Critical points of invariant functions on closed orientable surfaces

Generating Functions of Bipartite Maps on Orientable Surfaces

We compute, for each genus g > 0, the generating function Lg ≡ Lg(t; p1, p2, . . . ) of (labelled) bipartite maps on the orientable surface of genus g, with control on all face degrees. We exhibit an explicit change of variables such that for each g, Lg is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function Fg of...

Exact Cellular Decomposition of Closed Orientable Surfaces Embedded in R3

We address the task of covering a closed orientable surface embedded in < without any prior information about the surface. For applications such as paint deposition, the e ector (the paint atomizer) does not explicitly cover the target surface, but instead covers an o set surface | a surface that is a xed distance away from the target surface. Just as Canny and others use critical points to loo...

Embedding Digraphs on Orientable Surfaces

We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of “compatible Euler tours” or “orthogonal A-trails” by Andsersen at al [1] and by Bouchet [4]. This prior work has mostly been limited to embeddings of Eulerian digraph...

On the Critical Points of Harmonic Functions

Critical Points of Functions on Singular Spaces

We compare and contrast various notions of the “critical locus” of a complex analytic function on a singular space. After choosing a topological variant as our primary notion of the critical locus, we justify our choice by generalizing Lê and Saito’s result that constant Milnor number implies that Thom’s af condition is satisfied.