Nullhomotopic and Generating Knight's Tours on Non-Orientable Surfaces
Bradley Forrest, Zachary Lague
TL;DR
This paper explores the existence and characteristics of closed knight's tours on non-orientable surfaces like the Möbius strip and Klein bottle, focusing on their topological properties and fundamental group representations.
Contribution
It characterizes the board dimensions that allow nullhomotopic and generator-realizing knight's tours on non-orientable surfaces, advancing understanding of topological chess problems.
Findings
Identifies board dimensions supporting nullhomotopic tours.
Determines dimensions allowing tours that generate fundamental group elements.
Provides a topological classification of knight's tours on non-orientable surfaces.
Abstract
We investigate closed knight's tours on M\"obius strip and Klein bottle chess boards. In particular, we characterize the board dimensions that admit tours that are nullhomotopic and the board dimensions that admit tours that realize generators of the fundamental groups of each of the surfaces.
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Taxonomy
TopicsMathematics and Applications · Computational Geometry and Mesh Generation · Advanced Combinatorial Mathematics