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The Wrappingness and Trunkenness of Volume-Preserving Flows

Peter Lambert-Cole

Abstract

Link invariants of long pieces of orbits of a volume-preserving flow can be used to define diffeomorphism invariants of the flow. In this paper, we extend the notions of wrapping number and trunk and define invariants of links with respect to a fibration on a 3-manifold. Extending work of Dehornoy and Rechtman, we apply this to define diffeomorphism invariants wrappingness and trunkenness of volume-preserving flows on 3-manifolds and interpret these invariants as obstructions to the existence of a global surface of section for the flow. Finally, we construct flows and show that wrappingness and trunkenness are not functions of the helicity of a flow.

The Wrappingness and Trunkenness of Volume-Preserving Flows

Abstract

Link invariants of long pieces of orbits of a volume-preserving flow can be used to define diffeomorphism invariants of the flow. In this paper, we extend the notions of wrapping number and trunk and define invariants of links with respect to a fibration on a 3-manifold. Extending work of Dehornoy and Rechtman, we apply this to define diffeomorphism invariants wrappingness and trunkenness of volume-preserving flows on 3-manifolds and interpret these invariants as obstructions to the existence of a global surface of section for the flow. Finally, we construct flows and show that wrappingness and trunkenness are not functions of the helicity of a flow.
Paper Structure (21 sections, 12 theorems, 63 equations)

This paper contains 21 sections, 12 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Trunk and wrapping number
  4. Wrappingness and Trunkenness of vector fields
  5. Main results
  6. Obstruction to surfaces of section
  7. Independence of Helicity
  8. Existence of periodic orbits
  9. Acknowledgements
  10. Trunk and Wrapping number
  11. Main results
  12. Homeomorphism invariance
  13. Weak-$\ast$ convergence
  14. Existence of periodic orbits
  15. Independence of Helicity
  16. ...and 6 more sections

Key Result

Proposition 1.1

Let $\pi: Y \rightarrow S^1$ be a fibered 3-manifold (possibly with boundary) and $L \subset Y$ a link. Then

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.1
  • Conjecture 1.1
  • Definition 1.4
  • Definition 1.5
  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • ...and 21 more