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arxiv: 2606.27485 · v1 · pith:BE65XCVSnew · submitted 2026-06-25 · 🧮 math.GT

Contact cosmetic surgery on Legendrian knots in integer homology sphere L-spaces

Apratim Chakraborty , Swarup Kumar Das , Tanushree Shah This is my paper

Pith reviewed 2026-06-29 01:13 UTC · model grok-4.3

classification 🧮 math.GT
keywords contact cosmetic surgeryLegendrian knotsL-spacesHeegaard Floer theoryinteger homology spherescontact topologyknot surgeryLagrangian slice knots
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The pith

The contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots in integer homology sphere L-spaces, except possibly for Lagrangian slice knots.

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

This paper extends results on contact cosmetic surgeries from the three-sphere to integer homology sphere L-spaces. It establishes that such surgeries on non-trivial Legendrian knots cannot yield the same manifold with reversed orientation, except possibly when the knot is Lagrangian slice. The proof refines earlier S3 methods by adding obstructions drawn from Heegaard Floer homology. A reader would care because the result enlarges the class of manifolds in which contact structures are rigidly constrained by surgery data.

Core claim

We prove that the contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots in integer homology sphere L-spaces, with the possible exception of Lagrangian slice knots. Our argument adapts and refines techniques from the S3 case to the broader context of L-spaces, incorporating constraints arising from Heegaard Floer theory.

What carries the argument

Adaptation of contact surgery techniques from the three-sphere to L-spaces, using additional constraints from Heegaard Floer theory.

If this is right

  • Contact cosmetic surgeries are ruled out for every non-trivial Legendrian knot that is not Lagrangian slice.
  • The conjecture is settled for the entire class of integer homology sphere L-spaces.
  • Heegaard Floer obstructions suffice to replace the S3-specific arguments in this setting.
  • Only Lagrangian slice knots remain as potential exceptions requiring separate study.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same adaptation may succeed in other manifolds where Heegaard Floer homology is explicitly computable.
  • Lagrangian slice knots could be handled by combining Floer theory with other invariants such as knot Floer homology.
  • The result supplies a template for proving absence of cosmetic surgeries in contact manifolds beyond L-spaces.

Load-bearing premise

Techniques from the S3 case can be adapted and refined to L-spaces by incorporating constraints arising from Heegaard Floer theory.

What would settle it

Existence of a non-trivial Legendrian knot in an integer homology sphere L-space that is not Lagrangian slice yet admits a pair of contact cosmetic surgeries producing oppositely oriented manifolds.

Figures

Figures reproduced from arXiv: 2606.27485 by Apratim Chakraborty, Swarup Kumar Das, Tanushree Shah.

Figure 1
Figure 1. Figure 1: For a Legendrian knot K with tb = −1 we see a smooth −1/2 surgery (that is contact (1/2) surgery) and −1/n surgery for n > 2 (that is contact ((n − 1)/n) surgery) on the upper and lower left, respectively, and a smooth 1/n surgery (that is a contact ((n + 1)/n) surgery) on the right. On the right a and b are non-negative integers so that a+b = n (that is the second knot is the Legendrian push-off of the fi… view at source ↗
Figure 2
Figure 2. Figure 2: Computing the signature for Xn. This diagram is taken from Version 1 of [ES25] (arXiv). Lemma 6.3. For any choice of stablization in the definition of X−n we have c 2 1 (X−n) = 2−n. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: For a Legendrian knot K with tb = −2 we see a smooth −1 surgery (that is contact (1) surgery) and −1/n surgery for n > 1 (that is contact ((2n − 1)/n) surgery) on the upper and lower left, respectively (the stabilizations in the lower diagram can be of any sign), and a smooth 1/n surgery (that is a contact ((2n + 1)/n) surgery) on the right. On the right a and b are non-negative integers so that a + b = n … view at source ↗
Figure 4
Figure 4. Figure 4: For a Legendrian knot K with tb = −k < −2 we see a smooth −1/n surgery (that is contact ((kn − 1)/n) surgery) on the left, (the stabi￾lizations in the lower diagram can be of any sign). The red portion of the diagram should be ignored fo n = 1. On the right, we see smooth 1/n surgery (that is a contact ((kn + 1)/n) surgery) on the right. On the right a and b are non-negative integers so that a + b = n − 1 … view at source ↗
read the original abstract

We extend the study of contact cosmetic surgeries to Legendrian knots in integer homology sphere L-spaces . We prove that the contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots in this setting, with the possible exception of Lagrangian slice knots. Our argument adapts and refines techniques from the S3 case to the broader context of L-spaces, incorporating constraints arising from Heegaard Floer theory

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that the contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots in integer homology sphere L-spaces, with the possible exception of Lagrangian slice knots. The argument adapts and refines techniques from the S^3 case by incorporating constraints from Heegaard Floer theory on the knot Floer complex and d-invariants.

Significance. If the result holds, it extends the contact cosmetic surgery conjecture from the S^3 setting to the larger class of integer homology sphere L-spaces. The work shows how Heegaard Floer invariants can be used to resolve questions about Legendrian knots and contact surgeries, strengthening the case for the general conjecture while isolating the Lagrangian-slice case as the sole potential exception.

minor comments (2)
  1. The title uses '$L$-spaces' while the abstract uses 'L-spaces'; standardize the notation for L-spaces throughout the manuscript.
  2. Ensure that all references to prior S^3 results (e.g., the base techniques being adapted) are cited explicitly in the introduction and in the relevant technical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. The referee's summary accurately reflects the scope and methods of the work.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The manuscript adapts and refines existing S3-case techniques to integer homology sphere L-spaces by imposing Heegaard Floer constraints on the knot Floer complex and d-invariants. The abstract and skeptic summary indicate that the central claim follows directly from these external constraints once applied, with the Lagrangian-slice case isolated as an explicit exception. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the provided text; the argument remains independent of its own outputs by construction and relies on prior independent results from Heegaard Floer theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work appears to rest on standard background results in Heegaard Floer theory and contact geometry whose details are not visible here.

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Reference graph

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