed component. Thus Theorem 1 as stated is not established. Corollary 13, the geometric translation of the merge into 'add ⌊γ⌋', is also given by a descriptive proof rather than a formal verification, but the primary gap remains the local-system extension.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a concise geometric formula for the behavior of the immersed-curve invariant of knot Floer homology under cabling. Concretely, for a knot K in S^3 with associated immersed multicurve γ in the punctured torus, the paper claims that the immersed curve of the (p,q)-cable complement γ_{p,q} is obtained, after lifting to the universal cover, by applying an explicit plane map f_{p,q} to γ. The argument is based on a merge operation imported from the authors' earlier loop-calculus work with J. Rasmussen, translated into the immersed-curve language via a graphical toroidal-grid calculus. The paper also gives quick re-derivations of Hom's theorems on τ and ε under cabling, a re-proof of the L-space surgery criterion for cables, and a concordance application producing a Z^∞ summand from iterated (2,1)-cables of a knot with trivial Alexander polynomial and prescribed γ_0.","tokens_in":16963,"tokens_out":2649,"duration_ms":28532,"significance":"If the main theorem is fully established, the paper's formula is valuable: it turns cabling into a transparent geometric operation on immersed curves and immediately explains several known numerical cabling formulas. Its strengths are that the geometric picture is clear, the re-proofs of Hom's theorems are genuine consequences rather than assumed inputs, and the consistency checks against known results are convincing. The concordance application is also interesting, since it gives an independent family of topologically slice, smoothly independent knots from simple curve data. However, the paper currently depends in an essential place on an unproved local-system extension of the merge operation, so the full generality of the main theorem is not yet established.","major_comments":[{"comment":"Proposition 12 is load-bearing for Theorem 1, but its proof is not supplied: the text says only 'This is a straightforward computation' after referring to [6, Figures 10–12]. Since [6] does not handle nontrivial local systems, Proposition 12 is a new claim rather than an imported theorem. The concern is not cosmetic: in the local-system expansion, the extra arrows representing Φ interact with the c_k columns in the toroidal grid, and a basepoint, ordering, or contraction issue could change the holonomy on the image component. Without a written verification, Theorem 1 is unproved for any K whose HF-hat(M) has a nontrivial local system on a closed component. Please include the actual computation, or explicitly restrict the statement of Theorem 1 (and of Corollary 13) to trivial local systems.","section":"§1.2, Proposition 12"},{"comment":"The proof of Corollary 13 is descriptive rather than formal: it says 'This is the main thrust of Figure 13' and then asserts that the toroidal grid produces p copies of the word for ϑ with shifted c-indices, distinguishing only between components homologous to λ and nullhomologous components. Since Corollary 13 is used directly in the proof of Theorem 1 in §2.2, the argument should be written out as a verification that the grid assembly respects endpoints, periodicity, and the stated period pq. In particular, the claim that relative primality of p and q forces the new curve to make p vertical passes before closing deserves a precise justification.","section":"§1.2, Corollary 13"},{"comment":"The final step of the proof of Theorem 1 is a sequence of geometric shears described in words and pictures, but the map f_{p,q} is not given by an explicit formula or by a precise composition of piecewise-linear maps on the plane. In particular, the vertical shift (p-1)(q-1)/2 and the claim that f_{p,q} sends Z^2 to Z^2 are stated without calculation. This is not fatal, but a precise definition of f_{p,q} would make the theorem checkable and would also clarify the meaning of 'homotopic' for the lifted curves.","section":"§2.2, proof of Theorem 1"}],"minor_comments":[{"comment":"There are several typos: 'corresonding' in Theorem 1, 'the the' in the caption of Figure 7, and 'effects' in §2.1 should read 'affects'.","section":"Abstract and §2"},{"comment":"The definitions of letters a_k, b_k, c_k, d_k, and e are given informally through Figure 10 and the surrounding text; a short formal definition or an explicit reference to the corresponding notation in [6, Figure 1] would improve precision.","section":"§1.1"},{"comment":"The phrase 'a letter decorated by the trivial local system of dimension n corresponds to n parallel copies of the relevant curve segment' is clear, but the notation for the local system (V, Φ) on a cyclic word would benefit from an explicit convention about the order of composition of the endomorphisms.","section":"§1.1, local systems"},{"comment":"The captions in Figures 15 and 16 are long and partly repeat the text; labeling the key curves in the figures themselves would make the two-step shear argument easier to follow.","section":"Figure 16"}],"recommendation":"major_revision","confidential_remarks":"The central idea of the paper is sound and the geometric formula is likely correct, but the unproved local-system extension in Proposition 12 is exactly the point where the main theorem's full generality could fail. I would encourage the editor to request that the authors supply the missing computation or restrict the theorem's scope. The paper is short and well written; this is a fixable gap rather than a fundamental flaw."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe short version: this is a clean, useful note that gives a geometric formula for how the immersed-curve invariant of a knot complement transforms under cabling, and it mostly earns its claims. Theorem 1—the cable invariant is obtained by applying the explicit plane map f_{p,q} to the lift of the original curve—is the real new content. The paper also recovers Hom's epsilon/tau cabling theorems and the L-space surgery criterion in a few lines, which is a good sanity check, and it produces a new Z^∞ summand in the smooth concordance group from iterated (2,1)-cables of topologically slice knots with γ0(T2,3). That last application is neat and doesn't require heavy machinery once Theorem 1 is granted.\n\nWhat it does well: the geometric picture is persuasive, the translation from loop calculus to immersed curves is worked out carefully enough that a reader of [6] can follow, and the paper is honest about what is imported from earlier work. The quick re-proofs are real consequences, not circular: they use Theorem 1 to derive known results, and the logic runs the right direction. The citation pattern is appropriate; the heavy machinery lives in [4,5,6] and the authors say so.\n\nThe soft spot is exactly where the reader flagged. Theorem 1 is stated for arbitrary immersed multicurves, including components carrying nontrivial local systems. The proof relies on the merge operation from [6], and the extension to local systems is Proposition 12, whose proof is one sentence saying the computation in [6, Figures 10–12] carries over. That may be routine, but it is not displayed, and local systems are the sort of data that can misbehave in a gluing computation. Corollary 13, the geometric \"add ⌊γ⌋\" description, is also argued pictorially rather than formally. I don't think this is a load-bearing flaw—nothing suggests the formula fails, and the consistency checks would likely catch a wrong holonomy—but the full generality of Theorem 1 is not established by the text. A referee should ask for a real proof of Proposition 12 or a restriction of the statement to the class of curves for which the merge is verified.\n\nBottom line: this is for people working in Heegaard Floer homology, especially cable and concordance questions. It deserves a serious referee; with Proposition 12 supplied it would be a solid, publishable note. I'd bring it to reading group and would cite the formula if I worked in the area.","headline":"Clean geometric formula for cabling in immersed-curve Floer theory; the main theorem is credible and useful, but the local-system extension is asserted rather than proved.","tokens_in":17480,"tokens_out":3769,"would_cite":true,"duration_ms":36419,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K18","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Cabling a knot is one explicit shear of its immersed curves","keywords":["knot Floer homology","immersed curves","cabling","bordered Heegaard Floer homology","concordance invariants","L-space surgeries","merge operation","plane shear"],"falsifier":"Compute, from the bordered Floer complexes, the immersed multicurve for the (2,1)-cable of a knot whose complement's invariant includes a component with a nontrivial local system, and compare the result with the image of the original multicurve under f_{2,1}; a mismatch at the decorated component would refute the theorem as stated for arbitrary immersed multicurves.","tokens_in":16529,"feed_emoji":"🪢","tokens_out":6058,"duration_ms":62017,"temperature":0.7,"pith_summary":"This note establishes a formula for how the Heegaard Floer invariant of a knot complement—encoded as a collection of immersed curves in a punctured torus—changes when the knot is replaced by its (p,q)-cable. The formula says that the cable's immersed multicurve is obtained from the original by drawing p copies, shearing the plane along lines of slope q/p so that lattice points slide onto a common vertical line, and rescaling. This makes cabling a purely pictorial operation rather than a fresh three-manifold gluing computation. The authors use the formula to reprove known results on the tau and epsilon concordance invariants under cabling and to construct a new infinite family of linearly independent topologically slice knots.","feed_headline":"Cabling a knot is a single shear of its immersed curves","feed_subtitle":"One graphical recipe reproduces cable knot invariants and builds new concordance obstructions.","key_machinery":"The load-bearing object is the merge operation from the authors' earlier loop calculus, reinterpreted as a fractional plane shear. When one input to the merge is a curve made only of c_k segments, merging acts by shearing each vertical line of lattice points so that the horizontal axis is carried to the piecewise-linear curve that tracks the second input just below its lattice points. Cabling is the special case where that second input is a straight line of rational slope q/p; the shear then slides points along lines of slope q/p, and the bookkeeping of the cable framing turns this into the explicit plane map f_{p,q} appearing in the theorem.","core_discovery":"The paper proves that cabling a knot acts on its knot Floer data by an explicit homeomorphism of the punctured plane. Knot complements carry an invariant called an immersed multicurve: a finite set of immersed curves, possibly decorated with vector spaces, drawn in a punctured torus. The central theorem states that the multicurve for the (p,q)-cable is obtained from the multicurve for K by sliding each lattice point leftward along a line of slope q/p until it lands on a vertical line x = np, then compressing horizontally and stretching vertically by p, with a fixed vertical shift recorded. In drawing terms: lay p staggered copies of the original curve side by side, connect the loose ends, then slide the lattice pegs horizontally so they all line up vertically. The same map can be read as a periodic tiling of the plane by new tiles; the image of the original lattice under the map is the lattice of that tiling, and the cable's invariant is the image of the original curve under the tiling transformation.","pith_inferences":["The same plane-shear picture should describe how satellite operations act on immersed curves in general: a satellite operation is likely realized by a multivalued Lagrangian correspondence, and the cabling theorem is the first instance where that action is visibly a fractional plane shear.","The immersed-curve encoding loses only diagonal arrows from the full chain complex, so a refined curve-with-decorations invariant would be needed to lift the cabling formula to complete CFK^- complexes rather than just their curve shadows.","The asserted extension of the merge operation to nontrivial local systems could be tested explicitly on a small example with a nontrivial local system; this is the one step of the proof that is stated as a straightforward computation rather than written out.","The tiling reformulation may make the cabling action available to other invariants that admit curve or graph models in the punctured torus, provided those models satisfy the same merge rule."],"forward_implications":["The (p,q)-cable invariant can be computed by drawing p copies of the original curve, staggering them vertically by q, joining ends, and compressing horizontally; no new bordered Floer computation is needed.","The formulas of Hom for tau and epsilon under cabling are recovered directly from the first intersection of the first copy of the distinguished curve with the vertical axis.","For (2,1)-cables, the refined Phi_i counts obey parity rules: Phi_{2n} = Phi_n^{++} + Phi_n^{--} and Phi_{2n+1} = Phi_n^{+-} + Phi_{n+1}^{-+} for n at least 1.","Iterated (2,1)-cabling doubles the length of a unique maximal ++ left arc, yielding an infinite family of topologically slice knots that are linearly independent in the smooth concordance group.","If a knot's curve set contains a closed component enclosing two adjacent lattice points of different height modulo p, that knot cannot be a (p,q)-cable of another knot."],"supporting_citations":[{"why":"Supplies the immersed-curve description of Heegaard Floer homology for manifolds with torus boundary.","marker":"[4]"},{"why":"Fixes the framing conventions, the recipe deriving the curves from CFK^-, and the extraction of tau and epsilon.","marker":"[5]"},{"why":"Supplies the merge operation and its algebraic rules, imported into the proof as Proposition 11.","marker":"[6]"},{"why":"Provides the bordered trimodule calculation for P times S^1 that underlies the merge operation.","marker":"[3]"},{"why":"Supports the claim that the distinguished curve component is a concordance invariant.","marker":"[12]"},{"why":"Contains the earlier tau and epsilon cabling results that Theorem 1 is used to reprove.","marker":"[11]"},{"why":"Contains the earlier L-space surgery criterion for cables that is reproved from the curve picture.","marker":"[10]"},{"why":"Supplies the Phi_i concordance homomorphisms whose behavior under (2,1)-cabling is computed.","marker":"[1]"}],"fun_headline_variants":["Cabling is a single shear of the knot's curve","One shear maps cable invariants from knot data","Cabling: slide lattice pegs, stretch, done","Immersed curves reveal cabling as a simple shear","Cable knots: one geometric move captures it all"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on an earlier gluing construction being correct for curves with extra decorations; the paper asserts rather than proves that extension, so if that gluing construction fails for decorated curves, the cabling formula fails.","fun_headline_variants_meta":{"raw":{"variants":["Cabling is a single shear of the knot's curve","One shear maps cable invariants from knot data","Cabling: slide lattice pegs, stretch, done","Immersed curves reveal cabling as a simple shear","Cable knots: one geometric move captures it all"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1148,"prompt_tokens":798,"completion_tokens":350,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":414,"completion_tokens_details":{"reasoning_tokens":272}},"tokens_in":414,"tokens_out":350,"duration_ms":3897,"temperature":1.0,"reasoning_tokens":272,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:43:26.547285+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, from the bordered Floer complexes, the immersed multicurve for the (2,1)-cable of a knot whose complement's invariant includes a component with a nontrivial local system, and compare the result with the image of the original multicurve under f_{2,1}; a mismatch at the decorated component would refute the theorem as stated for arbitrary immersed multicurves.","supporting_citations":[{"cited_title":"A calculus for bordered Floer homology","cited_arxiv_id":"1508.05445","evidence_quote":"Supplies the merge operation and its algebraic rules, imported into the proof as Proposition 11."},{"cited_title":"Bordered Heegaard Floer homology and graph manifolds","cited_arxiv_id":null,"evidence_quote":"Provides the bordered trimodule calculation for P times S^1 that underlies the merge operation."},{"cited_title":"A survey on Heegaard Floer homology and concordance","cited_arxiv_id":null,"evidence_quote":"Supports the claim that the distinguished curve component is a concordance invariant."},{"cited_title":"Bordered Heegaard Floer homology and the tau-invariant of cable knots","cited_arxiv_id":null,"evidence_quote":"Contains the earlier tau and epsilon cabling results that Theorem 1 is used to reprove."},{"cited_title":"A note on cabling and L-space surgeries","cited_arxiv_id":null,"evidence_quote":"Contains the earlier L-space surgery criterion for cables that is reproved from the curve picture."}],"review_version":1}