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arxiv: 2604.02672 · v1 · submitted 2026-04-03 · 🧮 math.GT

Families of cosmetic surgeries

Qiuyu Ren This is my paper

Pith reviewed 2026-05-13 19:08 UTC · model grok-4.3

classification 🧮 math.GT
keywords cosmetic surgerieschirally cosmetic surgeriespurely cosmetic surgerieshyperbolic knotsDehn surgerychiral knotsmulti-cusped manifolds3-manifold topology
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The pith

Infinite families of chirally cosmetic surgeries exist on chiral hyperbolic knots, disproving conjectures that they do not appear.

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

The paper constructs infinite families of Dehn surgeries on chiral hyperbolic knots that produce homeomorphic manifolds with opposite orientations. These are known as chirally cosmetic surgeries. It also builds purely cosmetic surgeries, which preserve orientation, on hyperbolic manifolds with multiple cusps. The constructions disprove conjectures that such surgeries cannot occur in these restricted settings, including a specific problem from the K3 list. A reader would care because the examples show cosmetic surgeries are possible even when knots are chiral and manifolds are hyperbolic, affecting how we distinguish manifolds from their mirrors.

Core claim

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.

What carries the argument

Infinite families of specific hyperbolic knots and multi-cusped manifolds obtained through targeted Dehn fillings that preserve hyperbolicity while forcing the resulting manifolds to be homeomorphic to their mirrors or to each other.

If this is right

  • Conjectures that chirally cosmetic surgeries cannot occur on chiral hyperbolic knots are false.
  • Purely cosmetic surgeries can occur on hyperbolic manifolds with more than one cusp.
  • The distinction between chirally cosmetic and purely cosmetic surgeries is not as strict as some prior expectations suggested.
  • Problem 1.12(d) from the K3 list has counterexamples.

Where Pith is reading between the lines

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

  • The same construction methods might generate examples on other classes of knots or with different numbers of cusps.
  • The hints about relative frequency could guide searches for when orientation-reversing homeomorphisms arise after surgery.
  • These families might affect practical recognition algorithms for 3-manifolds by providing explicit cases where invariants must match despite apparent differences.
  • Further analysis of the surgery coefficients in the families could reveal patterns that apply to broader Dehn surgery questions.

Load-bearing premise

The knots and manifolds in the constructed families remain hyperbolic and chiral or multi-cusped after the chosen Dehn fillings.

What would settle it

A computation or geometric argument showing that one of the surgeries in the families produces a manifold that fails to be homeomorphic to its mirror image or that loses hyperbolicity would show the families do not work as claimed.

Figures

Figures reproduced from arXiv: 2604.02672 by Qiuyu Ren.

Figure 1
Figure 1. Figure 1: A 3-component amphichiral hyperbolic Brunnian link with symmetry group Z/6. The image was created by the program KnotJob [Sch]. (3) The symmetry group of the pair (S 3 , L) is Z/6, which is equal to the symmetry group of N = S 3\L; (4) A generator ϕ of the symmetry group is orientation-reversing on N, permuting the compo￾nents of L cyclically in the forward order. We postpone the proof of Proposition 4.1 a… view at source ↗
read the original 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.

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 manuscript constructs infinite families of chirally cosmetic surgeries on chiral hyperbolic knots and purely cosmetic surgeries on hyperbolic manifolds with multiple cusps. These families serve as counterexamples to conjectures that such surgeries do not occur, including Problem 1.12(d) in the K3 problem list. The paper also provides some hints on why chirally cosmetic surgeries appear more common than purely cosmetic surgeries on 1-cusped manifolds.

Significance. If the constructions hold, the result is significant because it supplies explicit infinite families of Dehn fillings that realize both chiral and pure cosmetic phenomena on hyperbolic objects, directly disproving several standing conjectures. The direct comparison of fundamental groups or explicit homeomorphisms, combined with standard large-parameter hyperbolicity arguments, provides concrete, falsifiable examples rather than conditional or parameter-dependent claims.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'give some hints' is vague; the corresponding section (likely §4 or the discussion of prevalence) should be referenced explicitly so readers know where the heuristic observations appear.
  2. [Construction sections] The statement that hyperbolicity holds 'for all but finitely many parameters' would benefit from a brief citation to the precise volume or cusp-geometry theorem invoked, even if standard.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; direct constructive existence proof

full rationale

The paper presents an explicit construction of infinite families of Dehn fillings on base hyperbolic knots and multi-cusped manifolds, showing the resulting filled manifolds are homeomorphic (chirally or purely cosmetic) by direct comparison of fundamental groups or explicit homeomorphisms. Hyperbolicity for large parameters follows from standard volume/cusp arguments that are independent of the target cosmetic property. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology; the central existence claim is self-contained against external benchmarks such as the hyperbolic Dehn filling theorem and does not rely on load-bearing self-citations for its validity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts from hyperbolic geometry and Dehn surgery theory. No free parameters are fitted to data. No new entities are postulated.

axioms (2)
  • standard math Hyperbolic knots and manifolds admit Dehn surgeries that preserve hyperbolicity for sufficiently large slopes (Thurston's hyperbolic Dehn surgery theorem).
    Invoked implicitly to ensure the resulting manifolds remain hyperbolic after the cosmetic surgeries.
  • domain assumption Chiral knots are not equivalent to their mirrors, allowing distinction between orientation-preserving and orientation-reversing homeomorphisms.
    Used to separate chirally cosmetic from purely cosmetic cases.

pith-pipeline@v0.9.0 · 5335 in / 1418 out tokens · 29275 ms · 2026-05-13T19:08:08.425178+00:00 · methodology

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Works this paper leans on

1 extracted references · 1 canonical work pages

  1. [1]

    295, Math

    [BKR] R.İ.Baykur,R.C.Kirby,andD.Ruberman,eds.,K3: A New Problem List in Low-Dimensional Topology, vol. 295, Math. Surveys Monogr. To appear, Providence, RI: American Mathematical Society. [BHW99] S. A. Bleiler, C. D. Hodgson, and J. R. Weeks,Cosmetic surgery on knots, Geom. Topol. Monogr.2 (1999), 23–34. [Cul+] M. Culler, N. M. Dunfield, M. Goerner, and J...

    work page arXiv 1999