{"id":"4845346c-fab5-4a1e-b366-e742b354b87a","arxiv_id":"2508.13369","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each rational slope p/q, two distinct knots exist whose p/q Dehn surgeries are orientation-preservingly homeomorphic, resolving Gordon's 1978 conjecture.","lead":"For every rational number p/q, the paper constructs two different knots whose p/q-framed Dehn surgeries are the same oriented 3-manifold. This proves a 1978 conjecture of Gordon that Dehn surgery functions on knots are never injective.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.8 does not verify that the band-summed blue component retains its p/q framing; if the framing shifts, the RBG cancellations may not yield the claimed common slope.","rationale":"The reader's weakest assumption identifies the same load-bearing point: Proposition 2.8 asserts the RBG property of the modified link without exhibiting the cancellation homeomorphisms or verifying the framings after the band-sum modification. This is indeed the most central issue, because if the framing of B is not preserved as p/q, the two knots produced by the RBG construction may not share the desired slope, and the main theorem would not follow. The paper's appeal to 'algebraic linking' being unchanged is insufficient, since framings can be modified by full twists without affecting linking numbers. I found no other concern of comparable weight. Secondary issues noted during reading: the nontriviality of C(R) in Lemma 3.11 is left implicit, and Lemma 3.6 appears to contain a linking-number typo (lk(bB, C(R)) should be q·s rather than q·r), but both are readily fixable and do not affect the overall strategy. The stress-test thus supports the CONDITIONAL verdict: the construction is plausible but the framing preservation in Proposition 2.8 needs a rigorous verification. If the proposed test confirms the framing, the paper's claim stands; if not, the construction would need adjustment.","tokens_in":17344,"tokens_out":35084,"duration_ms":326295,"concrete_test":"Perform the isotopy from the first to the second description of L (Figures 4, 11, 12) on the framing curve of B: start with a curve on ∂ν(B) representing the p/q framing, slide the band and satellite P_B(µ_G) from B to G, and express the resulting curve on ∂ν(bB) in the basis (µ_{bB}, λ_{bB}). Verify that it equals p µ_{bB} + q λ_{bB}. This is a finite diagrammatic computation; it can be checked first for a concrete parameter choice (e.g., p/q=2, r=3, s=2, t=4) and then for the general parameters. If the resulting framing is not p/q, Proposition 2.8 fails; if it is p/q, the RBG claim is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction requires the modified link L of §2.3.2 to be an RBG link with common slope p/q. Proposition 2.8 asserts this because 'we have not disturbed the red and blue (nor red and green) sublinking types or framings', but this is not demonstrated. The modification replaces B by a band sum B = bB #β P_B(µ_G). Although the second description of L has B = bB, the framing curve on B is not shown to map to the p/q curve on bB under the isotopy. Algebraic linking numbers are unchanged, but a framing can be changed by adding full twists without changing linking numbers. If the induced framing on bB is p/q + k for some nonzero integer k, then the pair (R,B) may not cancel to S^3, or the resulting knots KB and KG may share slope p/q + k rather than p/q, and Theorem 1 would not follow. Proposition 2.10 tracks the images of the meridians µ_B and µ_G but never tracks the longitudes/framings of B and G through the modification, so the key claim that the common slope remains p/q is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for every rational number p/q, there exist distinct knots K and K' in S^3 whose p/q-surgeries are orientation-preservingly homeomorphic, confirming a 1978 conjecture of Gordon. The construction uses a modified RBG link framework: starting from an RBG link of unknot, cable, and double-cable, the authors perform a band sum with a satellite pattern to break symmetry, obtaining two knots K_B and K_G that share a p/q-surgery. Distinctness is established by comparing the zeroth coefficient polynomials of the HOMFLYPT polynomial, using skein and linking trees to express the difference as a product of non-zero factors.","tokens_in":17500,"tokens_out":23163,"duration_ms":221273,"significance":"This is a landmark result in 3-manifold topology: it completes Gordon's program on the injectivity of Dehn surgery functions and shows that no rational slope is characterising for all knots. The construction is genuinely new in its use of dual and double dual knots to control the rational slope, and the paper provides explicit, parametric families of knots. The HOMFLYPT calculations are carefully structured, and Lemma 3.10's factorisation is a particularly clean way to avoid computing the full polynomials. The paper also makes good use of existing results (Ito's positivity theorem, Van Buskirk's lemma) without fitting any constants to data. If the framing issue discussed below is resolved, this will be an elegant and definitive contribution.","major_comments":[{"comment":"The proof that the modified link L remains an RBG link with common slope p/q is not sufficiently justified. The assertion that \"we have not disturbed the red and blue (nor red and green) sublinking types or framings\" is ambiguous and does not by itself demonstrate that the band-summed component B retains its p/q framing in the second description of Figure 4, nor that the induced framings on the knots K_B and K_G are p/q. Algebraic linking numbers are unchanged by the band sum, but a framing can change by full twists without changing linking numbers; the proof must track the longitude/framing curves of B and G through the band sum and the isotopy, or otherwise show that the common slope remains p/q. Proposition 2.10 tracks the images of the meridians µ_B and µ_G and the twistings of the satellite patterns, but it never tracks the longitudes/framings of B and G, so it does not fill this gap. This point is load-bearing: if the induced framing on bB is p/q + k for some nonzero integer k, then the pair (R,B) may not cancel to S^3, or the knots K_B and K_G may share slope p/q + k rather than p/q, and Theorem 1 would not follow.","section":"§2.3.2 (Proposition 2.8)"},{"comment":"Lemma 3.11 shows that C(R) can be chosen to be a positive braid knot, but Proposition 3.12 requires a non-trivial positive braid knot to conclude that Γ_C(R) is not a unit. The proof of Theorem 1 states that \"C(R) can be chosen to be a positive braid knot and thus its zeroth coefficient polynomial Γ_C(R) is not a unit,\" which does not follow without excluding the case where C(R) is the unknot. The authors should either prove that for every p/q with |p|>1 there exists a choice of (r,s) for which C(R) is a non-trivial positive braid knot, or handle separately the slopes for which all admissible choices give a trivial C(R). As written, the distinguishing argument may fail for slopes where C(R) is necessarily trivial.","section":"§3.4 (Lemma 3.11 and its use)"}],"minor_comments":[{"comment":"The definition of an RBG link is informal: the phrase \"and perhaps satisfying additional technical conditions\" leaves the precise requirements unclear. Since the paper relies on this framework, a precise definition with the exact conditions used in §2.3.2 would improve readability and verifiability.","section":"§2.3 (RBG links)"},{"comment":"In the paragraph before the skein-tree lemmas, the text reads \"we will write C(R) = C_{t,q}(T_{r,s}) and C(R) = C_{t,q}(T_{r,s})\"; the second expression appears to be a typo, presumably meant to denote the reverse of C(R). Please correct this to avoid confusion.","section":"§3.2 (Notation)"},{"comment":"The proof of Proposition 3.9 is left almost entirely to the reader. Given the complexity of the skein/linking tree in Figure 20, a brief outline of the key algebraic steps or a statement that the computation mirrors Proposition 3.8 step-by-step would be helpful.","section":"§3.3 (Proposition 3.9)"}],"recommendation":"major_revision","confidential_remarks":"This is an important result and the construction is elegant, but the gap in Proposition 2.8 concerning the framing of the modified RBG link is a genuine load-bearing issue. I am confident that it is fixable by a more detailed track of the longitudes, but as written the proof is not acceptable. The second issue about the non-triviality of C(R) is likely also fixable with a short argument. I would encourage the editor to request a revision rather than reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper proves Gordon's 1978 conjecture: for every rational slope p/q there exist distinct knots with orientation-preservingly homeomorphic p/q-surgeries. That is a genuine theorem, and the proof is largely convincing. The construction via RBG links and dual cables is elegant, and the use of the zeroth HOMFLYPT coefficient to distinguish the knots is a nice piece of work. The earlier partial results (|p|<=1, integral slopes, p≡1 mod q) are all subsumed.\n\nThe main thing that gave me pause is Proposition 2.8. The assertion that the modified link L is still an RBG link is justified in a few sentences, without exhibiting the homeomorphisms φ_B and φ_G after the band sum. That is the load-bearing step, and it deserves more detail. However, the specific stress-test concern about the framing on the blue component does not land. The framing on B is part of the framed link, and if the claimed isotopy from L to the second description exists, the framing coefficient p/q is preserved automatically—isotopy moves the framing curve along. So the worry that the framing might shift to p/q+k is not real. What is real is the need to verify that the isotopy can be chosen to avoid R and to keep the relevant sublinking types unchanged. That is probably doable, but it is not written out.\n\nThe other soft spots are minor: Proposition 3.9 leaves some tree computation to the reader, and the nontriviality of C(R) is implicit rather than stated. Neither threatens the theorem; the explicit formulas in Section 3.4 are checkable, and the positive braid argument for p>1 is clear.\n\nThe citation pattern looks fine. The result builds on Piccirillo's RBG framework and Ito's positivity theorem, and those are cited appropriately. No circularity.\n\nWho is this for? Anyone working in 3-manifold topology or Dehn surgery. It resolves a long-standing conjecture, so it will be read widely. The proof is somewhat intricate but the exposition is good, with helpful figures. I would send it to a serious referee. If I were refereeing, I would ask for a more detailed proof of Proposition 2.8 and a small expansion of the deferred computations, but I would expect the theorem to survive.\n\nMy recommendation: engage with it; send to peer review.","headline":"Gordon's conjecture is almost certainly proved; the construction is sound enough for a serious referee despite a sketchy Proposition 2.8.","tokens_in":18091,"tokens_out":7042,"would_cite":true,"duration_ms":72759,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57K30","57K31"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every rational slope, two distinct knots share the same Dehn surgery.","keywords":["Dehn surgery","knot surgery","RBG links","dual knots","HOMFLYPT polynomial","positive braid knots","non-characterising slopes","3-manifolds"],"falsifier":"Take the construction for a specific slope with $p>1$, such as $p/q=2/1$, compute the zeroth HOMFLYPT coefficient polynomials of the two explicit knots from their band-sum diagrams, and check whether the difference of the two polynomials is identically zero; if it is, the distinction argument would fail. A more direct check is to trace the band-sum modification and verify that the framed red-blue and red-green sub-surgeries of the modified link each still cancel to $S^3$; if either cancellation fails for some choice of parameters, the RBG premise is false.","tokens_in":17088,"feed_emoji":"🪢","tokens_out":13684,"duration_ms":130793,"temperature":0.7,"pith_summary":"This paper proves that the map sending a knot in the 3-sphere to the closed 3-manifold obtained by Dehn surgery along a fixed rational slope $p/q$ is never injective: for every rational number $p/q$, there exist distinct knots $K$ and $K'$ whose $p/q$-surgeries are orientation-preservingly homeomorphic. That settles a conjecture posed in 1978 and shows that the result of a rational surgery cannot, in general, identify the knot it came from. The proof constructs a candidate pair of knots for every slope from a three-component framed link whose blue-red and red-green sub-surgeries each cancel to the 3-sphere, and proves the candidates are distinct when $|p|>1$ by comparing their zeroth HOMFLYPT coefficient polynomials. The cases $|p|\\le 1$ were already covered by earlier constructions.","feed_headline":"For every rational slope, two knots share the same surgery","feed_subtitle":"A 1978 conjecture about Dehn surgery maps is proved for every rational slope.","key_machinery":"The load-bearing machinery is the dual-knot calculus inside Dehn surgery. In a $p/q$-surgery on a knot $K$, the core of the attached solid torus becomes a knot $K^*$ in the surgered manifold, and undoing the surgery by surgering along $K^*$ recovers $S^3$; after isotoping out of the solid torus, $K^*$ is the cable $C_{r,s}(K)$ for integers $r,s$ with $ps-qr=1$. Iterating once gives a double dual $K^{**}$, an iterated cable, and the paper arranges the gluing data so that the surgeries along $(K,K^*)$ and along $(K^*,K^{**})$ each cancel. This produces the initial RBG link, in which the third component becomes a knot with framing $p/q$. A band sum with a Whitehead-double satellite pattern breaks the symmetry between the two resulting knots while preserving the algebraic linking that fixes the slope; the explicit descriptions $K_B=[P'_B]$ and $K_G=[P'_G]$ then make the HOMFLYPT computation tractable.","core_discovery":"For a fixed rational slope $p/q$, the authors build a modified RBG link—a three-component framed link whose red-blue and red-green pairs each cancel to $S^3$ under surgery—and read off from it two knots, $K_B$ and $K_G$. The RBG structure gives explicit homeomorphisms showing $S^3_{K_B}(p/q)$ is orientation-preservingly homeomorphic to $S^3_{K_G}(p/q)$. The two knots come from band-summing the blue and green components with two asymmetric iterated Whitehead-double satellite patterns, $P_B$ and $P_G$, related by a component-exchanging isotopy. For $|p|>1$, the knots are shown to be non-isotopic because their zeroth HOMFLYPT coefficient polynomials differ: the difference reduces to a product of factors involving the coefficient polynomial of an iterated torus cable, and that polynomial is not a unit when the cable is a positive braid knot—a knot that closes a braid with all crossings in the same direction. Together with the known cases $|p|\\le 1$, this proves the theorem for every rational slope.","pith_inferences":["A natural test is to run the same skein-tree calculation on higher HOMFLYPT coefficient polynomials; if those also separate $K_B$ and $K_G$, the proof could be adapted to cover $|p|\\le 1$ without relying on earlier constructions.","Because the satellite patterns have winding number zero, the template may be adaptable to produce pairs of knots sharing a surgery that also have controlled genus or concordance properties, which would further clarify how non-unique surgery descriptions can be.","The distinction argument depends on the positivity of HOMFLYPT polynomials for positive braid knots, so the proof as written applies when the iterated cable can be arranged to be a positive braid; extending the template to other slopes would require a separate invariant.","One could test whether the two knots in a constructed pair are also distinguished by invariants such as the Alexander polynomial or knot Floer homology, which the paper does not compute."],"forward_implications":["For every rational slope $p/q$, the surgery function from knots to closed oriented 3-manifolds has no injective fibres: at least two distinct knots produce the same $p/q$-surgery manifold.","The 1978 conjecture that this map on knots is neither surjective nor injective is now fully settled, because non-surjectivity was already known.","The construction yields explicit pairs of knots for every slope, described as band sums of unknots with twisted Whitehead-double satellite patterns around iterated cables, so the pairs can be diagrammed and used in further examples.","For each knot in such a pair, the slope $p/q$ is a non-characterising slope, meaning knowledge of that surgery manifold does not determine the knot.","The construction can produce non-integer non-characterising slopes, in contrast with earlier families that mainly produced integer examples."],"supporting_citations":[{"why":"It states the conjecture that the $p/q$ surgery map on knots is neither surjective nor injective; the paper's theorem completes the non-injectivity half.","marker":"[Gor78]"},{"why":"It introduces the RBG link construction that the paper adapts to produce knots with a common rational surgery.","marker":"[Pic19]"},{"why":"It gives the first examples of distinct knots sharing a $\\pm 1$-surgery, covering the $|p|=1$ cases needed for the theorem.","marker":"[Lic77]"},{"why":"It extends the earlier construction to all slopes $\\pm 1/q$ and supplies the remaining $|p| \\le 1$ cases.","marker":"[Bra80]"},{"why":"It provides the positivity of the normalised HOMFLYPT polynomial for positive braid links, a key input for showing the constructed knots differ.","marker":"[Ito22]"},{"why":"It supplies the skein triple of positive braid links used to show the zeroth coefficient polynomial of a nontrivial positive braid knot is not a unit.","marker":"[VB85]"}],"fun_headline_variants":["Dehn surgery not injective for any rational slope","Every rational slope has two knots with identical surgery","Distinct knots, same surgery, for every rational slope","1978 conjecture confirmed: surgery map never one-to-one"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument depends on the claim that adding a small satellite band to one component of the link does not change the way the other two components cancel each other out to recover the 3-sphere, even though their relative geometry is altered.","fun_headline_variants_meta":{"raw":{"variants":["Dehn surgery not injective for any rational slope","Every rational slope has two knots with identical surgery","Distinct knots, same surgery, for every rational slope","1978 conjecture confirmed: surgery map never one-to-one"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1469,"prompt_tokens":796,"completion_tokens":673,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":412,"completion_tokens_details":{"reasoning_tokens":609}},"tokens_in":412,"tokens_out":673,"duration_ms":6759,"temperature":1.0,"reasoning_tokens":609,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:16:46.192878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the construction for a specific slope with $p>1$, such as $p/q=2/1$, compute the zeroth HOMFLYPT coefficient polynomials of the two explicit knots from their band-sum diagrams, and check whether the difference of the two polynomials is identically zero; if it is, the distinction argument would fail. A more direct check is to trace the band-sum modification and verify that the framed red-blue and red-green sub-surgeries of the modified link each still cancel to $S^3$; if either cancellation fails for some choice of parameters, the RBG premise is false.","supporting_citations":[],"review_version":2}