Homotopy classification of closed polygonal lines: results and problems

2025/08/22 by Alkin, E., Nikitenko, O., Skopenkov, A.
#55-02 #55M25 #57M05 #57M10 #57M15 #Algebraic Topology (math.AT) #FOS: Mathematics #History and Overview (math.HO)

paper · doi:10.48550/arxiv.2508.16287

Abstract

In this text we expose (as a sequence of problems) basic cases of some fundamental ideas and methods of mathematics. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal lines in a subset of the plane. Although these ideas and methods are parts of topology, they are used in many other areas including computer science. This text is expository and is accessible to mathematicians not specialized in the area (and to students).

Citations

• Invariants of almost embeddings of graphs in the plane
• Invariants of almost embeddings of graphs in the plane: results and problems
• Extendability of simplicial maps is undecidable
• Picture-Hanging Puzzles

Related

• Methods from Differential Geometry in Polytope Theory
• On The Topology of Polygonal Meshes
• Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
• Problems on Polytopes, Their Groups, and Realizations
• General polygonal line tilings and their matching complexes