Taming the Knight's Tour: Minimizing Turns and Crossings

We introduce two new metrics of simplicity for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with $9.5n+O(1)$ turns and $13n+O(1)$ crossings on a $n\times n$ board. We show lower bounds of $(6-\varepsilon)n$, for any $\varepsilon>0$, and $4n-O(1)$ on the respective problems of minimizing these metrics. Hence, we achieve approximation ratios of $19/12+o(1)$ and $13/4+o(1)$. We generalize our techniques to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for $(1,4)$-leapers. In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.