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arxiv: 2604.16857 · v1 · submitted 2026-04-18 · 🧮 math.GT

On hyperbolic L-space knots with braid index four and tunnel number two

Masakazu Teragaito This is my paper

Pith reviewed 2026-05-10 07:24 UTC · model grok-4.3

classification 🧮 math.GT
keywords L-space knotshyperbolic knotsbraid indextunnel numberstrongly invertibleSnapPy census
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The pith

An infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two is constructed.

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

The paper establishes the existence of infinitely many strongly invertible hyperbolic L-space knots that have braid index four and tunnel number two. Previously, only three such knots were known from the SnapPy census: t09284, t10496, and o9_34409. By providing an explicit infinite family that includes t10496 and o9_34409, the work shows that these properties can be satisfied simultaneously in an infinite collection. A sympathetic reader would care because this advances the classification of L-space knots and their geometric properties in low braid index.

Core claim

In this paper, we give the first infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two that includes t10496 and o9_34409.

What carries the argument

The infinite family of knots parametrized so that every member is hyperbolic, an L-space knot, strongly invertible, of braid index four, and of tunnel number two.

If this is right

  • There are infinitely many such knots rather than only the three previously known examples.
  • t10496 and o9_34409 belong to the same parametric family.
  • The combination of strong invertibility, hyperbolicity, and the L-space property occurs infinitely often at braid index four.
  • The tunnel number two condition is compatible with all the other properties in an infinite set.

Where Pith is reading between the lines

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

  • Similar explicit families might exist for other fixed braid indices or tunnel numbers.
  • These examples could be used to test conjectures relating L-space knots to their covering spaces or surgeries.
  • Computational checks on small members of the family could verify the pattern holds beyond the two included census knots.

Load-bearing premise

Every knot in the constructed infinite family satisfies the hyperbolic, L-space, strongly invertible, braid index four, and tunnel number two conditions simultaneously.

What would settle it

Any specific member of the family that is shown to be non-hyperbolic or not an L-space knot would disprove that the entire family satisfies the stated conditions.

Figures

Figures reproduced from arXiv: 2604.16857 by Masakazu Teragaito.

Figure 1
Figure 1. Figure 1: The link K0 ∪ c. By performing (−1/n)-surgery along c, K0 will be changed into Kn. This diagram also shows a strongly invertible position of the link. Lemma 2.1. For any n ≥ 0, Kn is strongly invertible. Proof [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Kirby–Rolfsen calculus for (8, 0)-surgery on K0∪c. The result is a lens space L(9, 7). Lemma 2.3 ([3, Theorem 1.13 and Lemma 6.1], [4, Lemma 2.1]). Let K ∪ c be a link of an L–space knot K and an unknot c with linking number w = lk(K, c) > 1. Let Kn be the image of K after (−1/n)-surgery on c for integers n. If there exists a slope r ≥ 2g(K) − 1 such that (r, 0)-surgery on K ∪ c is an L–space, then Kn is a… view at source ↗
Figure 3
Figure 3. Figure 3: The diagram after the tangle replacements (top left) and its deformation. The box with integer n contains right handed horizontal n half twists. Lemma 3.2. If n ≥ 3, then Kn has tunnel number two. Proof. First, the two arcs t1 and t2, as shown in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Further deformation. The resulting link (right) consists of two Montesinos tangles [−1/3, 1/3] and [−1/(n − 1), −1/2] [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An unknotting tunnel system {t1, t2} for Kn. Remark 3.3. It is well known that a torus knot has tunnel number one, so is K0. For K2, which is s682, it is not hard to see that it admits an unknotting tunnel. In fact, t1, shown in [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

There are only three known strongly invertible hyperbolic L-space knots with braid index four and tunnel number two. They are t09284, t10496 and o9_34409 in the SnapPy census. In this paper, we give the first infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two that includes t10496 and o9_34409.

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 asserts that only three strongly invertible hyperbolic L-space knots with braid index four and tunnel number two appear in the SnapPy census (t09284, t10496, o9_34409) and constructs the first infinite family of such knots, including t10496 and o9_34409.

Significance. An explicit infinite family satisfying all five conditions simultaneously would be a meaningful advance in the study of L-space knots, their geometric realizations, and the interplay between braid index, tunnel number, and strong invertibility. It would supply new examples for testing conjectures on L-space knots and potentially allow systematic computation of invariants across the family.

major comments (1)
  1. Abstract and introduction: the central claim is the existence of an infinite family in which every member is simultaneously hyperbolic, an L-space knot, strongly invertible, of braid index four, and of tunnel number two. No explicit construction, generating set, or verification argument is supplied in the provided text, so it is impossible to check whether the five conditions hold for all members or whether the family reduces to the three known examples by specialization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review of our manuscript. We address the major comment regarding the explicit construction of the infinite family below. We will make revisions to improve the clarity and completeness of the presentation.

read point-by-point responses

  1. Referee: Abstract and introduction: the central claim is the existence of an infinite family in which every member is simultaneously hyperbolic, an L-space knot, strongly invertible, of braid index four, and of tunnel number two. No explicit construction, generating set, or verification argument is supplied in the provided text, so it is impossible to check whether the five conditions hold for all members or whether the family reduces to the three known examples by specialization.

    Authors: We agree with the referee that the abstract and introduction lack sufficient detail on the construction and verification. While the full manuscript provides the explicit construction of the infinite family in subsequent sections, we will revise the introduction to include a summary of the generating set for the family and an outline of the arguments verifying that each member satisfies the five conditions (hyperbolicity, L-space knot, strong invertibility, braid index four, and tunnel number two). We will also explicitly show that the family is infinite and includes t10496 and o9_34409 as special cases without reducing to only the three known examples. This revision will make the claims verifiable from the introduction as well. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided document contains only an abstract stating the existence of an infinite family of knots with listed properties and noting three known examples; no equations, constructions, definitions, theorems, proofs, or self-citations appear in the visible text. Without any derivation chain or load-bearing steps to inspect, no reduction to inputs by construction can be exhibited, satisfying the rule that circularity is claimed only when a specific quoted reduction is shown.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no information on free parameters, axioms, or invented entities used in any construction.

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Reference graph

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