Toroidal Hitomezashi Patterns

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.

Figures (11)

• Figure 1: The symmetric toroidal Hitomezashi pattern $\mathsf{Cloth}_{8,8}(x,x)$ for $x=---+++++$, with Hitomezashi loops distinguished by different colors. The red and orange loops are nontrivial with homology class $(1,1)$, while the other six loops are trivial. The black dotted square represents a fundamental domain for the torus.
• Figure 2: Decomposition of \ref{['fig:00011111']} into two Hitomezashi patterns as in \ref{['defn:hitomezashi-original']}.
• Figure 3: Comparison between $\mathsf{Cloth}_{7,7}(x, x)$ and $\mathsf{Cloth}_{7,7}(x, x')$, where $x = --+++++$ and $x' = ---++++$. On the right, all green edges form a single toroidal Hitomezashi loop of homology class $(3, 1)$.
• Figure 4: The symmetric toroidal Hitomezashi pattern for $x=+-++-$, redrawn on the annulus. The blue points indicate the intersections that the first strand passes across in the sequence of triple point moves that exchanges the first and second strands.
• Figure 5: A link diagram for the torus link $T(5,5)$, equipped with an orientation. The underlying link-like graph is $G'(x)$, $x=+-++-$.

Theorems & Definitions (23)

• Definition 1.1
• Definition 1.2
• Definition 1.3: Combinatorial Definition of homology
• Theorem 1: Homology class
• Theorem 2: Loop length
• Theorem 3: Loop count
• Remark
• Definition 3.1
• Lemma 3.2
• proof