# Determination of topological properties of thin samples by the van der Pauw method

## TL;DR

This paper extends the van der Pauw method to determine the topological genus of flat samples by analyzing resistance measurements, enabling the identification of holes in the domain through electrical testing.

## Contribution

It introduces an extension of the van der Pauw method for samples with zero, one, or two holes, linking electrical measurements to topological properties.

## Key findings

- Maximum resistance behavior depends on the genus of the domain.
- Experimental results confirm the method's ability to detect holes in samples.
- Results align with Hurwitz's topological theorem on Riemann surfaces.

## Abstract

We solve the problem of determining basic topological properties of flat samples by performing measurements on their outer edge. The global maximum of four probe resistances shows a characteristic behaviour, which is dependent on the genus (i.e., the number of holes) of the domain. An extension of the van der Pauw method on domains having zero, one, or two holes is presented and discussed. A possibility of measuring topological properties of condensed matter is demonstrated. Experimental results for triply connected domains are presented and explained by continuous symmetry breaking caused by the presence of two holes. The results are consistent with the topological theorem of Hurwitz on the number of automorphisms of Riemann surfaces.