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From veering triangulations to dynamic pairs

Saul Schleimer, Henry Segerman

Abstract

From a transverse veering triangulation (not necessarily finite) we produce a canonically associated dynamic pair of branched surfaces. As a key idea in the proof, we introduce the shearing decomposition of a veering triangulation.

From veering triangulations to dynamic pairs

Abstract

From a transverse veering triangulation (not necessarily finite) we produce a canonically associated dynamic pair of branched surfaces. As a key idea in the proof, we introduce the shearing decomposition of a veering triangulation.
Paper Structure (47 sections, 24 theorems, 4 equations, 45 figures, 1 table)

This paper contains 47 sections, 24 theorems, 4 equations, 45 figures, 1 table.

Table of Contents

  1. Introduction
  2. Other work
  3. Future work
  4. Triangulations, train-tracks, and branched surfaces
  5. Ideal triangulations
  6. Train-tracks
  7. Branched surfaces
  8. Dynamics
  9. Dynamic pairs
  10. Complementary components
  11. The naive push-off
  12. Shearing regions, mid-bands, and the mid-surface
  13. Shearing regions
  14. Crimping
  15. The mid-surface
  16. ...and 32 more sections

Key Result

Lemma 2.2

In the universal cover ${\widetilde{M}}$, with ${\widetilde{B}}^\mathcal{V}$ and ${\widetilde{B}}_\mathcal{V}$ in dual position, every subray of every branch line of ${\widetilde{B}}^\mathcal{V}$ and of ${\widetilde{B}}_\mathcal{V}$ meets toggle tetrahedra. ∎

Figures (45)

  • Figure 2.1: An ideal triangulation of the complement of the figure-eight knot in the three-sphere. Each edge is equipped with a colour -- red (dotted) or blue (dashed) -- and an orientation. These determine the face pairings. The flattening (into the plane) makes the triangulation taut and transverse. Note that the taut structure and the orientation determine the veering structure and thus the colours.
  • Figure 2.2:
  • Figure 2.3: In both subfigures, above and below we have toggle tetrahedra while left and right we have, respectively, blue and red fan tetrahedra. A black arrow indicates a possible gluing from an upper face of the initial tetrahedron to a lower face of the terminal. Note that fan tetrahedra of different colours never share a face. Finally, inside each tetrahedron $t$ on the left (right) we draw the branched surface $B^t$ ($B_t$).
  • Figure 2.4:
  • Figure 2.5: Two positions of the upper branched surface in a tetrahedron.
  • ...and 40 more figures

Theorems & Definitions (123)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Lemma 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Corollary 3.4
  • ...and 113 more