• bridge trisection
• knotted surfaces

The aim of our thesis is to present the recent theory of bridge trisections of knotted surfaces.

Bridge trisections are the adaptation of the notion of bridge splitting to knotted surfaces and were produced a few years ago by J. Meier and A. Zupan, taking inspiration from the recent theory of trisections of 4-manifolds, developed by D. Gay and R. Kirby.

Meier and Zupan considered the simplest trisection of 4-sphere and proved that any knotted surface can be isotoped into a collection of 2-dimensional bridges contained in the three sectors of the trisection. In addition to the existence of bridge trisections, they proved that any two bridge trisections of a given surface are equivalent up to stabilizations/destabilizations. They also constructed a diagrammatic theory and a complete set of diagrammatic moves capable of describing any bridge trisection by a triple of planar diagrams. Meier and Zupan then turned their attention to closed surfaces embedded in any closed, connected and orientable 4-manifold and, thanks to the theory of trisections, were able to extend the existence result to this broader setting.