Right-left equivalent maps of simplified $(2, 0)$-trisections with different configurations of vanishing cycles

Abstract

Trisection maps are certain stable maps from closed $4$--manifolds to $\mathbf{R}^2$. A simpler but reasonable class of trisection maps was introduced by Baykur and Saeki, called a simplified $(g, k)$-trisection. We focus on the right-left equivalence classes of simplified $(2, 0)$-trisections. Simplified trisections are determined by their simplified trisection diagrams, which are diagrams on a genus-$2$ surface consisting of simple closed curves of vanishig cycles with labels. The aim of this paper is to study how the replacement of reference paths changes simplified trisection diagrams up to upper-triangular handle-slides. We show that for a simplified trisection $f$ satisfying a certain condition, there exists at least two simplified $(2, 0)$-trisections $f'$ and $f''$ such that $f, f'$ and $f''$ are right-left equivalent to each other but their simplified trisection diagrams are not related by automorphisms of a genus-$2$ surface and upper-triangular handle-slides.