相关论文: The Topology of Knight's Tours on Surfaces
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…
In this paper we are concerned with knight's tours on high-dimensional boards. Our main aim is to show that on the $d$-dimensional board $[n]^d$, with $n$ even, there is always a knight's tour provided that $n$ is sufficiently large. In…
The problem of existence of closed knight tours for rectangular chessboards was solved by Schwenk in 1991. Last year, in 2011, DeMaio and Mathew provide an extension of this result for 3-dimensional rectangular boards. In this article, we…
We study the closure of horocycles on rank 1 nonpositively curved surfaces with finitely generated fundamental group. Each horocycle is closed or dense on a certain subset of the unit tangent bundle. In fact, we classify each half-horocycle…
The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present paper, we provide a $5$-dimensional…
The author has constructed and enumerated tours of knight having various magic properties on 4 x n and 6 x n boards. 16 magic tours of knight have been discovered on 4 x 18 board, 88 on 4 x 20 board, 464 on 4 x 22 board, 2076 on 4 x 24…
The present paper aims to extend the knight's tour problem for $k$-dimensional grids of the form $\{0,1\}^k$ to other fairy chess leapers. Accordingly, we constructively show the existence of closed tours in $2 \times 2 \times \cdots \times…
We track the trajectories of individual horocycles on the modular surface. Our tracking is constructive, and we thus \emph{effectively} establish topological transitivity and even line-transitivity for the horocyclic flow. We also describe…
The problem of existence of closed knight's tours in $[n]^d$, where $[n]=\{0, 1, \dots, n-1\}$, was recently solved by Erde, Gol\'{e}nia, and Gol\'{e}nia. They raised the same question for a generalised, $(a, b)$ knight, which is allowed to…
We produce a sequence of finite dimensional representations of the fundamental group $\pi_1(S)$ of a closed surface where all simple closed curves act with finite order, but where each non--simple closed curve eventually acts with infinite…
In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various…
A knight's tour is often represented as a broken line connecting the centers of successively visited squares. We say that two knight moves form a cross if the midpoints of their respective segments coincide. We show that no knight tour…
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the…
In this paper we study different kinds of symmetries related to the domino tilings of chessboards.
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal…
The purpose of this paper is to survey the structure of closed and transitive transformation groups acting on a closed surface. In particular, we prove a number of relations between groups acting on the sphere that contain the rotation…
We study a class of finite-area, infinite-type translation surfaces, and find an explicit cylinder decomposition on these surfaces which do not manifest on finite-type translation surfaces. Each cylinder decomposition contains a special…
We consider algebraic actions of a cyclic group of order p on a K3 surface defined over an algebraically closed field of characteristic p. We classify possible loci of fixed points as well as possible quotient surfaces.
We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the…
We study the rotor-router walk with the clockwise ordering of outgoing edges on the semi-infinite cylinder. Imposing uniform conditions on the boundary of the cylinder, we consider growth of the cluster of visited sites and its internal…