Detecting right-veering diffeomorphisms

TL;DR

This paper addresses the problem of distinguishing tight from overtwisted contact structures by exploiting the right-veering criterion for open books. ItDevelops a combinatorial framework of extended towers built from a basis of arcs on a compact surface with boundary, enabling detection of left-veering arcs via regions and arc segments; the base-case uses a $6$-gon and an inductive step extends towers to more arcs while preserving key properties. The main contribution is a finite, basis-dependent algorithm (Theorem TowerCollection1) that decides right-veering by exhaustively checking a finite space of extended towers; the approach is connected to knot Floer homology through the invariant $b(K)$ and its relation to the contact class. The work thus provides a concrete, constructive method to certify right-veering open books and hints at deeper connections with knot Floer differentials.

Abstract

A result of Honda, Kazez, and Matić states that a contact structure is tight if and only if all its supporting open books are right-veering. We show a combinatorial way of detecting the left-veering arcs in open books, implying the existence of an algorithm that detects the right-veering property for compact surfaces with boundary.

Figures (48)

• Figure 1: A left-veering arc $\gamma$, which we want to detect using extended towers. The arcs $\alpha_i$, $\beta_j$, $\delta_k$ are arcs from a chosen basis $\mathcal{B}$. The arc is minimal with respect to $\mathcal{B}$, which means that its image is itself up until the point $x$ --that is, all segments up to $x$ are fixable-- and after that it goes to the left --that is, the last segment is left-veering. Finally, we will use the convention that the thicker lines represent the boundary of the surface.
• Figure 2: The extended tower $\{R\}$ is incomplete, showing that the arc $\beta = \alpha_1 + \alpha_2$ is left-veering.
• Figure 3: On the top left, a basis of arcs and their images, determining the monodromy. On the top right, the extended tower $\{R_1, R'_1\}$ is completed, so $\gamma_1$ is fixable. On the bottom left, $\{R_2\}$ is incomplete, so $\gamma_2$ is left-veering. This implies that $\gamma = \gamma_1 \cup \gamma_2$ is left-veering, as we can see on the bottom right.
• Figure 4: The arc-slide $\beta$ of two arcs $\alpha_1$ and $\alpha_2$. Observe we need to reverse the orientation of $\beta$ to obtain an orientation of the boundary of the $6$-gon.
• Figure 5: A (strictly) right-veering arc $\alpha$.

Theorems & Definitions (73)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Remark