Gluing and cutting cube tiling codes in dimension six

Abstract

Let $S$ be a set of arbitrary objects, and let $s\mapsto s'$ be a permutation of $S$ such that $s"=(s')'=s$ and $s'\neq s$. Let $S^d=\{v_1...v_d\colon v_i\in S\}$. Two words $v,w\in S^d$ are dichotomous if $v_i=w'_i$ for some $i\in [d]$, and they form a twin pair if $v_i'=w_i$ and $v_j=w_j$ for every $j\in [d]\setminus \{i\}$. A polybox code is a set $V\subset S^d$ in which every two words are dichotomous. A polybox code $V$ is a cube tiling code if $|V|=2^d$. A $2$-periodic cube tiling of $\mathbb{R}^d$ and a cube tiling of flat torus $\mathbb{T}^d$ can be encoded in a form of a cube tiling code. A twin pair $v,w$ in which $v_i=w_i'$ is glue (at the $i$th position) if the pair $v,w$ is replaced by one word $u$ such that $u_j=v_j=w_j$ for every $j\in [d]\setminus \{i\}$ and $u_i=*$, where $*\not\in S$ is some extra fixed symbol. A word $u$ with $u_i=*$ is cut (at the $i$th position) if $u$ is replaced by a twin pair $q,t$ such that $q_i=t_i'$ and $u_j=q_j=t_j$ for every $j\in [d]\setminus \{i\}$. If $V,W\subset S^d$ are two cube tiling codes and there is a sequence of twin pairs which can be interchangeably gluing and cutting in a way which allows us to pass from $V$ to $W$, then we say that $W$ is obtained from $V$ by gluing and cutting. In the paper it is shown that for every two cube tiling codes in dimension six one can be obtained from the other by gluing and cutting.