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Veering branched surfaces, surgeries, and geodesic flows

Chi Cheuk Tsang

Abstract

We introduce veering branched surfaces as a dual way of studying veering triangulations. We then discuss some surgical operations on veering branched surfaces. Using these, we provide explicit constructions of some veering branched surfaces whose dual veering triangulations correspond to geodesic flows of negatively curved surfaces. We construct these veering branched surfaces on (i) complements of Montesinos links whose double branched covers are unit tangent bundles of negatively curved orbifolds, and (ii) complements of full lifts of filling geodesics in unit tangent bundles of negatively curved surfaces, when the geodesics have no triple intersections and have ($n \geq 4$)-gons as complementary regions. As an application, this provides explicit Markov partitions of geodesic flows on negatively curved surfaces. In an appendix, we classify the drilled unit tangent bundles which admit a veering triangulation corresponding to a geodesic flow, by characterizing when there are no perfect fits.

Veering branched surfaces, surgeries, and geodesic flows

Abstract

We introduce veering branched surfaces as a dual way of studying veering triangulations. We then discuss some surgical operations on veering branched surfaces. Using these, we provide explicit constructions of some veering branched surfaces whose dual veering triangulations correspond to geodesic flows of negatively curved surfaces. We construct these veering branched surfaces on (i) complements of Montesinos links whose double branched covers are unit tangent bundles of negatively curved orbifolds, and (ii) complements of full lifts of filling geodesics in unit tangent bundles of negatively curved surfaces, when the geodesics have no triple intersections and have ()-gons as complementary regions. As an application, this provides explicit Markov partitions of geodesic flows on negatively curved surfaces. In an appendix, we classify the drilled unit tangent bundles which admit a veering triangulation corresponding to a geodesic flow, by characterizing when there are no perfect fits.
Paper Structure (22 sections, 19 theorems, 13 equations, 28 figures)

This paper contains 22 sections, 19 theorems, 13 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Veering triangulations
  4. Pseudo-Anosov flows
  5. Geodesic flows
  6. Montesinos links
  7. Veering branched surfaces
  8. Surgeries on veering branched surfaces
  9. Horizontal surgery
  10. Vertical surgery
  11. Variants of horizontal surgery: halved and concurrent
  12. Geodesic flows I: Genus zero orbifolds, Montesinos knots and links
  13. Case 1: $n=3, p_1=2, p_2=3, p_3 > 6$
  14. Case 2: $n=3, p_1=2, (p_2,p_3)>(4,4)$
  15. Case 3: $n=3, (p_1,p_2,p_3)>(3,3,3)$
  16. ...and 7 more sections

Key Result

Proposition 2.5

Let $\Delta$ be a veering triangulation of a 3-manifold $M$ and let $B$ be its unstable branched surface.

Figures (28)

  • Figure 1: A tetrahedron in a transverse veering triangulation. There are no restrictions on the colors of the top and bottom edges.
  • Figure 2: The local models for branched surfaces. The arrows indicate the maw coorientation of the branch locus.
  • Figure 3: Left: The portion of the unstable branched surface and the flow graph within each veering tetrahedron. Right: The portion of the flow graph on each sector of the unstable branched surface.
  • Figure 4: A perfect fit rectangle.
  • Figure 5: Definition of a Montesinos link.
  • ...and 23 more figures

Theorems & Definitions (72)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7: Landry-Minsky-Taylor LMT20
  • Definition 2.8
  • Proposition 2.9
  • ...and 62 more