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Order-detection and non-left-orderable surgeries on links

Adam Clay, Junyu Lu

TL;DR

This work develops a slope-detection framework for left-orderings to study Dehn fillings of 3-manifolds with multiple torus boundaries, extending prior single-boundary techniques. By proving cofinality results and constructing convex-subgroup refinements, the authors produce infinite families of Dehn-filled manifolds with non-left-orderable fundamental groups, and sharpen these results in the Whitehead/link case to align with the L-space conjecture. The approach yields concrete, verifiable predictions for regions of slope space and demonstrates applications to knots and multi-component links, including the Whitehead link and the figure-eight knot complement. Overall, the paper strengthens the connection between orderability, Dehn surgery, and L-space phenomena, offering new tools for detecting non-LO fillings and testing the L-space conjecture in more complex boundary settings.

Abstract

Beginning with a $3$-manifold $M$ having a single torus boundary component, there are several computational techniques in the literature that use a presentation of the fundamental group of $M$ to produce infinite families of Dehn fillings of $M$ whose fundamental groups are non-left-orderable. In this manuscript, we show how to use order-detection of slopes to generalise these techniques to manifolds with multiple torus boundary components, and to produce results that are sharper than what can be achieved with traditional techniques alone. As a demonstration, we produce an infinite family of hyperbolic links where many of the manifolds arising from Dehn filling have non-left-orderable fundamental groups. The family includes the Whitehead link, and in that case we produce a collection of non-left-orderable Dehn fillings that precisely matches the prediction of the L-space conjecture.

Order-detection and non-left-orderable surgeries on links

TL;DR

This work develops a slope-detection framework for left-orderings to study Dehn fillings of 3-manifolds with multiple torus boundaries, extending prior single-boundary techniques. By proving cofinality results and constructing convex-subgroup refinements, the authors produce infinite families of Dehn-filled manifolds with non-left-orderable fundamental groups, and sharpen these results in the Whitehead/link case to align with the L-space conjecture. The approach yields concrete, verifiable predictions for regions of slope space and demonstrates applications to knots and multi-component links, including the Whitehead link and the figure-eight knot complement. Overall, the paper strengthens the connection between orderability, Dehn surgery, and L-space phenomena, offering new tools for detecting non-LO fillings and testing the L-space conjecture in more complex boundary settings.

Abstract

Beginning with a -manifold having a single torus boundary component, there are several computational techniques in the literature that use a presentation of the fundamental group of to produce infinite families of Dehn fillings of whose fundamental groups are non-left-orderable. In this manuscript, we show how to use order-detection of slopes to generalise these techniques to manifolds with multiple torus boundary components, and to produce results that are sharper than what can be achieved with traditional techniques alone. As a demonstration, we produce an infinite family of hyperbolic links where many of the manifolds arising from Dehn filling have non-left-orderable fundamental groups. The family includes the Whitehead link, and in that case we produce a collection of non-left-orderable Dehn fillings that precisely matches the prediction of the L-space conjecture.
Paper Structure (10 sections, 20 theorems, 38 equations, 2 figures)

This paper contains 10 sections, 20 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organisation of the manuscript
  3. Background
  4. Left-orderings
  5. Slope detection and three-manifolds
  6. Cofinality, Dehn filling and slope detection
  7. Applications to knots and two-component links
  8. An infinite family of links
  9. Whitehead link
  10. Cofinal orderings of the figure-eight knot group

Key Result

Theorem 1.1

For each integer $n \geq 0$, let $\mathbb{L}_n$ denote the link depicted in Figure fig:linklnn. If $(r_1, r_2) \in (2n+2, \infty) \times (2, \infty)$ is a pair of rational numbers, then $(r_1, r_2)$-surgery on $\mathbb{L}_n$ produces a manifold which is non-LO.

Figures (2)

  • Figure 1: The two-component link $\mathbb{L}_n$; the block represents $n$ full twists
  • Figure 2: The three-component link $\mathbb{L}'$

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: claylu2024
  • Remark 2.2
  • Theorem 2.3: BC23
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 32 more