Bridge trisections and Seifert solids

We adapt Seifert's algorithm for classical knots and links to the setting of triplane diagrams for bridge trisected surfaces in the 4-sphere.Our approach allows for the construction of a Seifert solid that is described by a Heegaard diagram.The Seifert solids produced can be assumed to have exteriors that can be built without 3-handles; in contrast, we give examples of Seifert solids (not coming from our construction) whose exteriors require arbitrarily many 3-handles.We conclude with two classification results.The first shows that surfaces admitting doubly standard shadow diagrams are unknotted.The second says that a b-bridge trisection in which some sector contains at least b 1 patches is completely decomposable, thus the corresponding surface is unknotted.This settles affirmatively a conjecture of the second and fourth authors.