Families of cosmetic surgeries

Abstract: We construct infinite families of chirally cosmetic surgeries on chiral hyperbolic knots and purely cosmetic surgeries on hyperbolic manifolds with multiple cusps, disproving conjectures that these phenomena do not appear, including Problem 1.12(d) in the K3 problem list. We also give some hints regarding why chirally cosmetic surgeries appear to be more common than purely cosmetic surgeries on $1$-cusped manifolds.

• The paper presents a general construction that generates infinite families of chirally and purely cosmetic surgeries, challenging previous conjectures in 3-manifold topology.
• It employs geometric techniques including Dehn filling and controlled symmetries alongside group-theoretic analysis to systematically produce counterexamples.
• The findings imply that cosmetic surgeries are more prevalent in multi-cusped and chiral knot settings, reshaping the classification of exceptional Dehn fillings.

Infinite Families of Cosmetic Surgeries in Hyperbolic 3-Manifolds

Introduction and Background

The study of cosmetic surgeries, both purely and chirally, is central to the understanding of 3-manifold topology, particularly in the context of exceptional Dehn fillings on cusped hyperbolic manifolds. A Dehn filling is termed purely cosmetic if different filling slopes produce orientation-preservingly homeomorphic manifolds, and chirally cosmetic if they are orientation-reversingly homeomorphic. The long-standing cosmetic surgery conjecture posits that, except for clear geometric reasons, non-equivalent slopes cannot produce cosmetic fillings (2604.02672).

Historically, the conjecture held up under exhaustive computational and theoretical scrutiny for one-cusped hyperbolic manifolds and knot complements in $S^3$, with results like those of Ni–Wu constraining possible exceptions to tightly circumscribed cases. Similarly, conjectural constraints extend to multi-cusped settings and chirality, notably in the K3 list’s Problem 1.12(d), where it is expected that chiral, non-torus knots do not admit chirally cosmetic surgeries. Against this backdrop, the paper fundamentally alters the landscape by constructing infinite families that violate these expectations.

Main Results and Constructions

The core contribution is a general construction—parametrized in terms of manifolds with symmetries and specific long slopes—that generates infinite families of (1) chirally cosmetic surgeries on hyperbolic knots that are themselves chiral, and (2) purely cosmetic surgeries on multi-cusped hyperbolic manifolds, specifically demonstrating that the phenomenon is not restricted to isolated examples. These constructions systematically disprove several notable conjectures, including Problem 1.12(d) in the K3 list.

Specifically, two principal theorems are established:

• Theorem 1: For any $n\geq 3$, there exist $n$-cusped hyperbolic manifolds and pairs of slopes on the first cusp such that the corresponding Dehn fillings are purely cosmetic—not dictated by a manifold symmetry sending one set of slopes to the other.
• Theorem 2: There exist infinite families of asymmetric (chiral) knots in $S^3$ admitting chirally cosmetic hyperbolic surgeries, explicitly violating previously held conjectures around chirality and cosmetic surgery.

These results are obtained by a flexible model construction (see Section 2 of the paper): starting with a cusped hyperbolic manifold $N$ with toroidal boundaries and a symmetry $\phi$ that cyclically permutes $k$ cusps, then performing fillings along carefully chosen, long slopes subject to the action of $\phi$. Crucially, this method leverages the abundance of symmetries in multi-cusped or link exteriors, but the approach is shown—by contradiction using Euclidean structure and orbifold analysis—not to yield purely cosmetic surgeries for one-cusped hyperbolic manifolds, clarifying prior computational findings where chirally, but not purely, cosmetic surgeries appeared prevalent.

Methodology and Key Technical Ingredients

The constructions exploit both geometric and group-theoretic properties:

• For multi-cusped manifolds, the method is to realize a manifold as a finite abelian cover of an orbifold with rigid cusp shapes, ensuring the symmetry group behaves in a controlled, cyclic manner and the slopes on each cusp can be “cycled” using the deck group. By interleaved Dehn fillings along long slopes, Mostow rigidity ensures that resulting fillings are hyperbolic and can be orchestrated to yield homeomorphic (cosmetic) pairs with no symmetry accounting for the mapping between filled slopes.
• For the hyperbolic, chiral knot setting, the method uses complements of Brunnian links with highly nontrivial symmetry groups. By performing non-equivalent surgeries on link components using slopes dictated by a rotational orientation-reversing symmetry, chirally cosmetic surgeries on their images (which are asymmetric knots) are obtained, confirming the infinite nature of such counterexamples.

Vital in these constructions is the systemic use of hyperbolic manifolds with very controlled symmetry—achieved through careful choice of link exteriors and orbifolds—and analysis of the behavior of isometries and slope actions post-filling, supported by geometric finiteness and length spectrum arguments.

Implications and Contradictory Claims

The paper makes explicit, strong claims contradicting established conjectures:

• It constructs infinite families of chirally cosmetic surgeries on chiral, hyperbolic knots in $S^3$.
• It also constructs infinite families of hyperbolic manifolds with multiple cusps admitting purely cosmetic fillings not explained by symmetries of the ambient manifold.

These claims are supported by rigorous constructions and justification that no symmetry matches the observed cosmetic equivalences (e.g., detailed group action analysis and control of hidden symmetries, leveraging non-arithmeticity of the underlying orbifolds).

Moreover, the results clarify the empirical computational phenomenon: the scarcity of purely cosmetic surgeries in the one-cusped hyperbolic setting is explained by group-theoretic impossibility, while chirally cosmetic surgeries are substantially more prevalent due to the broader class of symmetries available (including orientation-reversing ones).

Theoretical and Practical Consequences

Theoretically, these findings demand a re-evaluation of the structure of exceptional Dehn fillings and cosmetic surgery phenomena. Previous conjectures that appeared robust to both computational verification and theoretical constraint are now effectively refuted in full generality for hyperbolic manifolds with more than one cusp and for chirally cosmetic surgeries among chiral hyperbolic knots.

Practically, this expands the class of potential exceptional phenomena in the census and classification of Dehn surgeries on cusped manifolds and knot complements. For computational experiments, it suggests that searches for cosmetic surgeries must be extended beyond symmetry-constrained pairs, especially in the multi-cusped and chirally cosmetic context, and validates the absence of purely cosmetic one-cusped examples as a consequence of deep geometric obstruction rather than simply computational limitations.

Future research directions include:

• Systematic classification of all possible cosmetic fillings arising from the general model construction, and clarifying possible exceptions outside this scheme.
• Investigating the frequency and distribution of such cosmetic pairs in the “generic” hyperbolic topological setting.
• Exploring the implications for Floer-theoretic and quantum invariants, where such surgery equivalences may manifest subtle structures.

Conclusion

This work establishes the existence of infinite families of both purely and chirally cosmetic surgeries in settings previously conjectured to forbid them, by explicit construction rooted in the topology and geometry of hyperbolic manifolds and their symmetries. These results necessitate a fundamental revision of the conjectural landscape surrounding cosmetic Dehn surgery, and mark a significant shift in both practical enumeration and theoretical understanding of 3-manifold topology (2604.02672).