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

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

• The paper proves the contact cosmetic surgery conjecture for nontrivial Legendrian knots in integer homology sphere L-spaces, except potentially for ±2 surgery when τ_Y(K)=0, g(K)=2, and maximal tb(K)=-1.
• The authors combine Heegaard Floer correction terms, an extended Ni–Wu obstruction, immersed-curve analysis, and d₃-invariant calculations to restrict possible cosmetic slopes to {±2} or {±1/q} before eliminating all but the stated exception.
• The results show that any Legendrian knot admitting cosmetic contact surgery must satisfy tb(L)+|rot(L)|≤−1, ruling out all examples with tb(L)≥0 and identifying finer invariants as necessary for the remaining cases.

This paper by Chakraborty, Das, and Shah extends the study of cosmetic surgeries to the contact category for Legendrian knots in integer homology sphere $L$-spaces. The central result is that the contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots in such manifolds, with a single narrow exceptional family: $\pm 2$ surgery on a Legendrian knot whose smooth type $K$ satisfies $\tau_Y(K)=0$, $g(K)=2$, and $\overline{tb}(K)=-1$. The argument combines Heegaard Floer correction terms, an extension of Ni–Wu's $\tau$-invariant obstruction to $L$-spaces, an extension of Hanselman's immersed-curve slope analysis, and explicit $d_3$-invariant computations from contact surgery diagrams.

Setting and main statement

Let $(Y,\xi)$ be a tight contact structure on an integer homology sphere $\pm 2$0-space (ZHS$\pm 2$1) with nonvanishing Heegaard Floer contact invariant; the Poincaré homology sphere $\pm 2$2 with its Stein-fillable tight structure is the motivating example. Contact $\pm 2$3-surgery on a Legendrian knot $\pm 2$4 corresponds topologically to smooth Dehn surgery of coefficient $\pm 2$5 relative to the Seifert framing. Two contact surgeries with distinct smooth coefficients are cosmetic if the resulting contact manifolds are contactomorphic; since contactomorphisms are orientation-preserving, the cosmetic/truly cosmetic distinction collapses in this category.

The paper formulates the contact cosmetic surgery conjecture: no Legendrian knot that is not smoothly an unknot admits cosmetic contact surgeries. The main theorem establishes it except possibly for $\pm 2$6 surgery on a knot type with $\pm 2$7, $\pm 2$8, $\pm 2$9. Local knots (those contained in a 3-ball) reduce immediately to the $K$0 analysis of Etnyre–Shah (Etnyre et al., 2024), so the substantive content concerns genuinely global knot types.

A separate refinement treats surgeries at fixed smooth coefficient: two contact $K$1-surgeries are weakly cosmetic if contactomorphic and strongly cosmetic if isotopic as contact structures. The authors pose, rather than conjecture, the question of which Legendrian knots admit weakly or strongly cosmetic surgeries, noting that Floer-theoretic constraints from the $K$2-space condition severely restrict strong cosmetically for $K$3-space knots realizing $K$4.

Extension of the Ni–Wu obstruction

The first structural result extends Ni–Wu's theorem from $K$5 to integer homology sphere $K$6-spaces: if $K$7 admits truly cosmetic surgeries, then $K$8. The proof adapts the mapping-cone rational surgery formula verbatim, using that $K$9 forces every $\tau_Y(K)=0$0-equivariant endomorphism of the tower $\tau_Y(K)=0$1 to be multiplication by $\tau_Y(K)=0$2, yielding integers $\tau_Y(K)=0$3 with monotonicity properties identical to the $\tau_Y(K)=0$4 case. A new $\tau_Y(K)=0$5-invariant formula is established:

$\tau_Y(K)=0$6

with the normalization $\tau_Y(K)=0$7 following from the connected sum formula and additivity of $\tau_Y(K)=0$8-invariants. Combining Wu's same-sign obstruction, the Boyer–Lines vanishing $\tau_Y(K)=0$9 via Casson–Walker and Casson–Gordon invariants, and the equality case of the $g(K)=2$0-invariant bound gives $g(K)=2$1, hence $g(K)=2$2.

The contact consequence is immediate: since $g(K)=2$3 agrees with Hedden's contact $g(K)=2$4 when the contact invariant is nontrivial, Plamenevskaya's bound $g(K)=2$5 yields the striking constraint that any Legendrian knot admitting a cosmetic contact surgery satisfies

$g(K)=2$6

In particular, all Legendrian knots with $g(K)=2$7 are eliminated outright.

Extending Hanselman's immersed curve analysis

The second major component extends Hanselman's classification of possible cosmetic slopes from $g(K)=2$8 to integer homology sphere $g(K)=2$9-spaces. Via the Hanselman–Rasmussen–Watson immersed curve invariant $\overline{tb}(K)=-1$0 of the bordered knot exterior, the key structural facts carry over: because $\overline{tb}(K)=-1$1 has rank one, exactly one distinguished component $\overline{tb}(K)=-1$2 meets the meridian line, wrapping once around the punctured cylinder, with the remaining components null-homologous and subject to a $\overline{tb}(K)=-1$3-rotation symmetry.

The condition $\overline{tb}(K)=-1$4 implies, via Hom's isolated-generator criterion (whose proof the authors verify works verbatim over general integer homology $\overline{tb}(K)=-1$5-spaces), that $\overline{tb}(K)=-1$6 splits off a tower summand and $\overline{tb}(K)=-1$7 is horizontal. From there:

• Equality of Floer ranks under the diffeomorphism forces $\overline{tb}(K)=-1$8, so cosmetic pairs have slopes $\overline{tb}(K)=-1$9.
• Vanishing of the Casson–Walker invariant of $\tau$0, equivalently of the Dedekind sum $\tau$1, forces $\tau$2.
• A relative grading computation—showing the grading shift $\tau$3 for generators on vertical segments, independent of the ambient $\tau$4—yields the sum rule $\tau$5, hence the rank inequality $\tau$6.

Consequences include: genus-one knots admit no truly cosmetic surgery; large-slope pairs ($\tau$7) force $\tau$8, $\tau$9, and $L$0; small-slope pairs ($L$1) satisfy a strengthened inequality $L$2 forcing $L$3. Thus the only candidate cosmetic slope pairs are $L$4 (requiring $L$5) and $L$6. This reduction is what makes the subsequent diagrammatic analysis finite.

The $L$7 computations

To compare candidate cosmetic contact surgeries, the authors need $L$8-invariants of contact manifolds presented by surgery on knots inside $L$9. A technical lemma, adapted from Kim's algorithmic work on Seifert surfaces in surgery presentations, shows that handle slides can be arranged so the knot has zero linking number with every component of the background surgery link presenting $d_3$0. The linking matrix then becomes block-diagonal, and Gompf's formula decouples:

$d_3$1

so only diagrams in $d_3$2 need be analyzed. This reduction is what allows the $d_3$3 computations of Etnyre–Shah to transfer.

The case analysis proceeds by Thurston–Bennequin invariant:

• $d_3$4: rotation number must vanish. For $d_3$5 surgery, $d_3$6 versus $d_3$7; for $d_3$8 ($d_3$9), $(Y,\xi)$0 while $(Y,\xi)$1. No cosmetic pair exists, except the $(Y,\xi)$2 case where $(Y,\xi)$3 does not distinguish the resulting contact structures—this is precisely the residual exception in the main theorem.
• $(Y,\xi)$4: rotation number must be $(Y,\xi)$5. For $(Y,\xi)$6 surgery, $(Y,\xi)$7 values are $(Y,\xi)$8 versus $(Y,\xi)$9 or $\pm 2$00; for $\pm 2$01, the $\pm 2$02 values on one side are always even while those on the other are $\pm 2$03 or odd, so no agreement occurs.
• $\pm 2$04: for $\pm 2$05 surgery, solving the quadratic $\pm 2$06 in the rotation parameter $\pm 2$07 yields no integer solutions; for $\pm 2$08, solving for $\pm 2$09 and imposing the constraint $\pm 2$10 on the unknot rotation parameter likewise yields no integral solutions.

The supporting linear algebra (determinants, signatures, and $\pm 2$11 computations for the relevant tridiagonal-type intersection matrices) is carried out in an appendix via cofactor formulas and Sylvester's criterion.

Limitations and open questions

The main theorem leaves open the family of knots with $\pm 2$12, $\pm 2$13, $\pm 2$14 under $\pm 2$15 surgery, where the $\pm 2$16-invariant fails to distinguish the resulting contact structures; resolving these cases requires finer invariants. The weakly/strongly cosmetic question at fixed smooth coefficient is posed but not resolved. The treatment of Legendrian unknots reveals a dichotomy—some contact surgeries have no cosmetic counterparts while others are contactomorphic to infinitely many distinct surgeries on the same unknot—but this phenomenon is described rather than fully classified. The extension of Hom's isolated-generator proposition to $\pm 2$17-spaces rests on the claim that its chain-complex-level proof transfers verbatim; the authors assert rather than reprove this. Finally, the entire framework assumes the ambient contact structure has nontrivial contact invariant, so overtwisted ambient settings lie outside scope.

Conclusion

The paper reduces the contact cosmetic surgery problem in integer homology sphere $\pm 2$18-spaces to a single exceptional configuration, by combining three independent lines of obstruction: the $\pm 2$19-invariant vanishing forced by Floer-theoretic $\pm 2$20-invariant bookkeeping, the slope restriction $\pm 2$21 or $\pm 2$22 from immersed curves, and exhaustive $\pm 2$23-invariant comparisons across all admissible Thurston–Bennequin values. The residual exception—$\pm 2$24 surgery on genus-two, $\pm 2$25-vanishing knots with maximal Thurston–Bennequin number $\pm 2$26—mirrors the state of the smooth problem and marks the precise boundary of current techniques.