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Order-detection, representation-detection, and applications to cable knots

Adam Clay, Junyu Lu

Abstract

Given a $3$-manifold $M$ with multiple incompressible torus boundary components, we develop a general definition of order-detection of tuples of slopes on the boundary components of $M$. In parallel, we arrive at a general definition of representation-detection of tuples of slopes, and show that these two kinds of slope detection are equivalent -- in the sense that a tuple of slopes on the boundary of $M$ is order-detected if and only if it is representation-detected. We use these results, together with new "relative gluing theorems," to show how the work of Eisenbud-Hirsch-Neumann, Jankins-Neumann and Naimi can be used to determine tuples of representation-detected slopes and, in turn, the behaviour of order-detected slopes on the boundary of a knot manifold with respect to cabling. Our cabling results improve upon work of the first author and Watson, and in particular, this new approach shows how one can use the equivalence between representation-detection and order-detection to derive orderability results that parallel known behaviour of L-spaces with respect to Dehn filling.

Order-detection, representation-detection, and applications to cable knots

Abstract

Given a -manifold with multiple incompressible torus boundary components, we develop a general definition of order-detection of tuples of slopes on the boundary components of . In parallel, we arrive at a general definition of representation-detection of tuples of slopes, and show that these two kinds of slope detection are equivalent -- in the sense that a tuple of slopes on the boundary of is order-detected if and only if it is representation-detected. We use these results, together with new "relative gluing theorems," to show how the work of Eisenbud-Hirsch-Neumann, Jankins-Neumann and Naimi can be used to determine tuples of representation-detected slopes and, in turn, the behaviour of order-detected slopes on the boundary of a knot manifold with respect to cabling. Our cabling results improve upon work of the first author and Watson, and in particular, this new approach shows how one can use the equivalence between representation-detection and order-detection to derive orderability results that parallel known behaviour of L-spaces with respect to Dehn filling.
Paper Structure (17 sections, 41 theorems, 92 equations, 2 figures)

This paper contains 17 sections, 41 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Definitions
  4. Dynamic realisations and representations
  5. Order-detection
  6. Representation-detection
  7. Equivalence of order- and representation-detection, and alternative definitions
  8. Gluing theorems
  9. JN-realisability and representation-detection
  10. Cable spaces, bases and gluing two pieces
  11. Notations and conventions
  12. Transition between bases and non-horizontal slopes
  13. Special cases of the gluing theorems for use when cabling
  14. Computations of relatively JN-realisable slopes in cable spaces
  15. Detected slopes and cabling
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose $M$ is a compact connected irreducible orientable $3$-manifold whose boundary is a union of incompressible tori $T_1, \ldots, T_n$; fix $J \subset K \subset \{1, \dots, n\}$ and $[\alpha_*] \in \mathcal{S}(M)$. Then $(J, K; [\alpha_*])$ is order-detected if and only if it is representation-d

Figures (2)

  • Figure 1: Cable knot complement obtained by gluing $C_{p,q}$ to ${S^3-n(K)}$
  • Figure 2: The planar surface $P$

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: See Theorem \ref{['thm:torusknot']} and Corollary \ref{['cor:cableintervals']}
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 68 more