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

TL;DR

This paper extends Kirchhoff's laws to higher dimensions using 2-simplicial complexes in 3D, and introduces a linear-time algorithm to construct a dual graph for solving electromagnetic problems.

Contribution

It introduces a novel higher-dimensional topological framework for electromagnetic circuits and a linear-time algorithm to construct the dual graph for problem-solving.

Findings

The dual graph is a standard graph for the complex's skeleton.

The construction algorithm is linear in the size of the 2-complex.

The approach simplifies solving higher-dimensional electromagnetic problems.

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.