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Characterizing slopes for Legendrian knots

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Pith's one-line read A rational slope that smoothly characterizes a knot also becomes a contact characterizing slope for its Legendrian representatives, whenever the Legendrian class is determined by tb and rotation number.

desk verdict New bridge criterion with real applications, but the load-bearing d3-sum step in Case 3.1 of Theorem 1.1 is asserted rather than proved; the paper deserves refereeing if the authors can close that gap. read the letter →

arxiv 2608.06079 v2 pith:MUXGAIG5 submitted 2026-08-06 math.GT math.SG

Youlin Li , Chi Zhang This is my paper · ORCID
classification math.GTmath.SG MSC 57K1057R17
keywords LegendrianknotscontactsurgerycharacterizingslopesThurston–Bennequininvariantrotationnumberd3-invariantDehnsimplicity
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 a transfer principle in 3-manifold topology and contact geometry: if a rational slope is known to characterize a knot in ordinary Dehn surgery—meaning that any knot whose surgery at that slope gives the same 3-manifold must be that knot—then the same slope becomes a characterizing slope for a Legendrian representative of the knot, provided the Legendrian knot is already determined by its two classical invariants, the Thurston–Bennequin number and the rotation number. The proof compares the $d_3$-invariants of every contact structure that can arise from contact surgery on the two candidate Legendrian knots. Summing these invariants over all expanded surgery diagrams cancels the linear terms in the rotation number and leaves a quadratic comparison, which forces the second knot to have the same classical invariants as the first. The criterion yields complete contact characterizing slopes for every Legendrian representative of the unknot, the trefoils, and the figure-eight knot, and partial results for the knots $5_2$, $\overline{5_2}$, and the $(5,\pm 2)$ torus knots. It also produces examples showing that not every non-zero rational number is a contact characterizing slope for a general Legendrian torus knot or for a general Legendrian $5_2$.

What carries the argument

The load-bearing object is the $d_3$-invariant of each contact 3-manifold produced by a contact $p/q$-surgery, written as a quadratic polynomial in the rotation number of the original Legendrian knot. The Ding–Geiges–Stipsicz algorithm converts a contact $p/q$-surgery diagram into finitely many diagrams of contact $(\pm1)$-surgeries on Legendrian links obtained from $L$ by push-offs and stabilizations; the Durst–Kegel formula then expresses $d_3$ as $\tfrac14 r^T Q^{-1}r + \tfrac12\sum_i \operatorname{sign}_i - \tfrac34\sigma(Q) - \tfrac12$, where $Q$ is the generalized linking matrix. The crucial structural observation is that summing $d_3$ over all expanded diagrams cancels the linear term in the rotation number, because reversing every stabilization sign pairs diagrams and replaces each coefficient $r_i-r$ by its negative. The sums for $L$ and $L'$ therefore agree in their quadratic and constant parts, so equality of the sums forces $r'=r$. In the exceptional opposite-slope case, the same formulas yield conditions such as $6|t+p|$ or $3|p+2t|$ being a sum of two squares, which the smooth surgery comparison rules out by $d$-invariant arguments and modular arithmetic.

What would settle it

Enumerate all expanded $(\pm1)$-surgery diagrams for a concrete case such as contact $\frac23$-surgery on a Legendrian unknot with $\operatorname{tb}=-1$, compute the summed $d_3$-invariant as a polynomial in $\operatorname{rot}(L)$, and check whether the linear coefficient is zero and the quadratic coefficient $(1,\dots,1)Q^{-1}(1,\dots,1)^T$ is non-zero; any failure there would break the comparison that forces $\operatorname{rot}(L')=\operatorname{rot}(L)$. Equivalently, a direct counterexample would be two Legendrian knots with the same $\operatorname{tb}$, different rotation numbers, and contactomorphic $p/q$-surgeries for a slope where $p/q+\operatorname{tb}$ is smoothly characterizing.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1. Let $L$ be a Legendrian representative of a knot $K\subset S^3$ in the standard contact structure, and let $p/q\neq 0$. If $p/q+\operatorname{tb}(L)$ is a smooth characterizing slope for $K$, and if the collection of contact manifolds obtained by contact $p/q$-surgery on $L$ is contactomorphic to the corresponding collection for another Legendrian knot $L'$, then $L'$ is smoothly isotopic to $L$, with $\operatorname{tb}(L')=\operatorname{tb}(L)$ and $\operatorname{rot}(L')=\operatorname{rot}(L)$. Consequently, whenever the Legendrian isotopy class of $L$ is uniquely determined by $\operatorname{tb}(L)$ and $\operatorname{rot}(L)$, the slope $p/q$ is a contact characterizing slope for $L$. The proof splits into two cases: when the two knots have the same Thurston–Bennequin invariant, a sum of $d_3$-invariants over all expanded surgery diagrams forces equality of rotation numbers; when the surgery slopes are opposite, so that $t+t'=-2p/q$, the same invariants reduce to sum-of-two-squares conditions that are contradicted by the smooth surgery homeomorphism and elementary modular arithmetic.

Load-bearing premise

The proof assumes that when the $d_3$-invariants of all contact structures in the surgery collection are added together, the linear terms in the rotation number cancel exactly because reversing every stabilization sign produces another diagram in the collection, and that the remaining quadratic coefficient is non-zero; neither the bijectivity of that sign-reversal nor the non-vanishing of $(1,\dots,1)Q^{-1}(1,\dots,1)^T$ is proven or cited.

Editorial extensions

If this is right

  • Every non-zero rational number is a contact characterizing slope for every Legendrian representative of the unknot, either trefoil, or the figure-eight knot (Corollary 1.3).
  • For Legendrian representatives of $5_2$, all non-integral rational slopes characterize, and every non-zero integer $p$ with $p+\operatorname{tb}(L)\leq 0$ characterizes (Corollary 1.4(1)).
  • For non-maximal Legendrian representatives of $\overline{5_2}$, all non-integral rational slopes characterize and every integer $p$ with $p+\operatorname{tb}(L)\geq 0$ characterizes; for maximal ones, $-1$ characterizes (Corollary 1.4(2),(3)).
  • For Legendrian representatives of $T_{5,-2}$ and $T_{5,2}$, slopes in the stated ranges, with the finite exceptions listed in Corollary 1.5, are contact characterizing.
  • The criterion also corrects a previously claimed contactomorphism: the contact $(+6)$-surgeries described in Example 1.9 of [CEK24] cannot actually be contactomorphic (Remark 3.3).

Reading between the lines

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

  • Beyond the paper: whenever new smooth characterizing slopes are discovered for a knot whose Legendrian representatives are determined by their classical invariants, Theorem 1.1 will immediately convert them into contact characterizing slopes; the bottleneck for future examples is Legendrian classification rather than surgery theory.
  • Beyond the paper: the proof only needs the multiset of $d_3$-invariants of the two surgery collections, not knowledge of which individual contact structures pair up, so the summing technique may certify non-contactomorphism in other settings where the contactomorphic correspondence is unknown.
  • Beyond the paper: the sum-of-two-squares obstructions in Lemmas 3.1 and 3.2 suggest that arithmetic properties of the surgery coefficient control when two Legendrian knots can have contactomorphic surgery collections; a systematic study of other slopes could reveal a general pattern.
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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

5 major / 4 minor

Summary. The paper proves a criterion linking smooth and contact characterizing slopes for Legendrian knots. Theorem 1.1 states that if p/q + tb(L) is a smooth characterizing slope for K and the contact p/q-surgery collections for L and L' are contactomorphic, then L' is smoothly isotopic to L, with tb(L') = tb(L) and rot(L') = rot(L). Under a uniqueness assumption on the classical invariants, p/q becomes a contact characterizing slope. Applications cover the unknot, trefoils, figure-eight knot, 5_2 and its mirror, and the torus knots T_{5,±2}. The proof uses the Ding–Geiges–Stipsicz algorithm, d3-invariant computations, and number-theoretic obstructions (sums of two squares) in Lemmas 3.1 and 3.2.

Significance. The criterion is an elegant bridge from the well-developed theory of smooth characterizing slopes to contact characterizing slopes, and the applications resolve [CEK24, Question 5.4] and provide many new contact characterizing slopes. The number-theoretic lemmas are essentially self-contained and yield clean obstructions. The proof is not circular: it builds on established black boxes such as smooth characterizing slope results, Legendrian simplicity, and d-invariant formulas. However, the central Case 3.1 argument contains several unproved assertions about the sum of d3-invariants over expanded surgery diagrams; until those are supplied, Theorem 1.1 is not fully established.

major comments (5)
  1. [§3.1, paragraph 'Suppose the contact structures in L(p/q) ...'] The sum D3(L, p/q) is defined as the sum of d3-invariants over all expanded contact (±1)-surgery diagrams produced by the Ding–Geiges–Stipsicz algorithm. A contactomorphism between L(p/q) and L'(p/q) gives a bijection between contact structures, but the paper does not show that the diagram set is a contactomorphism invariant, nor that each contact structure appears with the same multiplicity in the expanded diagram sets for L and L'. Without such a statement, the equality D3(L, p/q) = D3(L', p/q) is not justified.
  2. [§3.1, linear term of D3] The vanishing of the linear term in D3 is asserted because 'reversing the signs of all stabilizations from L to each Li yields a contact surgery diagram for which r̃_i − r = −(r_i − r)'. This sign-reversal symmetry of the particular expanded diagram set produced by the DGS algorithm is not proved; the algorithm's continued-fraction expansions and the choice of k in the positive-surgery case are not obviously invariant under reversing all stabilizations. A concrete verification or a citation is needed.
  3. [§3.1, conclusion 'admits a unique non-negative solution'] The conclusion that D3(L, p/q) = D3(L', p/q) forces r' = r requires that the quadratic coefficient N/4 · (1,...,1) Q^{-1} (1,...,1)^T be nonzero. The text only states that the 'leading coefficients' are equal; it never proves the coefficient is nonzero. For an indefinite symmetric matrix Q, this quadratic form can vanish for nonsingular Q (e.g., Q = [[1,1],[1,-1]] has 1^T Q^{-1}1 = 0), in which case the equation has multiple non-negative solutions and Theorem 1.1 does not follow. This is the unique step in Case 3.1 that distinguishes rotation numbers.
  4. [§3, first case p/q + tb(L) = 0] The formulas for e(L(−t)) and e(L'(−t)) are stated without proof after 'Using Theorem 2.12 and the algorithm...'. These formulas are load-bearing because they yield r = r'. Please provide the derivation, in particular for the t ≤ −1 case involving ±(t+1)^2, and specify the sign conventions and the indexing of the Spin^c structures.
  5. [Proposition 2.10] In Proposition 2.10, the symmetry d(S^3_{p/2}(K), i) = d(S^3_{p/2}(K), p+1−i) is asserted for i ∈ {0,...,p−1}; for i=0 the right-hand side is p+1, outside the stated range, and the cited formula [OS03, Prop 4.8] as reproduced in the paper yields d(i)=d(p−1−i) rather than d(i)=d(p+1−i) (for example, for p=5). Since the subsequent argument about the unique fixed point and the conclusion p=4l+1 depends on this symmetry, the index convention must be corrected and the proof re-verified.
minor comments (4)
  1. [Abstract and Introduction] There are several typographical errors, including 'charac terizing' in the abstract and 'Legedrian' on page 1; the manuscript would benefit from a careful proofreading pass.
  2. [Example 1.6] The notation S4+S−(L1) is hard to parse; please clarify the order and number of stabilizations applied to L1.
  3. [Theorem 2.12] The definition of the generalized linking matrix Q would be clearer if all entries were written explicitly and if the symmetry of Q in the (±1)-surgery case were stated, since Theorem 2.13 relies on Q being symmetric.
  4. [Proof of Corollary 1.4(3)] The citation [BEE12, Section 7.3] is used to assert that L(−1) and L̃(−1) are not contactomorphic; please give the precise statement from that reference, as the relevant result may not be immediately evident from the cited section.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem derives a contact-surgery statement from external smooth characterizing-slope results and independent d3-invariant computations.

full rationale

The paper's central result, Theorem 1.1, is not equivalent to any of its inputs by construction. The input is an externally established smooth characterizing-slope statement for K plus contactomorphism of the surgery collections L(p/q) and L'(p/q); the output is smooth isotopy of L' to L and equality of tb and rot. Smooth isotopy follows from the underlying homeomorphism of the surgeries and the cited black-box theorems [KMOS07, OS19, NZ23, BS24]. Equality of rotation numbers is then obtained by a separate calculation using d3-invariants of expanded Ding-Geiges-Stipsicz surgery diagrams, not by restating the assumptions. Corollary 1.2 uses Legendrian simplicity as an external classification input (e.g., [EF98, EH01, ENV13]), and it is applied only after Theorem 1.1 has already established equality of the classical invariants; this is an implication, not a definitional shortcut. No parameter is fitted to data and no quantity is renamed as a prediction. The cited results from the literature are used as black boxes rather than as premises supplied by the present authors. The only potentially problematic step is the unproved claim in Case 3.1 that the linear terms in the summed d3-invariants cancel and that the quadratic coefficient is nonzero; that is a gap in proof or an unsupported hypothesis, but not circularity, because the equality rot(L')=rot(L) is not assumed and is not forced by the smooth characterizing-slope input alone. Consequently, there is no self-definition, no fitted-input-called-prediction, and no load-bearing self-citation chain; the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters. The paper relies on standard theorems in 3-manifold topology and contact geometry as black boxes. Two auxiliary assumptions about the contact surgery expansion are introduced ad hoc in the proof and are not independently justified.

assumptions (5)
  • domain assumption The smooth characterizing slope theorems for unknot, trefoils, figure-eight, 5_2, and torus knots are correct.
    Used as black boxes in Corollaries 1.3-1.5.
  • domain assumption The d-invariant formula of Ni-Wu and the d3-invariant formulas of Ding-Geiges-Stipsicz and Durst-Kegel are correct.
    Underpin the computations in Propositions 2.9, 2.10 and Lemmas 3.1, 3.2.
  • domain assumption Legendrian simplicity of unknot, trefoils, figure-eight, 5_2 and torus knots is correct.
    Needed to apply the uniqueness assumption in Corollary 1.2.
  • ad hoc to paper The set of expanded contact (±1)-surgery diagrams for L(p/q) is closed under reversing all stabilization signs.
    Asserted in the proof of Theorem 1.1 to show the linear term in the d3-sum vanishes; not proven or cited.
  • ad hoc to paper The quadratic coefficient (1,...,1)Q^{-1}(1,...,1)^T in the d3-sum is nonzero for the slopes considered.
    Required for the uniqueness of the non-negative solution r'=r in Case 3.1; not established in the text.

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Pith. "Pith review of Characterizing slopes for Legendrian knots." pith.science (2026). https://pith.science/paper/MUXGAIG5

@misc{pith2026260806079,
  author       = {Pith},
  title        = {Pith review of: Characterizing slopes for Legendrian knots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MUXGAIG5}},
  note         = {Machine review of arXiv:2608.06079}
}
abstract

We establish a criterion relating smooth and contact characterizing slopes under a uniqueness assumption. Let $L$ be a Legendrian representative of a knot $K\subset S^3$ with standard contact structure, and assume that the isotopy class of $L$ is uniquely determined by its classical invariants: the Thurston--Bennequin invariant $tb(L)$ and the rotation number. Then, for any non-zero rational number $r$, if $r+tb(L)$ is a smooth characterizing slope for $K$, it becomes a contact characterizing slope for $L$. As applications, we study the characterizing slopes for Legendrian representatives of the unknot, trefoil, figure-eight knot, cinquefoil, $5_2$, and $\overline{5_2}$.

Figures

Figures reproduced from arXiv: 2608.06079 by the authors.

Figure 1
Figure 1. Here L1 is a Legendrian 52 with tb(L1) = 1 and rot(L1) = 0, L2 is a Legendrian P(3, −3, −8) with tb(L2) = −3 and rot(L2) = 2. Example 1.7. For a general Legendrian torus knot, not every non-zero rational number is a characterizing slope. For example, Let L1 be a Legendrian torus knot T−5,4 with Thurston-Bennequin invariant −20 and rotation number 1, and L2 be a Legendrian torus knot T−11,2 with Thurston-Bennequin in… view at source ↗
Figure 2
Figure 2. Contact p-surgery along L, where p ≥ 2. As to L, r = (r, r ± 1)T , Q =  t + 1 tp − t t tp − t − p  , b = (b1, b2) T , b1 = pr ± pt ∓ t t + p , b2 = −r ∓ t ∓ 1 t + p . As to L ′ , r ′ = (r ′ , r′ ± 1)T , Q′ =  −2p − t + 1 −2p 2 − pt + 2p + t −2p − t −2p 2 − pt + p + t  , b ′ = (b ′ 1 , b′ 2 ) T , b ′ 1 = −pr′ ± (2p + t)p ∓ (2p + t) p + t , b′ 2 = r ′ ∓ (2p + t) ± 1 p + t . If t + p > 0, σ(Q) = 0, σ(Q ′ ) = −2. Th… view at source ↗
Figure 3
Figure 3. Contact p 2 -surgery along L, where p ≥ 5. and r ′ has the following four cases: r ′1 = (r ′ , r′ + 1, r′ + 2)T , r ′2 = (r ′ , r′ + 1, r′ ) T , r ′3 = (r ′ , r′ − 1, r′ − 2)T , r ′4 = (r ′ , r′ − 1, r′ ) T . We solve the equations Qb = r and Q′b ′ = r ′ in each case. b 1 = (pr + pt − t p + 2t , − 2r + 2t + 1 p + 2t , − r + 2t + p+1 2 p + 2t ) T , b 2 = (pr + pt − 3t p + 2t , − 2r + 2t + 3 p + 2t , −r + p−3 2 p + 2t… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Contact 3 2 -surgery along L. t ′ =  −3 − t −5 − t  , Q′ =  −2 − t −3 − t −3 − t −6 − t  , r ′ =  r ′ r ′ 2  , r′ 2 = r ′ + 2, r′ , r′ − 2. Note that det(Q) = −2t − 3, tr(Q) = 2t − 2, det(Q′ ) = 2t + 3, tr(Q′ ) = −2t − 8. We solve the equations Qb = r and Q′b ′ =…
Figure 5
Figure 5. Figure 5: Contact p 2 -surgery along L, where p ≤ −1. Then Q =  t + p−1 2 t + p+1 2 t + p+1 2 t + p−1 2  , r =  ri ri  , b =  ri p+2t ri p+2t  , Q ′ =  −t − p+1 2 −t − p−1 2 −t − p−1 2 −t − p+1 2  , r ′ =  rj rj  , b ′ = −rj p+2t −rj p+2t ! . We have det(Q) = −(p + 2t)…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.