{"id":"afad2bea-a9ef-4599-a33b-5529f0d57750","arxiv_id":"2505.01217","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A rational homology solid torus is a Heegaard Floer homology solid torus if and only if the Dehn filling along its rational longitude contains a non-separating 2-sphere.","lead":"The paper proves that a rational homology solid torus is a Heegaard Floer homology solid torus exactly when one of its Dehn fillings contains a non-separating 2-sphere. This converts an algebraic property into a quick topological test and yields a complete classification of Seifert fibered examples.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim depends on the quoted geometric characterization from [HRW24, Prop. 7.11]; the proof of Proposition 4.1 does not spell out how the 'through the basepoint z' clause is used, so a mismatch there would undo Theorem 1.1.","rationale":"After working through the surgery exact sequence in Section 3 and the two cases in Proposition 4.1, no independent flaw was found. The proof is logically sound conditional on the quoted equivalence in [HRW24, Proposition 7.11], and the paper uses it in both directions of Theorem 1.1. The only real soft spot is the precise geometric content of condition (3) in that proposition, especially the qualifier 'through the basepoint z', which Proposition 4.1 never explicitly addresses. If the quoted condition is exactly as used, the theorem follows; if not, the central claim would be unsupported. This matches the reader's weakest-assumption analysis, and since no concrete error was identified, the verdict should remain ACCEPT with the same moderate confidence.","tokens_in":7728,"tokens_out":54565,"duration_ms":516638,"concrete_test":"Check the exact statement and proof of [HRW24, Proposition 7.11], and verify directly that condition (3) implies the two-case dichotomy used in Proposition 4.1: each component of dHF(M) after pulling tight is either homotopic to λ^j for some j or regularly homotopic to a curve disjoint from λ. In particular, confirm that for every nonzero j the cover argument in Case 2 goes through, and that the 'through the basepoint z' clause does not introduce components that prevent the regular homotopy from staying disjoint from the preimage of λ. If this verification fails, Proposition 4.1 and hence Theorem 1.1 need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The forward direction of Theorem 1.1 reduces to Proposition 4.1, whose hypothesis is exactly condition (3) of the quoted [HRW24, Proposition 7.11]: the immersed curve dHF(M) is supported in a neighborhood of the rational longitude through the basepoint z, after pulling tight. The proof splits the curve into components C_i. In Case 2, when C_i is homotopic to λ^j, it asserts that C_i is regularly homotopic to a curve disjoint from λ, using the cover Σ of the punctured torus corresponding to the subgroup <λ^j>. This step requires that the preimage of λ in Σ is a single embedded loop representing a generator of π1(Σ), and that an isotopy inside the cylinder can be chosen so that its projection avoids λ until the endpoint. The paper verifies neither property directly from the quoted condition, and the phrase 'through the basepoint z' suggests ν(λ) may not be a small annulus around λ but a larger set containing the basepoint, which could change the topology of the cover or introduce additional components. If the geometric content of condition (3) is weaker or differs from what Proposition 4.1 assumes, the equivalence 'HFST if and only if M(λ) contains a non-separating 2-sphere' would not follow. This is the load-bearing bridge identified by the reader; both directions of Theorem 1.1 pass through it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves a characterization of Heegaard Floer homology solid tori: a rational homology solid torus M is an HFST if and only if the Dehn filling of M along its rational longitude λ contains a non-separating 2-sphere. The forward direction is established in Section 3 via the surgery exact sequence with twisted coefficients and the Hanselman-Rasmussen-Watson characterization of HFSTs (Prop. 1.2). The reverse direction is proved in Section 4 using the immersed curve reformulation: assuming dHF(M) is supported near λ, the paper computes the twisted Floer homology of M(λ) as a tensor product of the solid torus module S with the bordered invariant, splitting according to components of the immersed curve. Section 5 applies Theorem 1.1 to classify Seifert fibered HFSTs.","tokens_in":8042,"tokens_out":44314,"duration_ms":416454,"significance":"The main theorem is an appealing and potentially very useful bridge between an algebraic condition (invariance under Dehn twists along the rational longitude) and a simple topological condition (a reducible Dehn filling). The proof is concise and relies on recent deep results rather than new technical machinery, which makes the paper accessible. If the gap in Proposition 4.1 is repaired, the result would be a significant contribution to the understanding of bordered Floer homology and L-space fillings. The paper also provides a clean classification of Seifert fibered HFSTs.","major_comments":[{"comment":"The proof that a component C_i homotopic to λ^j is regularly homotopic into a neighborhood of λ is under-specified. The argument uses the cover Σ corresponding to ⟨λ^j⟩ and asserts that C_i lifts to an embedded generator loop and that the preimage of λ is also an embedded generator loop, so any two such loops in the cylinder are regularly homotopic. However, the text does not explain how the 'through the basepoint z' clause in Prop. 1.2(3) is used, and if ν(λ) is not an annular neighborhood the topology of Σ could be different. The subsequent step applying Observation 2.3 to conclude ι1 [CFD(C_i) = 0 also requires the curve to be disjoint from the circle [0,1]×{0}, which a curve parallel to λ generally is not. Since both directions of Theorem 1.1 pass through Proposition 4.1, this is a load-bearing point that needs a detailed justification.","section":"Section 4, Proposition 4.1, Case 2"},{"comment":"The assertion 'By hypothesis, C_i is regularly homotopic to a curve which is disjoint from λ' is not immediate from the quoted hypothesis that dHF(M) is supported in a neighborhood of λ through the basepoint after pulling tight. The authors should explain why this neighborhood condition implies the existence of such a regular homotopy for components not homotopic to a power of λ; this is necessary for the admissibility and vanishing conclusion H_*(S⊠P) = 0.","section":"Section 4, Proposition 4.1, Case 1"}],"minor_comments":[{"comment":"The sentence 'Projecting that regular homotopy to the punctured torus gives (and stopping just before the end) gives the desired regular homotopy' contains a duplicated 'gives'.","section":"Section 4, Case 2"},{"comment":"The phrase 'an arc α in F which is the generator of H1(F,∂F) and geometrically dual to α' should read 'geometrically dual to η', since the arc is meant to be dual to the core curve η.","section":"Section 5, Möbius band case"},{"comment":"The final sentence of the proof, 'the dimension of S⊠P is at least this large,' is unclear; it should specify which homology group's dimension is being compared.","section":"Lemma 2.2"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is likely correct, but the proof of Proposition 4.1 contains a subtle step that needs to be written out carefully. The authors should verify and spell out the use of the 'through the basepoint' condition and the application of Observation 2.3; if that can be done, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look for anyone working on bordered Floer or L-space fillings. The main result is a clean iff: a rational homology solid torus M is an HFST exactly when filling along the rational longitude gives a non-separating 2-sphere. That's genuinely new; prior characterizations by Gillespie and HRW22 only covered the L-space filling regime. The forward direction is a neat surgery exact sequence argument. The reverse direction is a short reduction to the immersed curve characterization, and the Seifert fibered classification is a useful byproduct.\n\nThe one spot I'd want a referee to press is Proposition 4.1, Case 2. The proof that a component homotopic to λ^j is regularly homotopic into a neighborhood of λ is terse: it lifts to the cyclic cover, says any two generator loops in a cylinder are regularly homotopic, and projects back. I think the argument is sound; after pulling tight, the curve should sit in a small annulus around λ, and the cover argument is standard. The 'through the basepoint z' clause in the quoted HRW condition could matter if the neighborhood is larger than expected, but I don't see it as a real obstacle here. Still, it's the least spelled-out step in the paper and deserves a written expansion.\n\nThe citation pattern is fine. The paper leans on HRW24's Proposition 1.2 and AL19's vanishing theorem; both are independent anchors, and the self-citation to AL19 is legitimate because that theorem is established. No fitting, no invented parameters.\n\nBottom line: the paper will likely become the standard test for HFSTs. It's not a branch-reshaper, but it's a solid subfield result. I'd send it to a serious referee; my own verdict would be accept after asking for the Case 2 clarification.","headline":"A crisp new iff for Heegaard Floer solid tori, built from known machinery; the proof has one terse step that deserves referee pressure but nothing fatal.","tokens_in":8558,"tokens_out":7486,"would_cite":true,"duration_ms":79156,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K18"],"pacs":[],"model":"deepseek-v4-flash","headline":"A rational homology solid torus is a Heegaard Floer homology solid torus if and only if filling along its rational longitude yields a non-separating 2-sphere.","keywords":["Heegaard Floer homology","bordered Floer homology","solid torus","Dehn filling","immersed curves","rational longitude","non-separating 2-sphere","Seifert fibered spaces"],"falsifier":"One concrete check: take a rational homology solid torus $M$ whose rational-longitude filling $M(\\lambda)$ is a connected sum of lens spaces, so it contains a non-separating 2-sphere, and compute the immersed curve invariant $\\widehat{dHF}(M)$ after pulling tight. The theorem predicts the curve is supported in a neighborhood of $\\lambda$; if any strand escapes that neighborhood, the equivalence fails.","tokens_in":7551,"feed_emoji":"🍩","tokens_out":9489,"duration_ms":89723,"temperature":0.7,"pith_summary":"The paper establishes a purely topological detection criterion for a class of bordered 3-manifolds known as Heegaard Floer homology solid tori (HFSTs): a rational homology solid torus is an HFST exactly when the Dehn filling along its rational longitude contains a non-separating 2-sphere. HFSTs are defined by an algebraic invariance, namely that their bordered Floer invariant is unchanged by Dehn twisting along the rational longitude, so the theorem turns a difficult algebraic condition into a single 3-manifold-topology check. The criterion explains and unifies known examples, including knot complements in reducible manifolds, and it yields a complete classification of Seifert fibered HFSTs.","feed_headline":"One Dehn filling detects Floer solid tori","feed_subtitle":"If filling the rational longitude gives a non-separating 2-sphere, the bordered invariant is unchanged by Dehn twists.","key_machinery":"The load-bearing object is the immersed curve invariant $\\widehat{dHF}(M)$: an immersed 1-manifold, possibly with local systems, in the punctured torus that packages the bordered Floer homology of a torus-boundary manifold. The decisive identity is Proposition 1.2, quoted from [HRW24, Proposition 7.11], which says $M$ is an HFST if and only if $\\widehat{dHF}(M)$, after pulling tight, is supported in a neighborhood of the rational longitude $\\lambda$. The proof then uses the twisted-coefficient pairing theorem to convert that support condition into vanishing of $H_*(\\underline{\\mathcal{S}} \\boxtimes \\widehat{CFD}(M))$, with the auxiliary module $\\underline{\\mathcal{S}}$ built from the 0-framed solid torus over $\\mathbb{F}_2[[t,t^{-1}]]$, and the vanishing theorem of [AL19] that identifies this vanishing with the presence of a non-separating 2-sphere in $M(\\lambda)$.","core_discovery":"The central claim is Theorem 1.1: for a rational homology solid torus $M$ with rational longitude $\\lambda$, $M$ is a Heegaard Floer homology solid torus if and only if the Dehn filling $M(\\lambda)$ contains a non-separating 2-sphere. The proof passes through the immersed-curve reformulation of bordered Floer homology: $M$ is an HFST precisely when its immersed curve invariant $\\widehat{dHF}(M)$, pulled tight, is supported in a neighborhood of $\\lambda$. From that support condition the authors split the curve into components either non-parallel or parallel to $\\lambda$, show each component pairs trivially with a model module over the Laurent series ring $\\mathbb{F}_2[[t,t^{-1}]]$, and conclude that the twisted Floer homology of $M(\\lambda)$ vanishes, which by a known criterion is equivalent to $M(\\lambda)$ containing a homologically essential 2-sphere. The converse direction uses the surgery exact triangle with twisted coefficients to show that knot complements in reducible 3-manifolds with $b_1=1$ are always HFSTs.","pith_inferences":["The same surgery-exact-triangle mechanism may detect other algebraic invariance conditions: any bordered invariant that is unchanged under a rational-longitude twist should force a reducible filling, generalizing the HFST phenomenon to other Floer-theoretic settings.","The immersed-curve support condition suggests an algorithmic route: from a Heegaard diagram or surgery description, draw $\\widehat{dHF}(M)$ and check whether it lies in a neighborhood of $\\lambda$, giving a computable test for reducibility of $M(\\lambda)$.","The Seifert classification leaves open the analogous question for graph manifolds; the reducible-filling criterion gives a concrete first obstruction to test there."],"forward_implications":["HFST status can be checked by a single Dehn filling: fill along the rational longitude and look for a non-separating 2-sphere.","Every HFST arises as the complement of a knot of infinite order in a closed reducible 3-manifold with $b_1=1$, so the class is exactly the class of such knot complements.","For Seifert fibered rational homology solid tori, the HFST condition is completely classified: the base orbifold is a Möbius band, with or without cone points, or the manifold is $D^2(0;p/q,-p/q)$.","Together with earlier L-space filling results, the theorem closes the remaining case where no filling is an L-space: there, HFSTs are precisely detected by the reducible rational-longitude filling."],"supporting_citations":[{"why":"Supplies the equivalence between being an HFST and the immersed curve $\\widehat{dHF}(M)$ being supported in a neighborhood of the rational longitude, which both directions of Theorem 1.1 use.","marker":"[HRW24, Proposition 7.11]"},{"why":"Provides the twisted-coefficient pairing theorem used in Formula (1) to compute the Floer homology of $M(\\lambda)$ from bordered invariants.","marker":"[LOT18, Theorem 9.44]"},{"why":"Gives the criterion that vanishing of twisted Heegaard Floer homology over $\\mathbb{F}_2[[t,t^{-1}]]$ is equivalent to the presence of a non-separating 2-sphere.","marker":"[AL19, Theorem 7.11]"},{"why":"Supplies the twisted surgery exact triangle used in Proposition 3.1 to show knot complements in reducible manifolds are HFSTs.","marker":"[OSz04a, Theorem 9.21]"},{"why":"Establishes that admissible train tracks with local systems are represented by immersed curves, so the support condition can be read off a curve invariant.","marker":"[HRW24, Theorem 1.5]"}],"fun_headline_variants":["One Dehn filling with a 2-sphere detects Floer solid tori","If a filling gives a non-separating sphere, the torus is Floer","Characterizing Floer solid tori by a single Dehn filling","A non-separating sphere in the filling characterizes Floer solid tori","Solid tori are Floer iff one filling has a non-separating sphere"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing bridge is a previously established equivalence (cited as [HRW24, Proposition 7.11]) between being a Heegaard Floer homology solid torus and having the immersed curve invariant supported in a neighborhood of the rational longitude; if that equivalence has an unhandled edge case, the topological criterion in Theorem 1.1 would not follow.","fun_headline_variants_meta":{"raw":{"variants":["One Dehn filling with a 2-sphere detects Floer solid tori","If a filling gives a non-separating sphere, the torus is Floer","Characterizing Floer solid tori by a single Dehn filling","A non-separating sphere in the filling characterizes Floer solid tori","Solid tori are Floer iff one filling has a non-separating sphere"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000427,"raw_usage":{"total_tokens":2121,"prompt_tokens":814,"completion_tokens":1307,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":1199}},"tokens_in":430,"tokens_out":1307,"duration_ms":9864,"temperature":1.0,"reasoning_tokens":1199,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:22:47.075465+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check: take a rational homology solid torus $M$ whose rational-longitude filling $M(\\lambda)$ is a connected sum of lens spaces, so it contains a non-separating 2-sphere, and compute the immersed curve invariant $\\widehat{dHF}(M)$ after pulling tight. The theorem predicts the curve is supported in a neighborhood of $\\lambda$; if any strand escapes that neighborhood, the equivalence fails.","supporting_citations":[],"review_version":1}