Arc diagrams are simple, combinatorial objects associated to surfaces with boundary. They consist of homotopy classes of disjoint curves, and can be thought of as embedded graphs on suitably marked surfaces. This dissertation examines the behavior of weighted arc diagrams, that is, diagrams with nonnegative real numbers assigned to each arc, under branched covering maps. The lift of an arc under a branched covering map is an arc. Therefore, we can interpret a branched covering map as inducing a map on weighted arc diagrams by lifting. We are interested in the inverse problem: when can a weighted arc diagram be realized by lifting the arcs of a diagram under a suitable branched cover? This dissertation solves the problem in the case of a maximal diagram covering a disk with two marked boundary points, called a bigon. This dissertation presents an algorithm which decides whether a given maximal, weighted arc diagram may be realized by lifting a weighted arc diagram on the bigon, and which produces a branched cover and weighted arc diagram on the bigon which realize this diagram by lifting. It also proves that this algorithm is sound and complete.
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Citation
Peterpaul, Cyrus,
"Lifting Arc Diagrams Under Branched Covers: An Inverse Problem and its Algorithmic Solution"
(2019).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/qa83-b595
