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arxiv: 2409.09842 · v2 · submitted 2024-09-15 · 🧮 math.GT

The search for alternating surgeries

Kenneth L. Baker , Marc Kegel , Duncan McCoy This is my paper

Pith reviewed 2026-05-23 20:43 UTC · model grok-4.3

classification 🧮 math.GT
keywords alternating surgeryDehn surgeryhyperbolic knotsdouble branched coversalternating linkssurgery slopesgenus boundslens space surgeries
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The pith

The set of alternating surgery slopes on a knot is algorithmically computable, bounded by |p/q| ≤ 3g(K)+4 for hyperbolic knots.

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

The paper shows that alternating surgeries—those Dehn surgeries on a knot in the three-sphere whose result is the double branched cover of an alternating link—have a computable collection of slopes. A sympathetic reader would care because this converts the question of which slopes produce such covers into one that existing 3-manifold algorithms can decide in finite time, enabling exhaustive checks on knot tables. The authors also prove that hyperbolic knots admit only slopes satisfying the stated genus bound and apply the results to many explicit examples. These computations display new patterns, such as strongly invertible knots with exactly one alternating surgery and asymmetric knots with two.

Core claim

The central claim is that the set of alternating surgery slopes is algorithmically computable. For a hyperbolic knot any such slope p/q satisfies |p/q| ≤ 3g(K)+4, which strengthens the genus bounds that follow from the Goda-Teragaito conjecture on lens space surgeries. The authors compute the full sets for all hyperbolic knots in the SnapPy census and record phenomena including strongly invertible knots with a unique alternating surgery and asymmetric knots with two alternating surgery slopes.

What carries the argument

Alternating surgery, defined as a surgery on a knot in S^3 that yields the double branched cover of an alternating link. This object lets the problem reduce to decidable questions about alternating links via their branched covers.

If this is right

  • The set of alternating surgery slopes on any given knot can be computed by an algorithm.
  • Hyperbolic knots admit alternating surgery slopes only when |p/q| ≤ 3g(K)+4.
  • The same bound applies to lens space surgeries and improves the genus restriction from the Goda-Teragaito conjecture.
  • Strongly invertible knots may possess exactly one alternating surgery slope.
  • Asymmetric knots may possess exactly two alternating surgery slopes.

Where Pith is reading between the lines

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

  • The computability result could support systematic enumeration of all knots that admit any alternating surgery.
  • The explicit bound supplies a concrete test that can be checked on knot tables larger than the current census.
  • The distinction between knots with one versus two alternating slopes may relate to symmetry detection in other surgery problems.
  • The same reduction to branched covers might adapt to decide membership in other link classes beyond alternating links.

Load-bearing premise

Existing algorithms in 3-manifold topology can decide whether a given manifold is the double branched cover of an alternating link.

What would settle it

A single hyperbolic knot K of genus g together with an alternating surgery slope p/q satisfying |p/q| > 3g(K)+4 would disprove the bound.

Figures

Figures reproduced from arXiv: 2409.09842 by Duncan McCoy, Kenneth L. Baker, Marc Kegel.

Figure 1
Figure 1. Figure 1: Alternating diagrams of twelve alternating links are shown. Each has an arc along which a flat banding produces the unknot. The double branched covers of the exteriors of these arcs are the 12 census knots with a single alternating surgery. Each diagram is labeled with the SnapPy manifold of the census knot and the Hoste-Thistlethwaite name of the alternating link [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The incidence number of a crossing. is a choice of checkerboard colouring which gives every crossing incidence number µ = 1. Fix this choice of colouring and let R0, . . . , Rr denote the white regions in the plane. The white graph ΓD is obtained by taking one vertex for each region Ri and one edge between distinct regions Ri and Rj for every crossing between Ri and Rj in D. The Goeritz lattice of D is def… view at source ↗
Figure 3
Figure 3. Figure 3: A graph to show the relationship between the ti and the Tk. We have also shown how ρ0 and ρ1 occur as the number of ti equal to one and two, respectively (see Remark 8.8). The Tk are related to a changemaker vector via the following result. Lemma 8.7 ([McC17b, Lemma 2.10]) Let K be an L-space knot and let σ ∈ Z r be a changemaker vector compatible with ∆K(x). For 1 ≤ k < t0 we have Tk = max α∈Sk,r σ · α, w… view at source ↗
Figure 4
Figure 4. Figure 4: The outline of an algorithm to calculate Salt(K) for a non-trivial knot K. Lemma 11.2 Let K be a knot which admits positive alternating surgeries. If the stable coefficients of K satisfy ρ(K) = (2, . . . , 2 | {z } g ), then K is the torus knot T2,2g+1. Proof. Since the stable coefficients of a knot determine its Alexander polynomial, we see that K and T2,2g+1 must have the same Alexander polynomials. From… view at source ↗
Figure 5
Figure 5. Figure 5: The family of diagrams {Dn}n≥1 defining the knots Kn of Example 11.4. The chosen unknotting crossing is circled in red. The black block contains a twist of n ≥ 1 crossings. Example 11.4 For n ≥ 1 the (9n + 19 2 )-changemaker lattice L = ⟨e−1 − e0, e0 + e1 + 2e2 + 2e3 + 3e4 + · · · + 3en+3⟩ ⊥ ⊆ Z n+5 , admits a planar obtuse superbase B. By embedding the corresponding graph GB in the plane, one can construc… view at source ↗
Figure 6
Figure 6. Figure 6: The outline of a practical strategy to calculate Salt(K) for a hyperbolic knot K. The justification of the output follows directly by combining Theorems 1.2 and 8.5. Step 0: Restricting to L-space knots. Since any knot which admits an al￾ternating surgery is an L-space knot it suffices to consider L-space knots. From Dunfield’s data [Dun20a], we can extract a list of all L-space knot exteriors in the SnapP… view at source ↗
Figure 7
Figure 7. Figure 7: Two 15-crossing alternating diagrams of 2-component links. The double branched cover of the left link yields K(1,1,0,1,2)(1142, 1). The double branched cover of the right link yields K(2,1,0,1,1)(1066, 1). JSJ decomposition of a knot exterior [Bud06]. One of the components in the JSJ decomposition XK contains the boundary of XK. We will refer to this distinguished piece as the outermost piece of the decomp… view at source ↗
read the original abstract

Surgery on a knot in $S^3$ is said to be an alternating surgery if it yields the double branched cover of an alternating link. The main theoretical contribution is to show that the set of alternating surgery slopes is algorithmically computable and to establish several structural results. Furthermore, we calculate the set of alternating surgery slopes for many examples of knots, including all hyperbolic knots in the SnapPy census. These examples exhibit several interesting phenomena including strongly invertible knots with a unique alternating surgery and asymmetric knots with two alternating surgery slopes. We also establish upper bounds on the set of alternating surgeries, showing that an alternating surgery slope on a hyperbolic knot satisfies $|p/q| \leq 3g(K)+4$. Notably, this bound applies to lens space surgeries, thereby strengthening the known genus bounds from the conjecture of Goda and Teragaito.

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 / 1 minor

Summary. The paper defines an alternating surgery on a knot in S^3 as one whose result is the double branched cover of an alternating link. It claims that the set of alternating surgery slopes on any knot is algorithmically computable, derives several structural results on these slopes, computes the sets explicitly for many knots (including all hyperbolic knots in the SnapPy census), and proves that any alternating surgery slope r = p/q on a hyperbolic knot satisfies |p/q| ≤ 3g(K) + 4. The bound is noted to strengthen known genus bounds for lens space surgeries from the Goda-Teragaito conjecture. Examples illustrate phenomena such as unique alternating surgeries on strongly invertible knots and multiple slopes on asymmetric knots.

Significance. If the central claims hold, the work supplies both a theoretical framework for identifying surgeries yielding double branched covers of alternating links and a substantial body of explicit data from the SnapPy census. The slope bound |p/q| ≤ 3g(K) + 4 provides a concrete, genus-dependent restriction that improves on existing conjectural bounds for lens spaces. The computational enumeration, which reveals specific phenomena such as unique alternating surgeries, constitutes a useful contribution to the study of exceptional surgeries. The manuscript's use of existing 3-manifold algorithms for the census computations is a practical strength.

major comments (1)
  1. [Abstract] Abstract (main theoretical contribution): the claim that the set of alternating surgery slopes is algorithmically computable requires an effective, terminating procedure that decides whether a given 3-manifold is the double branched cover of an alternating link. While manifold recognition, hyperbolicity testing, and branched-cover reconstruction are known to be algorithmic, the manuscript does not cite or supply a decision procedure with proven termination for the subproblem of recognizing whether the link is alternating. This decidability gap is load-bearing for the computability assertion.
minor comments (1)
  1. The abstract refers to 'several interesting phenomena' observed in the census examples; a one-sentence summary of the most striking of these would improve readability for readers who do not consult the full tables.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this point concerning the algorithmic computability claim. We respond below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (main theoretical contribution): the claim that the set of alternating surgery slopes is algorithmically computable requires an effective, terminating procedure that decides whether a given 3-manifold is the double branched cover of an alternating link. While manifold recognition, hyperbolicity testing, and branched-cover reconstruction are known to be algorithmic, the manuscript does not cite or supply a decision procedure with proven termination for the subproblem of recognizing whether the link is alternating. This decidability gap is load-bearing for the computability assertion.

    Authors: We agree that the manuscript does not explicitly cite or describe a terminating decision procedure for recognizing whether a link is alternating. The computability claim in the abstract rests on the composition of known algorithms for the other steps (surgery computation, branched-cover reconstruction, and manifold recognition), but the alternating-link recognition subproblem requires additional justification. We will revise the manuscript to add a brief subsection outlining such a procedure, for instance by reference to diagram enumeration via normal surfaces (which terminates for a fixed link by crossing-number bounds) combined with verification of the alternating property on reduced diagrams. This will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on external algorithmic decidability assumptions

full rationale

The paper's core claims—that alternating surgery slopes are algorithmically computable and satisfy |p/q| ≤ 3g(K)+4—are framed as theoretical results derived from definitions of alternating surgeries (yielding double branched covers of alternating links) and known results in 3-manifold topology. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described structure. The computability assertion relies on effective decidability of manifold properties and alternating link recognition, but this is an external assumption rather than a circular reduction to the paper's own inputs. The bound is presented as a strengthening of prior genus conjectures without evident internal fitting or renaming. This qualifies as a normal non-finding per the evaluation criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard axioms of 3-manifold topology and knot theory with no new free parameters or invented entities.

axioms (1)
  • standard math Properties of double branched covers of links and alternating links are effectively decidable via existing 3-manifold algorithms
    Invoked to establish algorithmic computability of the surgery slopes.

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