{"id":"9c1da192-dfc6-4e6f-b172-6dd4104cc710","arxiv_id":"2509.09615","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Alternative link diagrams are visually prime: a non-prime link with an alternative diagram always has a visible decomposition circle, subsuming previous results for alternating, positive, and homogeneous braid links.","lead":"The paper extends open-book monodromy to all incompressible surfaces in 3-manifolds, using partial self-maps on arcs, and applies this to prove that alternative link diagrams are visually prime: any non-prime alternative link visibly splits in its diagram. It also gives a contact-geometry-free proof that strongly quasipositive surfaces are right-veering.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 6.2 rests on an unproved structural assertion: a fixed essential arc in a boundary-connected-sum vertex surface must cut out a disk with a subcollection of summing arcs. Proposition 3.7 does not imply this, and the boundary-isotopy step preserving fixedness is also unjustified.","rationale":"The reader's weakest assumption identifies the same load-bearing step in Theorem 6.2: the passage from a fixed essential arc in the vertex surface to a visible disk-and-summing-arc decomposition. I agree this is the point where the central proof is least secure. The surrounding machinery—Lemma 3.1, Theorem 4.2, and Proposition 5.1—is substantial and mostly coherent, but Theorem 6.2's final construction of an honest decomposition circle depends on a geometric assertion that is not derived from the cited proposition and is not obviously true. The additional boundary-isotopy claim is a further unproved sub-step. These gaps do not refute the theorem, and the paper may well be correct after filling them, so the reader's CONDITIONAL verdict is appropriate; my read does not change it.","tokens_in":33114,"tokens_out":15077,"duration_ms":196228,"concrete_test":"Take the boundary connected sum of two positive Hopf bands along a single arc s. Using the composition formula (1) and Corollary 3.5, enumerate all fixed essential arc classes that intersect s. For each such class, check whether it cuts out a disk together with a subarc of s (possibly after an isotopy that slides endpoints along the boundary). If any fixed essential arc fails this, the structural assertion in Theorem 6.2 is false. If all pass, supply the missing outermost-arc argument that derives the disk-cutting statement from Proposition 3.7; until that derivation exists, the step is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central application, Theorem 6.2, depends on the final reduction from a fixed essential arc a' in a vertex surface Sigma_v to an honest decomposition circle. After invoking Lemma 6.4 to write Sigma_v as a boundary connected sum along arcs {s_j}, the proof states: 'Then, by Proposition 3.7, a′ cuts out a disk D together with a subcollection of the s_j.' This does not follow from Proposition 3.7 as stated. Proposition 3.7 only shows that a fixed arc of a boundary connected sum restricts to fixed arcs in each summand. It says nothing about how those restrictions assemble globally; an essential fixed arc could in principle pass through several summands and connecting bands, with each restriction boundary-parallel in its irreducible summand, without cutting off a single disk together with some s_j. The required outermost-arc/innermost-disk argument is absent. The subsequent claim that an arc b' obtained from b by a free isotopy along the boundary is again fixed is also not justified, because fixedness is defined for arc classes with endpoints fixed pointwise and the partial monodromy is not shown to be invariant under endpoint-sliding along ∂Sigma. Since the constructed honest decomposition circle is ultimately c = b' (or c = s_j), this gap is load-bearing: if the asserted structure fails for some fixed essential arc, the proof of Theorem 6.2 does not produce the promised decomposition circle. The claim may still be true, but the proof as written is incomplete at exactly the point where the main theorem is established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of partial monodromies for arbitrary incompressible surfaces in 3-manifolds, generalizing the monodromy of fiber surfaces in open books. The main technical results are a composition formula for partial monodromies under Murasugi sums (Lemma 3.1), a primeness criterion for fixed arcs in tree-guided Murasugi sums (Theorem 4.2), a proof that strongly quasipositive surfaces are right-veering without contact geometry (Proposition 5.1), and a characterization of right-veering families closed under positive stabilization (Theorem 5.3). These tools are applied to prove that alternative link diagrams are visually prime (Theorem 6.2), subsuming previous visual-primeness results for alternating, positive, and homogeneous braid diagrams. The paper is ambitious and contains several substantial contributions, but the proof of the headline application has significant gaps.","tokens_in":33468,"tokens_out":16267,"duration_ms":179370,"significance":"If the results are correct, this is a meaningful step forward: the partial-monodromy framework extends open-book technology to all incompressible surfaces, and the visual-primeness theorem for alternative links would be a notable advance on Cromwell's conjecture. The paper also gives a new, topological proof that strongly quasipositive surfaces are right-veering and offers an alternative route to the Honda–Kazez–Matić tightness characterization. The composition formula and the primeness criteria are likely to be useful tools. However, the main application rests on a proof that is incomplete at a load-bearing point, and there is a false statement in the definitional discussion of alternative diagrams, so the significance is conditional on a successful revision.","major_comments":[{"comment":"The assertion “Then, by Proposition 3.7, a′ cuts out a disk D together with a subcollection of the sj” does not follow. Proposition 3.7 only states that a fixed arc in a boundary connected sum restricts to fixed arcs in each summand; it does not describe how those restrictions assemble globally. An essential fixed arc could in principle pass through several summands and connecting bands, with each restriction boundary-parallel, without cutting off a single disk together with some s_j. The required outermost-arc/innermost-disk argument is absent. Since the honest decomposition circle is eventually constructed from this structure, this gap is load-bearing.","section":"§6, proof of Theorem 6.2"},{"comment":"The step “b′ is obtained from b by a free isotopy along the boundary, so b′ is also fixed in Σv′” is unjustified. Fixedness (Definition 2.6) is defined for isotopy classes of arcs with endpoints fixed pointwise, and a free isotopy slides endpoints along ∂Σ, changing the arc class. The partial monodromy is not shown to be invariant under such endpoint-sliding. Therefore the conclusion that b′ is fixed in Σv′ does not follow from the fixedness of b. This affects the construction of c = b′, which is one of the two ways the proof aims to produce an honest decomposition circle.","section":"§6, proof of Theorem 6.2"},{"comment":"The statement “Dv is alternating if and only if it is positive or negative” is false. The closure of the positive 3-braid (σ1σ2)^4 is special (all Seifert circles are non-separating) and positive, but it is the (3,4) torus knot, which is not alternating. Thus the claimed equivalence between alternating and positive/negative special diagrams is not correct. This matters because the definition of alternative links in the paper is justified by this equivalence, and the equivalence to Kauffman's original definition is used to assert that the class studied is genuinely the class of alternative links. The authors need to correct this statement or adjust the definition of alternative diagrams accordingly.","section":"§6, Definition 6.1 and following paragraph"},{"comment":"The sentence “Since b′ ⊂ D, the discussion in Remark 2.5 implies that b′ is fixed in Σv” is not justified. Remark 2.5 explains how a partial monodromy can be recovered from a cutting system of arcs; it does not state that an arbitrary arc contained in a disk cut out by fixed arcs is itself fixed. A product disk witnessing fixedness of b′ would need to be constructed separately. This step is used to conclude that b′ is fixed in both Σv and Σv′, so it is essential to the proof.","section":"§6, proof of Theorem 6.2"}],"minor_comments":[{"comment":"The first line reads “For 3-manifolds M1 and M3”; this appears to be a typo for M1 and M2.","section":"§2.3, first paragraph"},{"comment":"The phrase “W.l.o.g. the endpoints of a are not on ∂Σ1 ∩ ∂Σ2 (if not, isotope them in the boundary, do the argument, and isotope back)” is another instance where endpoint-sliding is used without justifying invariance of fixedness. This may be repairable, but it needs an explicit argument.","section":"§3.2, proof of Proposition 3.10"},{"comment":"The introduction advertises a characterization (“if and only if”) of right-veering families, but Theorem 5.3 as stated gives only a sufficient condition in terms of 0-framed unknotted annuli; the full characterization appears only in Remark 5.4 via property (2). The presentation would be clearer if the theorem statement matched the stronger characterization or if the remark were clearly flagged as the converse.","section":"§5, statement of Theorem 5.3 and Remark 5.4"},{"comment":"In the proof of Lemma 6.5(i), the statement “A split union of links is prime if and only if all of the individual links are” is a standard fact, but the wording could be misinterpreted: a split union with two nontrivial components is not prime, and the equivalence should be stated with the convention for prime links. Please clarify the convention.","section":"§6, proof of Lemma 6.5"}],"recommendation":"major_revision","confidential_remarks":"The paper's core framework—partial monodromies and the Murasugi composition formula—is plausible and likely of lasting value. However, the main application to alternative links is not yet rigorously established: the proof of Theorem 6.2 contains at least two unjustified geometric assertions, and the false alternating/positive equivalence in Section 6 affects the definition of the class under study. These are not mere presentation issues; they require substantive mathematical fixes. I would not recommend rejection, because the gaps appear local and the overall approach may be repairable, but the paper should not be accepted until the proof of Theorem 6.2 is rewritten with full justification of the disk-cutting and endpoint-sliding claims, and the definition of alternative links is corrected or clarified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth taking seriously: it sets up a partial monodromy formalism for arbitrary incompressible surfaces and uses it to prove a genuinely new visual primeness result for alternative links, subsuming Menasco, Ozawa, and the authors' own homogeneous-braid theorem. The characterization of right-veering families (Theorem 5.3) and the contact-free proof that strongly quasipositive surfaces are right-veering are also solid, useful contributions. The composition formula for Murasugi sums (Lemma 3.1) is believable; the proof is long and figure-dependent, but the strategy is standard and the result is what experts would expect.\n\nThe main theorem (Theorem 6.2) is the issue. The proof's final reduction from a fixed essential arc in a vertex surface to an honest decomposition circle skips a step. After Lemma 6.4 decomposes the vertex surface as a boundary connected sum, the proof asserts that the fixed arc cuts out a disk together with a subcollection of the summing arcs, citing Proposition 3.7. Proposition 3.7 only says the restrictions to summands are fixed; it does not say how they assemble. An outermost-arc argument is needed, and it's not there. The subsequent claim that sliding an arc along the boundary preserves fixedness is also not justified, because fixedness is defined with endpoints fixed pointwise and the monodromy is not shown to be invariant under endpoint-sliding. Both gaps sit at the load-bearing junction. The theorem may be true—I suspect it is—but the proof as written is incomplete at exactly the point where the main application is established.\n\nOne more thing: the reader's flag about special diagrams being alternating iff positive or negative misses the mark. The proposed counterexample, the closure of (sigma1 sigma2)^4, is not special: the middle Seifert circle has crossings on both sides. The statement is standard and probably fine.\n\nI'd send this to a referee. The framework and the right-veering results are valuable and likely correct; the gap in Theorem 6.2 is repairable but needs real work. The paper is a good reading-group candidate, but go in knowing the final proof needs scrutiny.","headline":"Strong new framework and likely-true visual primeness theorem, but the proof of Theorem 6.2 has a real gap in the final reduction that needs fixing.","tokens_in":33968,"tokens_out":11038,"would_cite":false,"duration_ms":120995,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57R17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that alternative links are visually prime: if a non-prime link has an alternative diagram, the decomposition of the link as a connected sum is visible in that diagram.","keywords":["partial monodromy","incompressible surfaces","Murasugi sums","right-veering","strongly quasipositive","visual primeness","alternative links","Cromwell's conjecture"],"falsifier":"Find a non-prime link that admits an alternative diagram with no honest decomposition circle; equivalently, run a computer search over alternative diagrams of connected sums up to moderate crossing number and check whether every such diagram has a visible decomposition circle. A single counterexample would falsify Theorem 6.2, and a Murasugi sum of two strictly right-veering incompressible surfaces along an essential region that has an essential fixed arc would falsify Proposition 3.13 and Theorem 4.2.","tokens_in":32992,"feed_emoji":"🪢","tokens_out":6602,"duration_ms":59364,"temperature":0.7,"pith_summary":"This paper extends the classical monodromy of fiber surfaces and open books to every incompressible surface in a 3-manifold, where the monodromy becomes a partially defined self-map on isotopy classes of arcs. It proves a composition formula for Murasugi sums—the monodromy of the sum is the composition of the summand monodromies—and uses veering properties of arcs to detect when fixed arcs in a sum must come from fixed arcs in the summands. On this basis it shows that strongly quasipositive surfaces are right-veering, giving a contact-geometry-free proof. The payoff is a proof that alternative links are visually prime: if a non-prime link has an alternative diagram, the diagram contains an honest decomposition circle. This unifies and extends prior visual-primeness results for alternating links, positive links, and positive braids, and is a step toward Cromwell's conjecture.","feed_headline":"Non-prime alternative links reveal their decomposition in any diagram","feed_subtitle":"Partial monodromy for all incompressible surfaces makes visual primeness work beyond fibered links, advancing Cromwell's conjecture.","key_machinery":"The load-bearing object is the partial monodromy φΣ: A(Σ) → A(Σ), a partially defined self-map on the set of isotopy classes of properly embedded arcs of an incompressible surface, defined by product disks. The key identity is the Murasugi-sum composition formula φ = Φ1 ∘ Φ2, where Φi is the extension by the identity of the summand monodromy to the summed surface. Right-veeringness (every arc that goes to the other side satisfies a ≤ φ(a) in the boundary order) and left-veeringness are the veering properties that make the fixed-arc criterion work: in a tree-guided Murasugi sum with oppositely veering summands, fixed arcs restrict to fixed arcs (Theorem 4.2), and a surface built from strictly","core_discovery":"The paper's central claim is that monodromy data is available not only for fibered links but for all incompressible surfaces. The authors define the partial monodromy φΣ of an incompressible surface Σ in a 3-manifold as the partial self-map of the arc set A(Σ) that sends an arc a to the arc φ(a) on the other side of a product disk, if such a disk exists. For a Murasugi sum Σ of Σ1 and Σ2, they prove φ = Φ1 ∘ Φ2, where Φi extends φi by the identity, recovering the classical fibered composition formula as a special case. From this composition law they derive a primeness criterion (Proposition 3.10 and Theorem 4.2): in a tree-guided Murasugi sum of right-veering and left-veering summands, any f","pith_inferences":["If the tree decomposition of alternative diagrams is the template, one can test Cromwell's conjecture more broadly by seeking tree decompositions of arbitrary diagrams into veering pieces; the paper's approach suggests that visual primeness is a veering phenomenon, not a fiberedness phenomenon.","The partial-monodromy formalism may transfer other open-book tools—for instance, detecting homotopy ribbonness or considering arc complexes—to non-fibered surfaces, since the arc set is the only bookkeeping needed.","The right-veering characterization (Theorem 5.3) offers a concrete search target: in any family of surfaces closed under positive Hopf plumbing, the first obstruction to right-veeringness is an essential 0-framed unknotted annulus; one could look for such annuli in families arising from other diagram classes.","The proof of Proposition 5.1 converts a left-veering arc into a sobering arc with no interior intersections by positive Hopf plumbings; this suggests a constructive algorithm to exhibit overtwisted disks from non-right-veering open books, which could be made explicit."],"forward_implications":["Alternative links are visually prime: any alternative diagram of a non-prime link contains an honest decomposition circle (Theorem 6.2).","This recovers visual primeness of alternating links and positive links, and positive braids, as special cases; it also recovers the homogeneous-braid visual primeness result, since homogeneous braid diagrams are alternative.","Strongly quasipositive surfaces in S^3 are right-veering; they are strictly right-veering exactly when they are irreducible (Proposition 5.1).","A family of incompressible surfaces closed under positive Hopf plumbing is right-veering unless it contains a surface with an essential 0-framed unknotted annulus (Theorem 5.3).","All arborescent links are prime, and the tight/right-veering characterization of contact structures is reproved with minimal contact geometry (Corollary 5.5)."],"fun_headline_variants":["Monodromy for all surfaces proves visual primeness of alternative links","Beyond fibered links: primeness via partial monodromies","Partial monodromy criterion shows alternative links are visually prime","Alternative links pass visual primeness via surface monodromies"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of Theorem 6.2 relies on an unproven geometric assertion: a fixed essential arc in a vertex surface of the tree can be assumed to decompose that surface into a disk plus some of the boundary-connected-sum arcs; if the arc instead winds through the summands in a more complicated way, the constructed honest decomposition circle might not exist.","fun_headline_variants_meta":{"raw":{"variants":["Monodromy for all surfaces proves visual primeness of alternative links","Beyond fibered links: primeness via partial monodromies","Partial monodromy criterion shows alternative links are visually prime","Alternative links pass visual primeness via surface monodromies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00038,"raw_usage":{"total_tokens":1913,"prompt_tokens":862,"completion_tokens":1051,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":606,"completion_tokens_details":{"reasoning_tokens":989}},"tokens_in":606,"tokens_out":1051,"duration_ms":8966,"temperature":1.0,"reasoning_tokens":989,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T18:47:15.424874+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a non-prime link that admits an alternative diagram with no honest decomposition circle; equivalently, run a computer search over alternative diagrams of connected sums up to moderate crossing number and check whether every such diagram has a visible decomposition circle. A single counterexample would falsify Theorem 6.2, and a Murasugi sum of two strictly right-veering incompressible surfaces along an essential region that has an essential fixed arc would falsify Proposition 3.13 and Theorem 4.2.","supporting_citations":[],"review_version":1}