Trisections of 3-manifold bundles over $S^1$

TL;DR

This paper proves the existence of minimal genus trisections for certain 3-manifold bundles over S^1 with specific monodromy properties, extending previous results and providing an algorithm for diagram construction.

Contribution

It generalizes prior work by showing that nontrivial monodromies fixing a Heegaard surface still admit minimal genus trisections and offers an explicit algorithm for diagram creation.

Findings

Existence of genus 3g+1 trisections when monodromy fixes a genus g Heegaard surface.

Extension of previous results to nontrivial monodromies.

Algorithm for drawing trisection diagrams from Heegaard diagrams and monodromy descriptions.

Abstract

Let $X$ be a bundle over $S^1$ with fiber a 3--manifold $M$ and with monodromy $\varphi$. Gay and Kirby showed that if $\varphi$ fixes a genus $g$ Heegaard splitting of $M$ then $X$ has a genus $6g+1$ trisection. Genus $3g+1$ trisections have been found in certain special cases, such as the case where $\varphi$ is trivial, and it is known that trisections of genus lower than $3g+1$ cannot exist in general. We generalize these results to prove that there exists a trisection of genus $3g+1$ whenever $\varphi$ fixes a genus $g$ Heegaard surface of $M$. This means that $\varphi$ can be nontrivial, and can preserve or switch the two handlebodies of the Heegaard splitting. We additionally describe an algorithm to draw a diagram for such a trisection given a Heegaard diagram for $M$ and a description of $\varphi$.