Trisections of 3–manifold bundles over S1

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