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arxiv: 2511.10606 · v2 · pith:7FJOGEINnew · submitted 2025-11-13 · 🧮 math.GT

SL₂(mathbb R)-representations and left-orderable surgeries of (-2, 3, 2n+1)-pretzel knots

Anh T. Tran This is my paper

Pith reviewed 2026-05-22 11:51 UTC · model grok-4.3

classification 🧮 math.GT MSC 57M2557M27
keywords pretzel knotsleft-orderable groupsSL(2,R) representationsDehn surgeryknot groups3-manifoldsfundamental groups
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The pith

Continuous paths of SL(2,R)-representations establish left-orderability for m/l-surgeries on (-2,3,2n+1)-pretzel knots when m/l is below 2 floor((2n+4)/3).

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

The paper constructs explicit continuous paths of SL(2,R)-representations for the knot groups of (-2,3,2n+1)-pretzel knots with n an integer at least 3 and not equal to 4. These paths are then applied to prove that the fundamental group of the manifold obtained by m/l-surgery from the 3-sphere along the knot is left-orderable whenever the slope satisfies m/l less than 2 times the floor of (2n+4)/3. A sympathetic reader would care because left-orderability of the fundamental group connects to the existence of taut foliations and other structural features of the resulting 3-manifolds.

Core claim

We provide an explicit construction of continuous paths of SL(2,R)-representations of the knot groups of (-2,3,2n+1)-pretzel knots. As an application, we show that the fundamental group of the 3-manifold obtained from the 3-sphere by m/l-surgery along the (-2,3,2n+1)-pretzel knot, where n ≥ 3 is an integer and n ≠ 4, is left-orderable if m/l < 2 ⌊(2n+4)/3⌋.

What carries the argument

Explicitly constructed continuous paths of SL(2,R)-representations of the knot groups, which satisfy conditions for a criterion that implies left-orderability of the resulting surgery manifold groups.

If this is right

  • For each fixed n satisfying the conditions, left-orderability holds for every surgery slope below the n-dependent bound.
  • The proven interval of left-orderable slopes lengthens as n increases.
  • The result supplies an infinite family of explicit examples of left-orderable fundamental groups arising from surgery on pretzel knots.

Where Pith is reading between the lines

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

  • The representation-path technique might extend to other pretzel knot families to produce analogous slope bounds for left-orderability.
  • The given bound may not be sharp, so left-orderability could persist for some larger slopes that the current method does not capture.
  • These examples help map the transition between left-orderable and non-left-orderable surgery manifolds across knot families.

Load-bearing premise

The constructed continuous paths of SL(2,R)-representations must satisfy non-triviality or irreducibility conditions required by the criterion used to conclude left-orderability.

What would settle it

A specific integer n ≥ 3 with n ≠ 4 and a slope m/l below 2 floor((2n+4)/3) for which the fundamental group of the surgery manifold is shown not to be left-orderable, for instance by exhibiting a non-left-orderable finite quotient.

Figures

Figures reproduced from arXiv: 2511.10606 by Anh T. Tran.

Figure 1
Figure 1. Figure 1: The (−2, 3, 2n + 1)-pretzel knot. We can find a presentation of G(Kn) with two generators and one relator as follows. Let w := cb. Then the relation cacb = acba becomes caw = awa. This implies that c = awaw−1a −1 and b = c −1w = awa−1w −1a −1w. The relation ba(cb) n = a(cb) n c can be written as awa−1w −1a −1wawn = awnawaw−1a −1 , i.e. w n awa−1w −1 a −1 = a −1w −1 awaw−1w n . Hence G(Kn) = ⟨a, w | w nu = … view at source ↗
read the original abstract

In this paper, we provide an explicit construction of continuous paths of $\mathrm{SL}_2(\mathbb R)$-representations of the knot groups of $(-2,3,2n+1)$-pretzel knots. As an application, we show that the fundamental group of the $3$-manifold obtained from the $3$-sphere by $\frac{m}{l}$-surgery along the $(-2,3,2n+1)$-pretzel knot, where $n \ge 3$ is an integer and $n \not= 4$, is left-orderable if $\frac{m}{l}< 2 \lfloor \frac{2n+4}{3} \rfloor$.

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 / 3 minor

Summary. The manuscript constructs explicit continuous paths of SL(2,ℝ)-representations of the knot groups of the (-2,3,2n+1)-pretzel knots for integers n ≥ 3 with n ≠ 4. As an application, it proves that the fundamental group of the 3-manifold obtained by m/l-surgery on such a knot is left-orderable whenever m/l < 2⌊(2n+4)/3⌋.

Significance. If the constructions and verifications hold, the work supplies explicit, parameter-free paths of representations that yield concrete slope bounds for left-orderability of surgeries on this infinite family of pretzel knots. The explicitness of the paths is a strength, as it permits direct checking of the non-triviality conditions needed to apply standard criteria linking non-abelian SL(2,ℝ)-representations of surgered groups to left-orderability. This contributes concrete data to the program relating representation varieties of knot groups to left-orderability and the L-space conjecture.

major comments (1)
  1. [§5.3] §5.3, the descent argument after imposing the surgery relation: the claim that the path ρ_t remains non-trivial on the quotient group for all m/l below the stated bound requires an explicit verification that the image of the surgery curve avoids central elements (±I) uniformly in t and for the full range of n. The current argument invokes continuity of the path but does not supply a uniform lower bound on |tr(ρ_t(γ)) − 2| or an equivalent irreducibility check on the peripheral torus after quotienting; this step is load-bearing for the left-orderability conclusion.
minor comments (3)
  1. [Theorem 1.1] The statement of the main theorem (Theorem 1.1) should explicitly recall the precise left-orderability criterion (e.g., the variant of Boyer–Gordon–Watson or the SL(2,ℝ) non-abelian representation theorem) being applied, rather than citing it only in the introduction.
  2. [§4] Notation for the peripheral elements (meridian μ and longitude λ) is introduced in §2 but used without redefinition in the representation formulas of §4; a short reminder of the Wirtinger generators would improve readability.
  3. [Introduction] The exclusion of n=4 is stated without explanation; a brief remark on why the construction fails or requires separate treatment for this case would clarify the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for their positive assessment of its significance. We address the major comment below.

read point-by-point responses

  1. Referee: [§5.3] §5.3, the descent argument after imposing the surgery relation: the claim that the path ρ_t remains non-trivial on the quotient group for all m/l below the stated bound requires an explicit verification that the image of the surgery curve avoids central elements (±I) uniformly in t and for the full range of n. The current argument invokes continuity of the path but does not supply a uniform lower bound on |tr(ρ_t(γ)) − 2| or an equivalent irreducibility check on the peripheral torus after quotienting; this step is load-bearing for the left-orderability conclusion.

    Authors: We agree that the current exposition in §5.3 would benefit from a more explicit verification of non-triviality on the quotient. In the revised version we will add a new lemma immediately following the descent argument. The lemma uses the explicit matrix forms of ρ_t from §4 to compute tr(ρ_t(γ)) as a continuous function of t (a rational function in the parameter t whose coefficients depend on n but remain bounded for n ≥ 3, n ≠ 4). By evaluating the minimum of |tr(ρ_t(γ)) − 2| over the compact interval t ∈ [0,1] and over the discrete set of admissible n, we obtain a uniform positive lower bound that depends only on the slope threshold 2⌊(2n+4)/3⌋. This bound is strictly positive for all m/l below the stated threshold, ensuring ρ_t(γ) ≠ ±I uniformly in t. We will also include a short irreducibility check for the peripheral torus representation after quotienting. These additions make the passage to the surgered group fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit construction stands independently

full rationale

The paper presents an explicit construction of continuous paths of SL(2,R)-representations of the knot groups for the specified pretzel knots, then applies them to conclude left-orderability of surgery manifolds for slopes below the given bound. No quoted step reduces the central claim to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation whose content is itself unverified within the paper. The derivation relies on direct construction rather than tautological renaming or imported uniqueness theorems from the same authors, rendering it self-contained with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts from algebraic topology and representation theory of knot groups; no free parameters, new postulated entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • standard math Standard properties of fundamental groups of knot complements and 3-manifolds obtained by Dehn surgery.
    Invoked implicitly when passing from knot group representations to properties of surgery manifolds.
  • domain assumption Existence of a criterion that converts suitable SL(2,R)-representations into left-orderability of the surgery group.
    The application step in the abstract presupposes such a linking theorem from the literature.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

    math.GT 2026-04 unverdicted novelty 5.0

    All Dehn fillings of these pseudo-Anosov mapping tori have left-orderable fundamental groups via analysis of taut foliations and their branching behavior.

Reference graph

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