Higher dimensional electrical circuits and the matroid dual of a nonplanar graph

Abstract

In this paper we describe a physical problem, based on electromagnetic fields, whose topological constraints are higher dimensional versions of Kirchhoff's laws, involving $2-$ simplicial complexes embedded in $\mathbb{R} ^3$ rather than graphs. However, we show that, for the skeleton of this complex, involving only triangles and edges, we can build a matroid dual which is a graph. On this graph we build an `ordinary' electrical circuit, solving which we obtain the solution to our original problem. Construction of this graph is through a `sliding' algorithm which simulates sliding on the surfaces of the triangles, moving from one triangle to another which shares an edge with it but which also is adjacent with respect to the embedding of the complex in $\mathbb{R} ^3.$ For this purpose, the only information needed is the order in which we encounter the triangles incident at an edge, when we rotate say clockwise with respect to the orientation of the edge. The dual graph construction is linear time on the size of the $2-$ complex.