Mutual arc presentations and braided open books

TL;DR

This work develops a topological framework connecting canonical fiberedness to braided open books via mutual arc presentations. It introduces mutual arc presentations as a generalization of arc presentations and shows that canonically fibered links admit such presentations, enabling their braiding relative to open books. The authors then demonstrate that braided open-book structures are preserved under operations like Stallings plumbing and satellites, using Rampichini diagrams as a combinatorial tool. These results yield new infinite families of fibered links that bind braided open books, including examples not canonically fibered, and provide elementary proofs and constructions related to longstanding questions in the field.

Abstract

We show that every canonically fibered link in $S^3$ is the binding of a braided open book in $S^3$, addressing a question of Montesinos and Morton. We introduce mutual arc presentations as our main technical tool, which we consider to be of independent interest. We prove that any fibered link admitting such a presentation is the binding of a braided open book. Furthermore, new examples of fibered links serving as bindings of braided open books are obtained via connected sum and cabling operations, thereby providing examples of bindings of braided open books that are not canonically fibered.

Figures (12)

• Figure 1: Implications among conjectures and Theorem \ref{['thm:harer']}.
• Figure 2: From a generalized arc presentation of $L$ in $\operatorname{OB}$ to a braiding of $L$ relative $\operatorname{OB}$.
• Figure 3: The deformation of $L$ and $B$ near an intersection point. a) $L$ and $B$ intersect in an isolated point. One segment is colored red, the other blue. b) A sphere around the intersection point. Red curves indicate the intersections between fibers of one open book with the sphere. Blue curves are the intersections between the fibers of the other open book and the sphere. c) Deforming the red segment of the binding to a curve that is transverse to the blue fibers. d) Deforming the blue segment of the other binding to be transverse to the red fibers.
• Figure 4: Two ways of smoothing a vertex of a $4$-valent planar graph.
• Figure 5: Constructing the binding of an unbook and deforming the short and parallel edges.

Theorems & Definitions (23)

• Conjecture 1: montesions_morton
• Conjecture 2: bode:braided
• Conjecture 3
• Definition 2.1
• Definition 2.2
• Definition 2.3: Definition 2.1 in baader
• Definition 2.4
• Remark 2.6
• proof
• proof