### Abstract

A relative trisection of a smooth, compact, oriented 4-manifold with boundary X is a decomposition of X into three diffeomorphic pieces which have “nice” intersection properties. The trisection induces an open book decomposition on the boundary, which is a surface bundle over $S^1$ in the compliment of a link in $\partial X$. It is known that every such 4-manifold admits a trisection and that any two trisections can be made isotopic after suitable “stabilization” operations. In this talk, I will show that any two diffeomorphic relative trisections of the 4-ball which induce isotopic open books on the boundary 3-sphere are in fact isotopic trisections. An interesting feature of the argument is that we do not show that the original diffeomorphism is isotopic to the identity! I will give a good deal of background on trisections, trisection diagrams, and open books. If time permits, I will discuss some practical features of relative trisections which allow us to classify low-“complexity” relative trisections. This work is joint with Patrick Naylor.