The Topology of Knight's Tours on Surfaces
Bradley Forrest, Kara Teehan
TL;DR
This paper explores the topological properties of closed knight's tours on cylindrical and toroidal chessboards, classifying their homotopy classes and identifying conditions for different fundamental group realizations.
Contribution
It characterizes the dimensions of cylindrical and toroidal chessboards that admit specific types of closed knight's tours based on homotopy classes.
Findings
Classified dimensions of cylindrical boards with tours representing the identity element
Identified conditions for tours representing generators of the fundamental group
Extended results to toroidal chessboards
Abstract
We investigate the homotopy classes of closed knight's tours on cylinders and tori. Specifically, we characterize the dimensions of cylindrical chessboards that admit closed knight's tours realizing the identity of the fundamental group and those that admit closed tours realizing a generator of the fundamental group. We also produce analogous results for toroidal chessboards.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology