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

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Apratim Chakraborty, Swarup Kumar Das, Tanushree Shah

Geometric Topology · Submitted 25 Jun 2026 · 50 references

Pith's verdict: The paper adapts the S3 proof to integer homology sphere L-spaces via knot Floer and d-invariant constraints and the argument goes through cleanly for non-trivial Legendrian knots except the Lagrangian-slice case.

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arXiv ID2606.27485 (version 1)
Pith IDBE65XCVS, Pith's permanent record of this paper and its review
CitationApratim Chakraborty, Swarup Kumar Das, Tanushree Shah. "Contact cosmetic surgery on Legendrian knots in integer homology sphere $L$-spaces." arXiv:2606.27485 (2026). https://pith.science/paper/BE65XCVS
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The paper

The paper claims

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

The reading

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.

What carries the argument

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

Extended reading notes

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.

Load-bearing premise

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

Editorial extensions

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.

Reading between the lines

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.
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

Figures
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 first with n stablizations of one sign or the other). This diagram is taken from Version 1 of [ES25] (arXiv). We begin with n = 2 and let X±2 be the 4-manifolds given by the surgery…
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
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 − 1 (that is the second knot is the Legendrian push-off of the first with n − 1 stablizations of one sign or the other). This diagram is taken from Version 1 of [ES25] (arXiv) hence
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 (that is the second knot is the Legendrian push-off of the first with n − 1 stablizations of one sign or the other). This diagram is taken from Version 1 of [ES25] (arXiv). Lemma…

Figures reproduced from arXiv:2606.27485 by the authors.

References

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 15 sentences without a confident label

Compares with 1 paper
  • For smooth $(±2)$–surgeries, both the contact surgery diagrams and the corresponding $d_3$–computations are identical to those appearing in the work of the third author and Etnyre in version 2 of [ref] .

    On contact cosmetic surgery · Limiting contact cosmetic surgeries · 0.90

Cites as background 3 papers
  • We summarize the bordered Heegaard Floer machinery of Lipshitz-Ozsváth-Thurston [ref] and its reformulation by Hanselman-Rasmussen-Watson [ref] via immersed curves in the punctured torus, restricting attention to the case of torus boundary, which is the case we will need.

    Bordered Heegaard Floer homology: Invariance and pairing · Bordered Heegaard Floer homology and the immersed curve invariant · 0.85

  • We summarize the bordered Heegaard Floer machinery of Lipshitz-Ozsváth-Thurston [ref] and its reformulation by Hanselman-Rasmussen-Watson [ref] via immersed curves in the punctured torus, restricting attention to the case of torus boundary, which is the case we will need.

    Bordered Floer homology for manifolds with torus boundary via immersed curves · Bordered Heegaard Floer homology and the immersed curve invariant · 0.85

  • We summarize the bordered Heegaard Floer machinery of Lipshitz-Ozsváth-Thurston [ref] and its reformulation by Hanselman-Rasmussen-Watson [ref] via immersed curves in the punctured torus, restricting attention to the case of torus boundary, which is the case we will need.

    Heegaard Floer homology for manifolds with torus boundary: properties and examples · Bordered Heegaard Floer homology and the immersed curve invariant · 0.85

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