Obtaining the Chamanara Surface from the van der Corput sequence

TL;DR

This paper explores how graphs derived from the van der Corput sequence can be embedded into the Chamanara surface, revealing connections between sequence graphs and translation surfaces.

Contribution

It demonstrates that graphs from the van der Corput sequence embed into the Chamanara surface, extending the understanding of sequence graphs and their geometric embeddings.

Findings

Graphs from the Kronecker sequence embed into the torus.

Graphs from the van der Corput sequence embed into the Chamanara surface.

Possible removal of one edge facilitates embeddings.

Abstract

We investigate a family of $4$-regular graphs constructed to test for the presence of combinatorial structure in a sequence of distinct real numbers. We show that the graphs constructed from the Kronecker sequence can be embedded into the torus, while the graphs constructed from the binary van der Corput sequence can be embedded into the Chamanara surface, in both cases with the possible removal of one edge. These results allude to a general theory of sequence graphs which can be embedded into particular translation surfaces coming from interval exchange transformations.