Toroidal Hitomezashi Patterns

TL;DR

This paper studies toroidal Hitomezashi patterns and loops, providing structural results on their length, count, and homology classes, with special focus on symmetric patterns and connections to knot theory.

Contribution

It introduces a comprehensive analysis of Hitomezashi patterns on a torus, including optimal bounds and a novel link to knot theory for symmetric patterns.

Findings

Optimal bounds for loop length and count

Classification of homology classes of loops

Connection between symmetric patterns and knot theory

Abstract

Extending a proposal of Defant and Kravitz [Discrete Mathematics, \textbf{1}, 347 (2024)], we define Hitomezashi patterns and loops on a torus and provide several structural results for such loops. For a given pattern, our main theorems give optimal residual information regarding the Hitomezashi loop length, loop count, as well as possible homology classes of such loops. Special attention is paid to toroidal Hitomezashi patterns that are symmetric with respect to the diagonal $x = y$, where we establish a novel connection between Hitomezashi and knot theory.