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

Abstract

Let $X$ be a bundle over $S^1$ with fiber a 3--manifold $M$ and with monodromy $φ$. Gay and Kirby showed that if $φ$ 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 $φ$ 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 $φ$ fixes a genus $g$ Heegaard surface of $M$. This means that $φ$ 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 $φ$.