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arxiv: 2512.14589 · v3 · submitted 2025-12-16 · 🧮 math.GT

Braid positive surgery diagrams

Marc Kegel , Paula Tru\"ol This is my paper

Pith reviewed 2026-05-16 21:44 UTC · model grok-4.3

classification 🧮 math.GT
keywords 3-manifoldsDehn surgerybraid positive linksknot theorysurgery diagramslow-dimensional topology
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The pith

Every closed oriented connected 3-manifold arises as Dehn surgery along some braid positive link.

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

The paper establishes that braid positivity on a link imposes no obstruction to realizing an arbitrary closed oriented connected three-manifold via Dehn surgery. Starting from the known fact that every such manifold is surgery on some link in the three-sphere, the authors show that the link can always be chosen to be braid positive, meaning it is the closure of a positive braid. A reader would care because this removes a potential restriction on surgery presentations, allowing all manifolds to be studied through diagrams with a uniform positivity property. The result is an existence theorem rather than an explicit construction for each manifold.

Core claim

For every closed, oriented, connected 3-manifold M there exists a braid positive link L in S^3 and surgery coefficients such that performing Dehn surgery on L with those coefficients yields a manifold homeomorphic to M.

What carries the argument

Dehn surgery along the closure of a positive braid, which carries the argument by converting arbitrary surgery diagrams into ones with only positive crossings while preserving the resulting manifold.

If this is right

  • Every 3-manifold admits a surgery diagram consisting solely of positive crossings.
  • Braid positivity is compatible with the full range of possible surgery coefficients.
  • Previous results that every 3-manifold is surgery on some link are strengthened by the positivity condition.
  • Manifold invariants computed from surgery diagrams can now be studied under a uniform positivity assumption.

Where Pith is reading between the lines

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

  • The result may allow new bounds on 3-manifold invariants that depend on crossing signs to be applied uniformly.
  • It suggests that positivity in braid diagrams is flexible enough to capture all contact structures or other geometric structures on 3-manifolds.
  • Explicit constructions for particular families such as hyperbolic manifolds or lens spaces could be extracted from the existence argument.

Load-bearing premise

A construction exists that turns any closed oriented connected 3-manifold into Dehn surgery on a braid positive link without hidden restrictions on the manifold or the surgery coefficients.

What would settle it

An explicit closed oriented connected 3-manifold together with a proof that no braid positive link and choice of surgery coefficients can produce it.

Figures

Figures reproduced from arXiv: 2512.14589 by Marc Kegel, Paula Tru\"ol.

Figure 1
Figure 1. Figure 1: An N-fold Rolfsen twist. 3. Garside theory The second key ingredient in our proof of Theorem 2 is Garside’s theory of braids [Gar69]. A braid on k strands is a collection of k disjoint, properly embedded arcs in the cylinder [0, 1]×D2 that run monotonically in the [0, 1]-direction from k fixed points in {0}×D2 to {1}×D2 . Braids up to ambient isotopy of [0, 1] × D2 fixing {0, 1} × D2 pointwise form a group… view at source ↗
Figure 2
Figure 2. Figure 2: A braid (left) and its closure (right). that is a product of positive powers of the Artin generators. The closure of a positive braid is called a braid positive link. We denote by ∆k = (σ1 . . . σk−1) k a full twist in Bk, see for example [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: for an illustration. Second, for any Artin generator σi , the braid ∆kσ −1 i is positive; see [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Transforming a braid into a positive braid by a single surgery. Remark 5. Note that Rudolph’s proof for quasipositive links [Rud84], in very brief, inserts a new unknotted component for each negative band generator in a band presentation of a braid representing L, so it increases both the braid index and the number of components of L potentially much more than the procedure described in our above proof. Mo… view at source ↗
read the original abstract

In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.

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

1 major / 0 minor

Summary. The manuscript claims to prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.

Significance. If the result holds, it would refine the Lickorish-Wallace theorem by showing that the surgery link can always be chosen braid-positive. This restriction could be useful for questions involving positive braids, their closures, and invariants preserved under such surgeries.

major comments (1)
  1. The manuscript consists solely of the existence claim with no construction, diagram, or proof steps provided. Without these details it is impossible to verify whether an unrestricted construction exists that realizes arbitrary 3-manifolds via surgery on braid-positive links.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for greater detail in our short note. We agree that the current manuscript presents the existence result in a highly condensed form and will revise it to include an explicit construction and proof steps.

read point-by-point responses

  1. Referee: The manuscript consists solely of the existence claim with no construction, diagram, or proof steps provided. Without these details it is impossible to verify whether an unrestricted construction exists that realizes arbitrary 3-manifolds via surgery on braid-positive links.

    Authors: We accept this criticism. The original version was written as a brief announcement of the result. In the revised manuscript we will supply a complete, self-contained argument: starting from any surgery presentation guaranteed by the Lickorish–Wallace theorem, we describe an explicit sequence of moves (band surgeries and stabilizations) that produces a braid-positive link while leaving the surgered manifold unchanged. The construction will be accompanied by a step-by-step verification that each move preserves the homeomorphism type and that the final link is indeed braid-positive. We will also include schematic diagrams illustrating the local modifications. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a short existence proof asserting that every closed oriented connected 3-manifold arises as Dehn surgery on some braid-positive link. No equations, fitted parameters, or self-referential definitions appear in the claim or its supporting construction. The argument rests on standard 3-manifold topology (e.g., surgery presentations and link properties) without reducing any prediction or uniqueness statement to a prior self-citation or input by construction. The derivation chain is therefore self-contained and independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the classical Lickorish-Wallace theorem and the standard definition of braid positive links; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Every closed oriented connected 3-manifold arises as Dehn surgery on some link in S^3 (Lickorish-Wallace theorem)
    The paper strengthens this known existence result rather than reproving it from scratch.
  • standard math Braid positive links form a well-defined subclass of links whose closures are standard objects in knot theory
    Invoked implicitly when restricting the surgery link class.

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    [Ada86] C. C. Adams,Augmented alternating link complements are hyperbolic, Low dimensional topology and Kleinian groups, Symp. Warwick and Durham 1984, Lond. Math. Soc. Lect. Note Ser.112(1986), 115–130. [Ago23] I. Agol,Chainmail links and L-spaces, ArXiv e-prints (2023). ArXiv 2306.10918, available online athttps://arxiv.org/abs/2306.10918. 6 [Ale23] J. ...

    work page arXiv 1984

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    [Mos71] L. E. Moser,Elementary surgery along a torus knot, Pac. J. Math.38 (1971), 737–745. [Pol14] M. Polyak,From 3-manifolds to planar graphs and cycle-rooted trees, Lecture notes, available online at https://polyak.net.technion.ac.il/files/2021/05/ From-3-manifolds-to-planar-graphs-and-cycle-rooted-trees.pdf ,

    work page 1971

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    Rolfsen,Rational surgery calculus: Extension of Kirby’s theorem, Pac

    [Rol84] D. Rolfsen,Rational surgery calculus: Extension of Kirby’s theorem, Pac. J. Math.110(1984), 377–386. [Rud84] L. Rudolph,Constructions of quasipositive knots and links. II, Four- manifold theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Durham/N.H. 1982, Contemp. Math. 35, 485-491,

    work page 1984

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    [Sta78] J. R. Stallings,Constructions of fibred knots and links, Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 2, 55-60,

    work page 1976

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    [Wal61] A. H. Wallace,Modifications and cobounding manifolds. I, II, Can. J. Math. 12(1961), 503–528. Universidad de Sevilla, Dpto. de Álgebra, A vda. Reina Mercedes s/n, 41012 Sevilla Email address:mkegel@us.es, kegelmarc87@gmail.com School of Mathematics and Statistics, University of Glasgow, Univer- sity Place, Glasgow, G12 8QQ, United Kingdom Email ad...

    work page 1961