Lifting Arc Diagrams Under Branched Covers: An Inverse Problem and its Solution

TL;DR

This paper introduces a combinatorial method called a lifting picture to solve the inverse problem of lifting arc diagrams via branched covers, providing both brute force and efficient solutions.

Contribution

It presents a novel combinatorial representation for branched covers and offers algorithms to determine liftability of arc diagrams, including an efficient method for triangulations.

Findings

Brute force solution for the general case.

Efficient solution for triangulated arc diagrams.

Algorithmic approach to the inverse lifting problem.

Abstract

A branched covering map of surfaces induces a map in the opposite direction between their arc complexes. We represent a branched covering map combinatorially using what we call a lifting picture, and use this representation to computably solve the membership problem of the set of weighted arc diagrams on a given surface which can be obtained by lifting a weighted arc diagram on a bigon. We provide a brute force solution in the general case, and an efficient solution when the input arc diagram is a triangulation.