We introduce the concept of a bridge trisection of a neatly embedded surface in a compact
four-manifold, generalizing previous work with Alexander Zupan in the setting of closed
surfaces in closed four-manifolds. Our main result states that any neatly embedded surface
in a compact
four-manifold
can be isotoped to lie in bridge trisected position with respect to any trisection
of
. A bridge trisection
of
induces a
braiding of the link
with respect to the open-book decomposition of
induced by
, and we show that the
bridge trisection of
can be assumed to induce any such braiding.
We work in the general setting in which
may
be disconnected, and we describe how to encode bridge trisected surface
diagrammatically using shadow diagrams. We use shadow diagrams to show how
bridge trisected surfaces can be glued along portions of their boundary, and we
explain how the data of the braiding of the boundary link can be recovered from a
shadow diagram. Throughout, numerous examples and illustrations are given. We
give a set of moves that we conjecture suffice to relate any two shadow diagrams
corresponding to a given surface.
We devote extra attention to the setting of surfaces in
,
where we give an independent proof of the existence of bridge trisections and develop
a second diagrammatic approach using tri-plane diagrams. We characterize bridge
trisections of ribbon surfaces in terms of their complexity parameters. The process of
passing between bridge trisections and band presentations for surfaces in
is
addressed in detail and presented with many examples.
