arXiv subjects
Publications and source records attributed to Ben Coté.
Crossword puzzles lend themselves to mathematical inquiry. Several authors have already described the arrangement of crossword grids and associated combinatorics of answer numbers. In this paper, we present a new graph-theoretic representation of crossword puzzle grid designs and describe the mathematical conditions placed on these graphs by well-known crossword construction conventions.
The complement of a hyperplane arrangement in $\mathbb{C}^n$ deformation retracts onto an $n$-dimensional cell complex, but the known procedures only apply to complexifications of real arrangements (Salvetti) or the cell complex produced depends on an initial choice of coordinates (Björner-Ziegler). In this article we consider the unique complex euclidean reflection group acting cocompactly by isometries on $\mathbb{C}^2$ whose linear part is the finite complex reflection group known as $G_4$ in the Shephard-Todd classification and we construct a choice-free deformation retraction from its hyperplane complement onto an elegant $2$-dimensional complex $K$ where every $2$-cell is a euclidean equilateral triangle and every vertex link is a Möbius-Kantor graph. Since $K$ is non-positively curved, the corresponding braid group is a CAT(0) group, despite the fact that there are non-regular points in the hyperplane complement, the action of the reflection group on $K$ is not free, and the braid group is not torsion-free.