{"id":"4fee044a-2c34-4fa8-afda-79c5fa10ffaf","arxiv_id":"2508.03394","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A 2-torsion-free framed instanton homology for rational surgeries forces the knot to be an instanton L-space knot, sharpening small-surgery SU(2)-abelian obstructions.","lead":"The paper proves that if a rational Dehn surgery on a knot has no 2-torsion in its framed instanton Floer homology, then the knot must be an instanton L-space knot with surgery slope greater than 2g-1. This yields new obstructions: 5-surgery and 11/2-surgery are SU(2)-abelian only for the unknot and the right-handed trefoil.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.7's slope-zero rank inequality is the pivotal unverified step; its octahedral proof contains an opaque rank computation and unit-ambiguous diagram chasing, leaving the main theorem's genus bound dependent on a not-fully-pinned-down argument.","rationale":"The reader's verdict of CONDITIONAL is appropriate, and my stress-test identifies the same load-bearing assumption: Proposition 2.7. This proposition is not merely a technical lemma; it is the bridge between the slope-zero first differentials and the genus calculation that produces an instanton L-space knot. The proof is a formal diagram chase whose key displayed rank identity is at least misprinted and, as written, requires an extra injectivity condition that is not proved. The surrounding arguments in Propositions 2.5 and 2.8 are also diagrammatic and unit-sensitive, so a single sign or convention error in the bypass picture could propagate. I do not claim the result is false; rather, the paper does not yet make the crucial step independently checkable. The proposed concrete test would settle the issue by re-deriving the rank inequality from the formal exact triangles alone and, if necessary, by computing a concrete genus-one example. Since the reader already conditioned acceptance on independent verification of this step, no verdict change is needed.","tokens_in":25282,"tokens_out":13737,"duration_ms":150396,"concrete_test":"Re-derive the final rank estimate in Proposition 2.7 algebraically from the exact triangles (3.14)-(3.15), without referring to Figure 5: explicitly write the four-term complex with the maps alpha, eta, theta, beta, verify that the differentials satisfy d^2=0, apply [KM16, Lemma 10.3], and check whether the lower bound obtained is exactly 1/2(dim L1 - dim L3) or contains an additional term dim(im alpha intersect ker beta). If an extra term appears, compute whether it is forced to vanish by the established relations; if it is not, the claimed inequality is unsupported. As a second check, instantiate the diagram in the case K = right-handed trefoil (g=1, tau_I=1) and compute both sides of Proposition 2.7 at h=1/2 using the bypass exact triangles (3.9) and the classification of tight contact structures; if the inequality fails, the sign convention for the positive differential is wrong.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.4 reduces through Proposition 2.4 to three propositions, of which Proposition 2.7 is the most delicate. After Proposition 2.5 and the vanishing of the slope-p/q first differentials give tau_I(K)=g and the extreme-grading ranks of d0_{1,+-} equal 1, Proposition 2.7 is exactly what upgrades this to dim KHI(-S^3_0(K), rK_0)=4g, which then yields dim I^#(S^3_{2g}(K))=2g and the L-space conclusion. The proof of Proposition 2.7 in Section 5.1 is an octahedral diagram chase relying on the exact triangles (3.14)-(3.15), on [KM16, Lemma 10.3], and on commutative diagrams that hold only up to multiplication by a unit (Remark 3.1). In the decisive displayed calculation, the line 'dim im beta - dim im beta' appears to omit the quotient 'im beta / im(beta composed with alpha)'; the stated equality to 'rk beta - (dim(L1/im alpha) - dim ker beta)' would require im alpha intersect ker beta = 0, which is not established. The subsequent lower bound rk(d0_{1,+}) >= rk beta - dim(L1/im alpha) is salvageable from the surjectivity of the induced map, but reaching the final 1/2(dim L1 - dim L3) uses the octahedral dimension relations whose sign conventions are not independently checked. A sign error in which bypass map is called positive, or a unit ambiguity that turns one of the equalities into an inequality with an extra term, would invalidate Proposition 2.7 and with it Proposition 2.4 and the main theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a rational-surgery extension of the authors' earlier 2-torsion result for framed instanton homology. Theorem 1.4 states that if I^#(S^3_r(K);Z) has no 2-torsion for a positive rational slope r, then K is an instanton L-space knot and r > 2g(K)-1, with stronger bounds when r is an integer or half-integer. The proof strategy is to convert the absence of 2-torsion into a rational dual-Floer-simplicity condition, then study the first differentials on the instanton knot homology of dual knots, especially for slope 0. The main technical engine is Proposition 2.4, which is derived from Propositions 2.5, 2.7, and 2.8 via an octahedral diagram calculus for bypass maps. The paper also derives applications to SU(2)-abelian surgeries, including a resolution of the small-surgery question for slopes 5 and 11/2 up to the unknot and trefoil, and a partial result for slope 7.","tokens_in":25588,"tokens_out":24453,"duration_ms":285305,"significance":"If the proof is correct, the paper is a significant advance: it extends the 2-torsion-free surgery obstruction from integral to all rational slopes, gives new small-surgery obstructions that sharpen results of Kronheimer--Mrowka, Baldwin--Sivek, and Baldwin--Li--Sivek--Ye, and introduces a slope-0 first-differential framework that may be useful beyond this paper. The authors are explicit that the central new ingredient is Proposition 2.7, which plays the role of an immersed-curve local-maximum/local-minimum statement in a setting where immersed curve theory is not yet available. The paper is well structured and connects cleanly to prior work; the reduction to established invariants such as tau_I and t_2, and the use of Gordon's cabling formula, give the argument independent grounding. However, the load-bearing propositions are proved through an informal bypass-map and octahedral-diagram calculus in which maps are defined only up to units and key rank computations are not fully pinned down; the proof therefore needs substantial repair before the main theorem can be regarded as established.","major_comments":[{"comment":"The decisive displayed rank computation contains an algebraic error and an unjustified equality. The line `dim im(β∘α) = dim im β - dim im β` should involve the quotient `im β / im(β∘α)`, and the subsequent equality to `rk β - (dim(L1/im α) - dim ker β)` requires `im α ∩ ker β = 0`, which is not established. The final lower bound may be salvageable from surjectivity of the induced map, since the kernel of that induced surjection has dimension `dim ker β - dim(ker β ∩ im α) ≥ 0`, but the written proof does not exhibit this. Because this computation is the step that upgrades the rank information to the genus bound, it must be corrected and fully justified.","section":"§5.1, proof of Proposition 2.7"},{"comment":"There is an apparent inconsistency in the grading shift of the slope-0 first differentials. Formula (2.5) and Remark 3.5 say that for p/q = 0 the first differentials preserve the Alexander grading, while the octahedral proof of Proposition 2.7 uses `d0_{1,+} = β∘α : L(0,h) -> L(0,h+1)` and compares dim KHI(h) with dim KHI(h+1). Moreover, Proposition 2.8 asserts that `d0_{1,+}` has positive rank at the top grading `h = g-1/2`; if the shift were +1, its target would be zero, contradicting positivity. The authors need to state the actual grading shift for slope 0 and reconcile (2.5), Proposition 2.7, and Proposition 2.8.","section":"§2 and §5.1, grading shift for slope 0"},{"comment":"The proof of Proposition 2.7 relies on the octahedral exact triangles (3.14) and (3.15), which are asserted from diagram chasing or [LY24, Proposition 3.6], and on dimension identities (5.3) obtained from [KM16, Lemma 10.3]. All the commutative diagrams in the octahedron hold only up to multiplication by a unit (Remark 3.1), and the signs of the bypass maps are convention-dependent. Since a sign error or a unit ambiguity that introduces an extra term in one of the rank identities would invalidate Proposition 2.7 and therefore Theorem 1.4, the octahedral data should be stated with explicit maps, domains, codomains, and grading shifts, and the diagram chasing should be either proved in the text or supplied with a complete reference.","section":"§3.2, octahedral diagram"},{"comment":"The proof claims that because the instanton chain complex of an SU(2)-abelian surgery has exactly p generators and Euler characteristic p, there is no differential over any coefficients. This implication is not valid as stated: differentials can cancel generators in pairs while preserving the Euler characteristic. The conclusion that t2(S^3_r(K)) = 0 needs an independent grading argument, for example that all reducible generators lie in the same mod-2 grading, or a citation to the precise theorem in [BS23] or [BLSY24]. Since Corollaries 1.9 and 1.11 depend on this step, this is a load-bearing point.","section":"§5.3, proof of Corollary 1.8"},{"comment":"The proof of the half-integer bound in Theorem 1.4 uses `[Bha24, Theorem 1.1]` for the exact triangle computing `I^#(S^3_{(4g-1)/2}(K);F2)`. This is an unpublished preprint. The authors should state the status of [Bha24] and either include the exact statement of the theorem used or otherwise make the proof independent of an unavailable reference; as written, the half-integer portion of the main theorem is conditional on an external verification.","section":"§5.2, proof of the half-integer case"}],"minor_comments":[{"comment":"The diagram in (5.2) is very hard to parse as typeset; the domains and codomains of the arrows β1, η1, θ2, and α2 are not visually clear. Please redraw it with explicitly labeled maps and targets, and check that the composition `β1∘θ2 = η1∘α2` has a common target.","section":"§5.1, proof of Proposition 2.7"},{"comment":"The final sentence of the proof of Lemma 5.1 says `t2(Y#L(p,q);F2) = 0`, but t2 is defined as a difference of F2- and C-dimensions, not as an F2-valued quantity. This should read `t2(Y#L(p,q)) = 0`.","section":"§5.2, Lemma 5.1"},{"comment":"In the statement of Proposition 2.5, the display `τ = τI(K) = -τI(K)` appears to be missing the mirror notation; it should presumably read `τ = τI(\\bar K) = -τI(K)`. Please clarify the notation.","section":"§2, Proposition 2.5"},{"comment":"Several central references are to arXiv preprints, including [Bha24], [LY21], [LY24], and [LY25b]. Please indicate which of these are published or accepted, and update the citation data accordingly, so that the reader can verify the results on which the proof depends.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"I see the main risk as the internal consistency of the slope-0 grading conventions and the correctness of the octahedral rank computation in Proposition 2.7. If those are repaired, the paper's results are likely correct and significant. I would not reject on the current evidence, but the proof needs to be made precise before publication. The dependence on [Bha24] should also be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new: it pushes the 2-torsion technique from integral surgeries to all rational slopes, proving that a 2-torsion-free framed instanton homology forces K to be an instanton L-space knot with r > 2g(K) - 1. The cabling reduction via Gordon's formula and the connected-sum lemma is clean, and Corollary 1.9 answers the 5-surgery and 11/2-surgery cases for SU(2)-abelian surgeries, sharpening earlier work by Kronheimer-Mrowka, Baldwin-Sivek, and Baldwin-Li-Sivek-Ye. The paper is honest about its reliance on unpublished preprints and clearly flags where things are informal.\n\nThe soft spot is exactly where the reader and stress-test point: Proposition 2.7. The statement is reasonable, but the proof in Section 5.1 is an octahedral diagram chase with maps defined only up to units. The displayed rank computation contains a line 'dim im beta - dim im beta' that looks like a typo and obscures the actual quotient being taken. The equality leading to rk beta - (dim(L1/im alpha) - dim ker beta) is not justified in the text, and reaching the final 1/2(dim L1 - dim L3) depends on sign and grading conventions that are not independently checked. This is not a fabricated flaw; it is in the paper, and as written the proof of Proposition 2.7 is incomplete. That said, I did not find an actual contradiction, and the surrounding structure—Propositions 2.5, 2.8, and the reduction to integral cases—is much more solid. The central theorem probably holds, but it is not fully established on the page.\n\nThe small-surgery corollaries are argued carefully and the use of known classification results for knots of genus up to 2 is appropriate. The paper will be important to anyone working in instanton Floer homology or Dehn surgery, and the main theorem is a real advance.\n\nRecommendation: send it to a serious referee, but make sure the referee understands that the proof of Proposition 2.7 needs to be rewritten and checked line by line. This is not a desk-reject; it is a revise-and-resubmit with the burden on the authors to make the slope-zero argument reproducible.","headline":"Strong new results in instanton Floer homology, but the proof of the pivotal slope-zero rank inequality (Prop 2.7) is written too loosely to verify without substantial referee work.","tokens_in":26226,"tokens_out":2080,"would_cite":true,"duration_ms":25322,"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":"The paper proves that a rational surgery on a nontrivial knot with 2-torsion-free framed instanton homology forces the knot to be an instanton L-space knot, and uses this to rule out SU(2)-abelian surgeries at slopes 5 and 11/2 except for…","keywords":["instanton Floer homology","2-torsion","Dehn surgery","L-space knots","rational surgeries","SU(2) representations","sutured instanton homology","tau invariant"],"falsifier":"The theorem would be refuted by exhibiting a non-trivial knot $K$ and a rational slope $r\\le 2g(K)-1$ with $I^\\sharp(S^3_r(K);\\mathbb{Z})$ free of 2-torsion. A more local test is to compute the rank of the slope-zero differential $\\tilde d^0_{1,+}$ in a grading $h$ with $\\tau_I(K)\\neq 0$ and compare it with the dimension gap in Proposition 2.7; a violation would show exactly where the main argument breaks.","tokens_in":25017,"feed_emoji":"🪢","tokens_out":15067,"duration_ms":153850,"temperature":0.7,"pith_summary":"The paper extends an earlier integral-surgery result to every positive rational slope: if the framed instanton homology $I^\\sharp(S^3_r(K);\\mathbb{Z})$ of a surgery on a non-trivial knot $K$ has no 2-torsion, then $K$ must be an instanton L-space knot, meaning some positive surgery on it has the minimal possible instanton homology, and the slope satisfies $r>2g(K)-1$. This matters because 2-torsion is a sensitive probe: its absence is strong enough to force the knot into a very restricted topological class, and it yields new restrictions on which surgeries can have no irreducible $SU(2)$ representation of the fundamental group. As consequences, every non-trivial knot has 2-torsion for small slopes, and an $SU(2)$-abelian surgery at slope $5$ or $11/2$ can occur only for the unknot or the right-handed trefoil.","feed_headline":"No 2-torsion in a rational surgery forces an L-space knot","feed_subtitle":"The same 2-torsion test rules out SU(2)-abelian surgeries at slopes 5 and 11/2 except for the unknot and right-handed trefoil.","key_machinery":"The load-bearing object is the pair of first differentials $\\tilde d^{p/q}_{1,+}$ and $\\tilde d^{p/q}_{1,-}$ on the instanton knot homology $KHI(-S^3_{-p/q}(K),\\tilde K_{-p/q})$, defined by compositions of bypass maps, equivalently contact gluing maps attached to basic slices of $[0,1]\\times T^2$. For slope zero these differentials preserve the Alexander grading, and the paper uses their ranks as a stand-in for counts of local maxima and minima of an immersed curve invariant that has not yet been constructed in instanton theory. Proposition 2.7 gives rank lower bounds from dimension gaps between adjacent Alexander gradings, Proposition 2.5 transfers those bounds to arbitrary slopes, and a diagram chase through an octahedral exact triangle proves Proposition 2.7.","core_discovery":"The paper's central claim is Theorem 1.4: for a non-trivial knot $K\\subset S^3$ and $r\\in\\mathbb{Q}_+$, if $I^\\sharp(S^3_r(K);\\mathbb{Z})$ has no 2-torsion, then $K$ is an instanton L-space knot and $r>2g(K)-1$. For positive integer slopes the bound improves to $r\\ge 2g(K)-1+t_2(S^3_1(K))$, where $t_2(Y)=\\frac{1}{2}(\\dim I^\\sharp(Y;\\mathbb{F}_2)-\\dim I^\\sharp(Y;\\mathbb{C}))$, and $t_2(S^3_1(K))\\ge 1$; for positive half-integer slopes the bound improves to $r>2g(K)$. The proof reduces rational slopes to integral slopes by the cabling diffeomorphism $S^3_{pq}(K_{p,q})\\cong L(p,q)\\#S^3_{p/q}(K)$, then uses the first differentials on sutured instanton knot homology to show that absence of 2-torsion forces the knot to be an instanton L-space knot.","pith_inferences":["The octahedral proof of Proposition 2.7 suggests that once an immersed curve description of instanton knot homology exists, the rank inequalities should become equalities and should yield the sharper slope bound that the paper leaves open for $r\\in(5,7)$.","The cabling reduction raises a testable converse: if an integral surgery on the cable $K_{p,q}$ is 2-torsion-free, is the rational surgery on $K$ also 2-torsion-free? The paper proves one direction, and the reverse would make torsion-freeness exactly cable-invariant.","Because no closed oriented 3-manifold with $t_2(Y)=1$ or $2$ is known, computing $t_2$ for the remaining genus-3 candidates from the 7-surgery classification would either produce the first examples or eliminate that case."],"forward_implications":["Every non-trivial knot $K$ has 2-torsion in $I^\\sharp(S^3_r(K);\\mathbb{Z})$ for every rational slope with $0<|r|\\le 2g(K)-1$.","If $S^3_r(K)$ is $SU(2)$-abelian and the Alexander polynomial of $K$ satisfies $\\Delta_K(\\zeta^2)\\neq 0$ for every $p$-th root of unity $\\zeta$, then $r>2g(K)-1$, with stronger bounds for integer and half-integer slopes.","For $r\\in\\frac{1}{2}\\mathbb{Z}$ with $|r|<6$, an $SU(2)$-abelian surgery occurs only for the unknot or the right-handed trefoil at $r=\\pm 5,\\pm 11/2$.","An $SU(2)$-abelian 7-surgery forces the knot to be the unknot, the right-handed trefoil, or a genus-3 instanton L-space knot with $t_2(S^3_1(K))\\le 2$.","Together with the Euler characteristic computation for framed instanton homology, the main theorem indicates that among all knot surgery manifolds, the only one with one-dimensional $\\mathbb{F}_2$ framed instanton homology would be $S^3$ itself."],"supporting_citations":[{"why":"Provides the concordance invariant $\\nu^\\#$ and the theorem that an instanton L-space surgery on a cable forces the companion knot to be an instanton L-space knot, used in the rational-slope reduction.","marker":"[BS21]"},{"why":"Establishes the instanton L-space framework and the genus bound $g(K)\\le (r+1)/2$ for L-space surgeries, which the proof uses throughout.","marker":"[BS23]"},{"why":"Proves the prior integral-surgery 2-torsion results, including the implication from $t_2(S^3_n(K))=0$ to dually Floer simplicity and the lower bound $t_2(S^3_1(K))\\ge 1$.","marker":"[LY25a]"},{"why":"Gives the cabling diffeomorphism $S^3_{pq}(K_{p,q})\\cong L(p,q)\\#S^3_{p/q}(K)$ that reduces rational slopes to integral slopes.","marker":"[Gor83]"},{"why":"Introduces the first differentials and the spectral sequences whose collapse expresses rationally dually Floer simplicity.","marker":"[LY21]"},{"why":"Provides the bypass maps, exact triangles, Alexander gradings, and commutative diagrams underlying the proofs of Propositions 2.5 and 2.7.","marker":"[LY22]"},{"why":"Supplies the equivalent definitions of the instanton tau invariant and its identification with $\\tau^\\#$ used in Corollary 4.5 and the V-shaped dimension sequences.","marker":"[GLW24]"},{"why":"Gives the Euler characteristic $\\chi(I^\\sharp(S^3_{p/q}(K);\\mathbb{C}))=p$ and the surgery exact triangles used to pass between integral surgery homologies.","marker":"[Sca15]"},{"why":"Provides the surgery exact triangle used for the half-integer slope bound, relating the dual knot of the $(2g-1)$-surgery to the $(4g-1)/2$-surgery.","marker":"[Bha24]"}],"fun_headline_variants":["Rational surgery with no 2-torsion: knot must be L-space","2-torsion-free instanton homology forces L-space knots in rational surgeries","Small-surgery obstruction: slopes 5 and 11/2 yield unknot or right-handed trefoil","Improved bound: r>2g-1 for 2-torsion-free rational surgeries","Instanton 2-torsion test now covers all rational surgeries"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the rank inequalities of Proposition 2.7, which assert that the slope-zero first differentials are large enough to account for every dimension drop between adjacent Alexander gradings; if a sign or grading convention in the bypass picture is wrong, the inequalities fail and the theorem no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Rational surgery with no 2-torsion: knot must be L-space","2-torsion-free instanton homology forces L-space knots in rational surgeries","Small-surgery obstruction: slopes 5 and 11/2 yield unknot or right-handed trefoil","Improved bound: r>2g-1 for 2-torsion-free rational surgeries","Instanton 2-torsion test now covers all rational surgeries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001087,"raw_usage":{"total_tokens":4584,"prompt_tokens":1024,"completion_tokens":3560,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":640,"completion_tokens_details":{"reasoning_tokens":3450}},"tokens_in":640,"tokens_out":3560,"duration_ms":32465,"temperature":1.0,"reasoning_tokens":3450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:29:23.631911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorem would be refuted by exhibiting a non-trivial knot $K$ and a rational slope $r\\le 2g(K)-1$ with $I^\\sharp(S^3_r(K);\\mathbb{Z})$ free of 2-torsion. A more local test is to compute the rank of the slope-zero differential $\\tilde d^0_{1,+}$ in a grading $h$ with $\\tau_I(K)\\neq 0$ and compare it with the dimension gap in Proposition 2.7; a violation would show exactly where the main argument breaks.","supporting_citations":[],"review_version":1}