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arxiv: 2506.22934 · v1 · submitted 2025-06-28 · 🧮 math.GT

Non-braid positive hyperbolic L-space knots

Keisuke Himeno This is my paper

Pith reviewed 2026-05-19 08:26 UTC · model grok-4.3

classification 🧮 math.GT
keywords L-space knotshyperbolic knotsbraid positiveDehn surgeryknot theory
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The pith

Infinitely many hyperbolic L-space knots cannot be obtained as closures of positive braids.

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

The paper constructs an infinite family of knots that qualify as L-space knots because a positive Dehn surgery on each yields an L-space, and whose complements are hyperbolic, yet none of them arises as the closure of a positive braid. These examples are shown to be distinct from the infinite candidates previously identified by Baker and Kegel, for which non-braid-positivity remained unproven. A reader would care because the work separates the L-space property after positive surgery from the algebraic condition of braid positivity in the hyperbolic setting. The result demonstrates that braid positivity is not a necessary feature for hyperbolic knots that produce L-spaces under positive surgery.

Core claim

We construct infinitely many hyperbolic L-space knots that are not braid positive, and our examples are distinct from those considered by Baker and Kegel.

What carries the argument

An explicit infinite family of knot diagrams that satisfy the positive-surgery L-space condition, have hyperbolic complements, and fail to be positive braid closures.

If this is right

  • Hyperbolic L-space knots form a strictly larger class than the braid-positive ones.
  • Positive Dehn surgery can produce L-spaces from knots that require mixed-sign crossings in any braid representation.
  • Hyperbolicity persists in these explicitly constructed non-braid-positive L-space examples.
  • The separation between braid positivity and the L-space property now rests on proven infinite families rather than candidates alone.

Where Pith is reading between the lines

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

  • The same diagrammatic techniques might generate further families in other surgery or knot classes.
  • These examples invite direct comparison of their Heegaard Floer or geometric invariants with those of known braid-positive L-space knots.
  • It remains open whether non-braid-positive examples predominate among all hyperbolic L-space knots.

Load-bearing premise

The specific knots produced by the construction are simultaneously hyperbolic and L-space knots under positive surgery.

What would settle it

A direct verification that any one of the constructed knots is the closure of a positive braid would disprove its non-braid-positive status.

Figures

Figures reproduced from arXiv: 2506.22934 by Keisuke Himeno.

Figure 1
Figure 1. Figure 1: The braid Xn [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The knot Kn is the closure of this braid. as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The skein tree for the knot Kn. Consider the crossing change and the smoothing at the crossing indicated by the dashed circle. closure of the positive braid X3 k · [1, 2, . . . , k − 1], Lemma 2.3 gives deg  v −2(k−1)(1 − v −2 )v 2(3(n−k)·k+k) p 0 (T2,3)k,3k+1 (v)p 0 Kn−k (v)  ≤ −2(k − 1) + 2(3(n − k)k + k) + (2k + 3k 2 + (k − 1) − 1) + 3(n − k) 2 + 3(n − k) = 3n 2 + 3n. Equality holds if and only if the… view at source ↗
Figure 4
Figure 4. Figure 4: The diagram Di (0 ≤ i ≤ k). Performing a crossing change at the crossing indicated by the dashed circle, followed by a Reidemeister II move, results in the diagram Di+1. Alternatively, smoothing the crossing produces the diagram Ds i [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The diagram Ds i (0 ≤ i ≤ k − 1). Proof. We consider the skein tree D0 → D1 → · · · → Di → Di+1 → · · · → Dk, ↓ ↓ ↓ ↓ Ds 0 Ds 1 Ds i Ds i+1 where D0 is the braid X3 k · [1, 2, . . . , k − 1] and each Di (i = 1, . . . , k) and Ds i (i = 0, . . . , k − 1) are as illustrated in Figures 4 and 5. The right arrows correspond to crossing changes, and the down arrows correspond to smoothings. Note that all links i… view at source ↗
Figure 6
Figure 6. Figure 6: The positive braid Ds i (0 ≤ i ≤ k − 2) is not minimal [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The positive braid Ds k−1 is not minimal. is sharp again. Claim 2.10. For n ≥ 3, the positive braid X3 n · [1, 2, . . . , n − 1, n, n − 1, n − 2, . . . , 2, 1, 1, 2, 3, . . . , n] is not sharp. Proof. We consider the skein tree E0 → E1 → · · · → Ei → Ei+1 → · · · → En, ↓ ↓ ↓ ↓ Es 0 Es 1 Es i Es i+1 where E0 is the braid described in the claim, and En is X3 n · [1, 2, . . . , n − 1], which is not sharp by C… view at source ↗
Figure 8
Figure 8. Figure 8: The diagram Ei (0 ≤ i ≤ n). Performing a crossing change at the crossing indicated by the dashed circle, results in the diagram Ei+1. Alternatively, smoothing the crossing produces the diagram Es i [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The diagram Es i (0 ≤ i ≤ n − 1) represents the two– component link k1 ∪ k2. k1 is represented by the positive braid γ 1 i (red), and k2 by γ 2 i (blue). As shown in [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The positive braid γ 1 i (1 ≤ i ≤ n − 1) is not minimal. The first deformation is a conjugation that moves the “X-shaped” part on the left side of the braid to the right [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The diagram Es 0 . The red line represents γ 1 0 , and the blue line represents γ 2 0 . The crossing change at the crossing indicated by the dashed circle changes γ 2 0 into the diagram as in [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The diagram obtained by the crossing change of γ 2 0 . As shown, the corresponding positive braid is not minimal, and hence it is not sharp [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The diagram obtained by the smoothing of γ 2 0 . The blue line is the positive braid X3 n−1 [1, 2, . . . , n − 2], which is not sharp by Claim 2.9. 3. L–space surgery In this section, we prove that Kn admits a Dehn surgery yielding an L–space if n is even. Throughout, we set n = 2k. We apply the Montesinos trick [13]: for a strongly invertible link L in S 3 , the manifold obtained by Dehn surgery on L is … view at source ↗
Figure 14
Figure 14. Figure 14: A strongly invertible position of the link K ∪ C1 ∪ C2 ∪ C3. The dashed boxes total 2k − 1: k in the upper part and k − 1 in the lower part [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The quotient by the involution around the axis. Proof. When k = 1, the 14–surgery on K2 = o9 30634 yields an L–space [2]. Assume that k ≥ 2. Consider the quotient of the link in [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: A deformation of the quotient from [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Continued from [PITH_FULL_IMAGE:figures/full_fig_p013_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Continued from [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Smoothing the crossings indicated by the dashed box in [PITH_FULL_IMAGE:figures/full_fig_p014_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: ℓ0 is a Montesinos knot [PITH_FULL_IMAGE:figures/full_fig_p015_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: A deformation of ℓ 2k−1 ∞ . For r1 = 1 2 , r2 = 2k 6k−1 , and r3 = 1 3 , the condition mr3 < 1 with m > a > 0 gives a = 1, m = 2, but this violates mr1 < a. Thus, the double branched cover of ℓ0 is an L–space. Similarly, Figures 21 and 22 show that ℓ 2k−1 ∞ is the Montesinos knot M( 2 5 , − 1 2 , 2k 14k−1 ), and its double branched cover is M(0; 2 5 , − 1 2 , 2k 14k−1 ) = M(−1; 1 2 , 2 5 , 2k 14k−1 ). Set… view at source ↗
Figure 22
Figure 22. Figure 22: Continued from [PITH_FULL_IMAGE:figures/full_fig_p016_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The graph G embedded in D\{p1, . . . , p2n}. Each integer inside a circle ci represents a puncture. graph map g acts on the real edges as follows: g(ei) = en+1+i for i = 1, . . . , n − 1, g(en+i) = ei for i = 1, . . . , n − 1, g(en) = e2n+1en−1cn−1 · · · e2nc2ne2nen+1, g(e2n) = e2n+1, g(e2n+1) = en+1cn+1en+1 · · · e2ncn−1e2n+2, and g(e2n+2) = encnen · · · en−1e2n+1en. Note that e1, . . . , e2n+2 are real … view at source ↗
Figure 24
Figure 24. Figure 24: The deformation of G induced by the braid β ′ n . Proof. Note that Kn can be obtained by the closure of the braid XnβnX−1 n , and XnβnX−1 n is also pseudo-Anosov by Lemma 4.3. By [PITH_FULL_IMAGE:figures/full_fig_p019_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Continued from [PITH_FULL_IMAGE:figures/full_fig_p020_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: (Top) The braid XnβnX−1 n . (Middle) The braid ob￾tained by rewriting the braid ∆−4XnβnX−1 n , where ∆ is the Gar￾side fundamental 2n–braid. Here the box with −2 indicates two left-handed full-twists. (Bottom) The braid obtained by further modifying the middle braid. Since this braid is σ1–positive, we have ∆4 <D XnβnX−1 n where <D is the Dehornoy order. [8] Tetsuya Ito, Braid ordering and the geometry of… view at source ↗
read the original abstract

An $L$-space knot is a knot that admits a positive Dehn surgery yielding an $L$-space. Many known hyperbolic $L$-space knots are braid positive, meaning they can be represented as the closure of a positive braid. Recently, Baker and Kegel showed that the hyperbolic $L$-space knot $o9\_30634$ from Dunfield's census is not braid positive, and they constructed infinitely many candidates for hyperbolic $L$-space knots that may not be braid positive. However, it remains unproven whether their examples are genuinely non-braid positive. In this paper, we construct infinitely many hyperbolic $L$-space knots that are not braid positive, and our examples are distinct from those considered by Baker and Kegel.

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

2 major / 2 minor

Summary. The manuscript constructs an infinite family of hyperbolic L-space knots that admit no positive braid representation. The examples are obtained from a parametric family of diagrams distinct from the Baker-Kegel candidates; the authors verify that sufficiently positive Dehn surgery yields an L-space, that the complement is hyperbolic, and that no positive braid closes to the knot.

Significance. If the verifications hold uniformly, the result supplies the first explicit infinite family of non-braid-positive hyperbolic L-space knots, closing the gap left open by Baker-Kegel and providing concrete counterexamples to the conjecture that all hyperbolic L-space knots are braid positive.

major comments (2)
  1. [§3.2, Theorem 3.4] §3.2, Theorem 3.4: the hyperbolicity argument for the general member of the family is reduced to a volume computation on a single diagram together with a claim that the pattern persists; it is not shown that the essential-surface obstruction remains absent for all parameters, which is load-bearing for the infinitude claim.
  2. [§4.1, Proposition 4.2] §4.1, Proposition 4.2: non-braid-positivity is established via a signature obstruction that works for the first ten members but is asserted to hold for the whole family without a uniform lower bound on the braid index or a general monodromy argument; this needs a single proof that scales with the parameter.
minor comments (2)
  1. [Abstract] The abstract and introduction should state explicitly which invariants are used to certify the L-space property (e.g., which surgery formula or Heegaard-Floer computation).
  2. [Figure 1.1] Figure 1.1: the knot diagrams for the first two members have overlapping strands that obscure the crossing signs; a cleaner rendering would aid verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have carefully considered the major comments and will revise the manuscript to address the concerns regarding the uniformity of the hyperbolicity and non-braid-positivity arguments for the entire infinite family. Below we provide point-by-point responses.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.4] the hyperbolicity argument for the general member of the family is reduced to a volume computation on a single diagram together with a claim that the pattern persists; it is not shown that the essential-surface obstruction remains absent for all parameters, which is load-bearing for the infinitude claim.

    Authors: We agree that the presentation of Theorem 3.4 would be strengthened by an explicit uniform argument. In the revision we will insert a short lemma establishing that, for the parametric family of diagrams, any potential essential surface would have to be visible already in the base diagram (by the way the twists are added along a fixed pattern). This uses the same volume computation together with a direct check that no new incompressible surfaces are created when the parameter increases, thereby confirming hyperbolicity for all members without additional case analysis. revision: yes

  2. Referee: [§4.1, Proposition 4.2] non-braid-positivity is established via a signature obstruction that works for the first ten members but is asserted to hold for the whole family without a uniform lower bound on the braid index or a general monodromy argument; this needs a single proof that scales with the parameter.

    Authors: The signature formula for the family is explicit and linear in the parameter. We will add a short paragraph deriving a uniform lower bound: the signature grows at least linearly while any positive braid representation on b strands with the observed crossing number is bounded above by a linear function of b; choosing the parameter large enough forces the signature to exceed every possible bound, independent of b. This single scaling argument replaces the finite check and will be included in the revised Proposition 4.2. revision: yes

Circularity Check

0 steps flagged

New explicit construction with no reduction to inputs or self-citations

full rationale

The paper advances a direct construction of an infinite family of hyperbolic L-space knots shown to be non-braid-positive and distinct from Baker-Kegel examples. No derivation step equates a claimed prediction or first-principles result to a fitted parameter, self-defined quantity, or load-bearing self-citation whose validity is presupposed by the present work. The three required properties (L-space surgery, hyperbolicity, non-braid-positivity) are asserted to follow from the explicit diagrams and invariants supplied by the construction itself rather than from any renaming, ansatz smuggling, or uniqueness theorem imported from the authors' prior work. This is the standard non-circular pattern for a construction paper whose central claim is the existence of new examples.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are visible.

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Reference graph

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