2022/10/18 by Joseph, Jason, Meier, Jeffrey, Miller, Maggie +1
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2210.09669
We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane 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.