Boundary Measurement Matrices for Directed Networks on Surfaces

TL;DR

This paper studies a boundary measurement map for directed graphs on surfaces, showing its independence from fundamental domain choices and providing a combinatorial formula for Grassmannian coordinates.

Contribution

It proves the boundary measurement map's independence from fundamental domain choices and derives a combinatorial formula for Grassmannian Plücker coordinates.

Findings

Boundary measurement map is independent of fundamental domain choice.

Provides a rational function formula for Plücker coordinates.

Expresses coordinates in terms of paths and cycles in the graph.

Abstract

Franco, Galloni, Penante, and Wen have proposed a boundary measurement map for a graph on any closed orientable surface with boundary. We consider this boundary measurement map which takes as input an edge weighted directed graph embedded on a surface and produces on element of a Grassmannian. Computing the boundary measurement requires a choice of fundamental domain. Here the boundary measurement map is shown to be independent of the choice of fundamental domain Also, a formula for the Pl\"ucker coordinates of the element of Grassmannian in the image of the boundary measurement map is given. The formula expresses the Pl\"ucker coordinates as a rational function which can be combinatorially described in terms of paths and cycles in the directed graph.