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Morse diagrams, Murasugi sums, and the mapping class group

Jack Brand, David Gay, Joan Licata

Abstract

A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition.

Morse diagrams, Murasugi sums, and the mapping class group

Abstract

A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition.

Paper Structure

This paper contains 14 sections, 15 theorems, 15 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Acknowledgments
  4. Combinatorial Morse structures
  5. Open books and contact structures
  6. Murasugi sums and open books
  7. Combinatorial Morse structures
  8. Morse diagrams
  9. Connect sums of open books
  10. Murasugi sums for Morse structures
  11. Stabilisation
  12. Morse diagram for overtwisted contact manifolds
  13. Once-punctured torus pages
  14. Mapping class group of the one-holed torus

Key Result

Theorem 1.1

The splice $X^1*X^2$ is a Morse diagram for the open book $(\Sigma_{1} * \Sigma_2,\overline{\phi}^{1}\circ \overline{\phi}^{2})$. $\blacktriangleleft$$\blacktriangleleft$

Figures (12)

  • Figure 1: Identifying a pair of Murasugi polygons in surfaces $\Sigma_i$ (left, right) produces the Murasugi sum $\Sigma_1*\Sigma_2$ (center). The Murasugi polygon on the left is shown with its starlike graph, as described in Section \ref{['sec:sums']}.
  • Figure 2: Distinct handle structures on the thrice-punctured sphere.
  • Figure 3: The left and right pictures are local images of handle structures $\mathcal{H}_i$ and $\mathcal{H}_{i+1}$ that are related by an arc slide of the blue co-core over the red one; the central figure shows the singular handle structure $S_{i+1}$ connecting them.
  • Figure 4: Three elementary slices on a two-holed torus. Observe that the elementary slices in the top row can be stacked in any order because the initial and final handle structures agree, but the elementary slice in the bottom row can only be stacked on top of one of the slices from the top row.
  • Figure 5: A sequence of handle structures and the corresponding planar Morse diagram.
  • ...and 7 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Example 2.7
  • ...and 28 more