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Filling braided links with trisected surfaces

Jeffrey Meier

Abstract

We introduce the concept of a bridge trisection of a neatly embedded surface in a compact four-manifold, generalizing previous work with Alexander Zupan in the setting of closed surfaces in closed four-manifolds. Our main result states that any neatly embedded surface $\mathcal{F}$ in a compact four-manifold $X$ can be isotoped to lie in bridge trisected position with respect to any trisection $\mathbb{T}$ of $X$. A bridge trisection of $\mathcal{F}$ induces a braiding of the link $\partial\mathcal{F}$ with respect to the open-book decomposition of $\partial X$ induced by $\mathbb{T}$, and we show that the bridge trisection of $\mathcal{F}$ can be assumed to induce any such braiding. We work in the general setting in which $\partial X$ may be disconnected, and we describe how to encode bridge trisected surface diagrammatically using shadow diagrams. We use shadow diagrams to show how bridge trisected surfaces can be glued along portions of their boundary, and we explain how the data of the braiding of the boundary link can be recovered from a shadow diagram. Throughout, numerous examples and illustrations are given. We give a set of moves that we conjecture suffice to relate any two shadow diagrams corresponding to a given surface. We devote extra attention to the setting of surfaces in $B^4$, where we give an independent proof of the existence of bridge trisections and develop a second diagrammatic approach using tri-plane diagrams. We characterize bridge trisections of ribbon surfaces in terms of their complexity parameters. The process of passing between bridge trisections and band presentations for surfaces in $B^4$ is addressed in detail and presented with many examples.

Filling braided links with trisected surfaces

Abstract

We introduce the concept of a bridge trisection of a neatly embedded surface in a compact four-manifold, generalizing previous work with Alexander Zupan in the setting of closed surfaces in closed four-manifolds. Our main result states that any neatly embedded surface in a compact four-manifold can be isotoped to lie in bridge trisected position with respect to any trisection of . A bridge trisection of induces a braiding of the link with respect to the open-book decomposition of induced by , and we show that the bridge trisection of can be assumed to induce any such braiding. We work in the general setting in which may be disconnected, and we describe how to encode bridge trisected surface diagrammatically using shadow diagrams. We use shadow diagrams to show how bridge trisected surfaces can be glued along portions of their boundary, and we explain how the data of the braiding of the boundary link can be recovered from a shadow diagram. Throughout, numerous examples and illustrations are given. We give a set of moves that we conjecture suffice to relate any two shadow diagrams corresponding to a given surface. We devote extra attention to the setting of surfaces in , where we give an independent proof of the existence of bridge trisections and develop a second diagrammatic approach using tri-plane diagrams. We characterize bridge trisections of ribbon surfaces in terms of their complexity parameters. The process of passing between bridge trisections and band presentations for surfaces in is addressed in detail and presented with many examples.

Paper Structure

This paper contains 31 sections, 50 theorems, 62 equations, 39 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some conventions
  4. Lensed cobordisms
  5. Compressionbodies
  6. Heegaard splittings and Heegaard-page splittings
  7. Trivial tangles
  8. Bridge splittings
  9. Disk-tangles
  10. Open-book decompositions and braidings of links
  11. Formal definitions
  12. The four-ball setting
  13. Preliminaries and a precise definition
  14. Band presentations
  15. Bridge-braiding band presentations
  16. ...and 16 more sections

Key Result

Lemma \oldthetheorem

Let $(\Sigma; H_1, H_2)$ be a $(m,n)$--standard Heegaard splitting with $H_i\cong H_{g,\bold p,\bold f}$. Then, where $(H'_1)^j\cong(H'_2)^j\cong H_{p_j,f_j}$, for each $j = 1, \ldots, n$, and $(\Sigma";H_1",H_2")$ is the standard genus $g-p$ Heegaard surface for $\#^m(S^1\times S^2)$.

Figures (39)

  • Figure 1: Three examples of trivial tangles inside lensed compressionbodies.
  • Figure 2: (A) A $(1,2)$--standard Heegaard diagram for the standard Heegaard double $Y_{4,(1,0),(2,1)}$. (B) A schematic showing the standard Heegaard double $Y_{2,1,1}$, containing a $(3,4)$--bridge splitting for an unlink. The unlink has no flat components and four vertical components.
  • Figure 3: A schematic of the disk-tangle $\mathcal{D}_{1,2}$, which contains one flat component and two vertical components. Note that the 3--component unlink on the boundary is in $(3,2)$--bridge position with respect to the standard Heeggaard double $Y_{0,0,1}$ for the 3--sphere.
  • Figure 4: Markov stabilization, depicted as the banding of a braid to a meridian of the binding.
  • Figure 5: A schematic illustration of a standard Heegaard double, with orientation conventions for the constituent pieces of $\partial Z_1$ indicated.
  • ...and 34 more figures

Theorems & Definitions (139)

  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • ...and 129 more