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REVIEW 2 major objections 3 minor 24 references

Dehn surgery functions are never injective

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For every rational slope, two distinct knots share the same Dehn surgery.

desk verdict Gordon's conjecture is almost certainly proved; the construction is sound enough for a serious referee despite a sketchy Proposition 2.8. read the letter →

arxiv 2508.13369 v1 pith:RZMH6YJO submitted 2025-08-18 math.GT

Kyle Hayden , Lisa Piccirillo , Laura Wakelin This is my paper · ORCID
classification math.GT MSC 57K1057K3057K31
keywords DehnsurgeryknotRBGlinksdualknotsHOMFLYPTpolynomialpositivebraidnon-characterisingslopes3-manifolds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

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.

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 (2)
  1. [§2.3.2 (Proposition 2.8)] 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.
  2. [§3.4 (Lemma 3.11 and its use)] 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.
minor comments (3)
  1. [§2.3 (RBG links)] 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.
  2. [§3.2 (Notation)] 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.
  3. [§3.3 (Proposition 3.9)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is self-contained and the shared-slope claim is proved from explicit dual-cable lemmas rather than imported from the cited RBG framework.

full rationale

The paper's central claim is established by an explicit construction: the knots KB and KG are defined from a modified RBG link, and the shared p/q-surgery is verified by tracking the relevant gluing maps and framings through Lemmas 2.3, 2.4, 2.5, and Proposition 2.7. Proposition 2.8 then asserts that the band-sum modification preserves the RBG conditions because the red-blue and red-green sublinking types and framings are unchanged; this is a geometric preservation claim, not a restatement of the theorem. The later HOMFLYPT computation uses standard skein relations, Proposition 3.1 for the zeroth coefficient polynomial, and the external positivity result of Ito [Ito22] together with Van Buskirk's lemma [VB85]. These are independent mathematical inputs that do not assume the target theorem. The citation to [Pic19] for the RBG construction is contextual and not load-bearing: the needed cancellation homeomorphisms and framing computations are proved in the paper itself. There are no fitted parameters, no data subset from which a prediction is extracted, and no uniqueness theorem imported from the same authors to force the choice. The terseness of Proposition 2.8's preservation argument is a potential correctness concern, but it is not a circularity: even if the argument were incomplete, the claimed reduction would not be equivalent by definition to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the construction uses integer choices r and s from Bezout's identity, which are not fitted values. The paper introduces no new physical or mathematical entities. It relies on standard Dehn surgery theory, the RBG link formalism, and two external positivity results for positive braid links.

assumptions (4)
  • domain assumption For a positive braid link, the normalized HOMFLYPT polynomial has non-negative coefficients.
    Ito's theorem, cited as [Ito22, Theorem 2], is used in Proposition 3.12 to show that the zeroth coefficient polynomial of a non-trivial positive braid knot is not a unit.
  • domain assumption Every non-trivial positive braid knot admits a skein triple consisting of positive braid links.
    Van Buskirk's lemma, cited as [VB85, Lemma 2], is used in the proof of Proposition 3.12 to express the normalized zeroth coefficient polynomial as a sum of positive polynomials.
  • standard math Standard properties of the HOMFLYPT zeroth coefficient polynomial, including the skein relation and the product formula for link components.
    These background facts, from [Ito22], are used throughout Section 3 to compute the difference between the polynomials of K_B and K_G.
  • domain assumption Injectivity of the p/q-surgery function is equivalent to injectivity of the -p/q-surgery function.
    Stated at the start of Section 3.4 without proof; it lets the authors assume p/q >= 0. It follows from taking mirror images and reversing orientation.

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Pith. "Pith review of Dehn surgery functions are never injective." pith.science (2026). https://pith.science/paper/RZMH6YJO

@misc{pith2026250813369,
  author       = {Pith},
  title        = {Pith review of: Dehn surgery functions are never injective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RZMH6YJO}},
  note         = {Machine review of arXiv:2508.13369}
}
abstract

We prove that, for each fixed rational number $p/q \in \mathbb{Q}$, there exists a pair of distinct knots whose $p/q$-surgeries are orientation-preservingly homeomorphic. This confirms a 1978 conjecture of Gordon.

Figures

Figures reproduced from arXiv: 2508.13369 by the authors.

Figure 1
Figure 1. The template knot [P] is the band sum of the satellite knot P(C−t,−q(Tr,s)) with an unknot U situated such that Tr,s = Cr,s(U). Here, r, s, t are integers such that ps − qr = 1 and t = −s(1 − qr). The knots KB and KG are given by [P ′ B ] and [P ′ G], respectively, where P ′ B and P ′ G are the iterated twisted Whitehead double patterns shown. When |p| > 1, we distinguish our pairs of knots by comparing their (zerot… view at source ↗
Figure 2
Figure 2. Perspectives on duals and double duals. Lemma 2.5. The double dual knot K∗∗ ⊂ S 3 is isotopic to the iterated cable knot Cqr2,p(Cr,s(K)) ⊂ S 3 K∗ and the double dual gluing map is given by A ∗∗ =  p(1 + q 2 r 2 ) r(1 + pqrs) q s  . Proof. Performing Dehn surgery on K∗ with the matrix A∗ from Lemma 2.3 should map the core c ∗ of V ∗ to the image of λV ∗ in ∂S 3 K(p/q)K∗ , which is simply qr2µK∗ + pλK∗ . We can thus… view at source ↗
Figure 3
Figure 3. The initial RBG link Lb = Rb ∪ Bb ∪ Gb. As discussed in Remark 2.6, Lb has the properties of an RBG link: the surgeries (B, b ˆb) and (R, b rˆ) cancel to leave a surgery on a single green knot KGb and the surgeries (R, b rˆ) and (G, b gˆ) cancel to leave a surgery on a single blue knot KBb. In the following lemma, we confirm that both of these new knots have the same framing p/q. Proposition 2.7. The link Lb is an R… view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: The modified RBG link, drawn in two different ways. To define the satellite pattern, let Dk m denote the k-clasped, m-twisted Whitehead double pattern as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: The k-clasped m-twisted Whitehead doubling pattern Dk m in a (parametrised) solid torus, viewed as the exterior of the unknot U. Here, k, m ∈ Z denote numbers of full right-handed twists between parallel strands. PG U PB U [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: The patterns PB = D1 0 ◦ D2 0 and PG = D2 0 ◦ D1 0 , each lying in a solid torus expressed as the exterior of the unknot U. At the level of L, the isotopy from PB to U straightens out B at the expense of G (without affecting R), as on the right-hand side of [PITH_FULL…
Figure 7
Figure 7. Figure 7: The knots KB and KG. Since we have broken the symmetry of Lb, we can be optimistic that this new link L presents distinct knots KB and KG. To this end, we now give concrete descriptions of KB and KG (see also [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: The twisted patterns P ′ B = D1 0 ◦ D2 qt and P ′ G = D2 0 ◦ D1 qt. Let us briefly digress to discuss these twisted patterns. Given a pattern knot P in the solid torus D 2 × S 1 , the n-twist pattern Pn is the image of P after applying n meridional Dehn twists to D 2 ×…
Figure 9
Figure 9. Figure 9: The result of applying an n-fold twist to the union of the pattern Dk (black) and its 0-framed pushoff (red). Here, k and n refer to numbers of full twists. The n-twist of Dl 0 ◦ Dk 0 can be obtained from the n-twist of L by introducing a clasp with −l twists. The n-tw…
Figure 10
Figure 10. Figure 10: represent the image of the meridian under a gluing map that sends the meridian to a curve with respectively positive and negative longitudinal component. Here, we assume that all strands in the grey region are coherently oriented. (a) (b) (c) [PITH_FULL_IMAGE:figures…
Figure 11
Figure 11. Figure 11: (a) The band sum of Bb and µG. (b) After pushing the green solid torus into the blue solid torus V ∗ , the curve µG becomes a meridian of the core of V ∗ . (c) Pushing µG back out of the red solid torus to a curve lying on ∂V ∗ . (d) This is a (−t, −q)-cable, i.e. Ct,…
Figure 12
Figure 12. Figure 12: (a) The band sum of Gb and µB. (b) After pushing µB into the blue solid torus V , it becomes a (s, −q)-cable of a copy c ′ of the core curve c ⊂ V . Pushing Gb into the red solid torus yields its core. (c) Redrawing Cs,−q(c ′ ). (d) The result of pushing the red solid…
Figure 13
Figure 13. Figure 13: A skein triple. Let L be an oriented link in S 3 . Recall that the HOMFLYPT polynomial PL(z, v) ∈ Z[z ±1 , v±1 ] of L is defined by a skein relation for skein triples (L+, L0, L−) as in [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: Another diagram of the template knot [P] = U#βP(C(R)). Rather than compute the entire polynomials for KB = [D1 ◦D2 qt] and KG = [D2 ◦D1 qt], we will express them in terms of the polynomials for three simpler knots: the unknot U, the iterated torus knot C(R), and the k…
Figure 15
Figure 15. Figure 15: The local skein triple for Lemma 3.2, depicted for k = 2. To simplify our calculations, we now present a series of lemmas describing the behaviour of the zeroth coefficient polynomial for such skein triples. We also describe the behaviour of the zeroth coefficient pol…
Figure 16
Figure 16. Figure 16: The patterns in the skein triple for Lemma 3.4, depicted for (k, l) = (1, 2). 3.3. Computing the polynomials. We now express the zeroth coefficient polynomials for KB and KG in terms of the polynomials of the simpler knots C(R) and [H] discussed above. We will encode …
Figure 17
Figure 17. Figure 17: A decomposition of KB. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: A further decomposition of one of the stages of [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: The full skein/linking tree for KB. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]
Figure 20
Figure 20. Figure 20: The full skein/linking tree for KG. 3.4. Distinguishing the polynomials. It is difficult to evaluate the polynomials ΓC(R) and Γ[H] which appear in our formulae, but in fact it will be fine to continue working in terms of these polynomials. The following lemma gives a…

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