A hyperbolic approach to exp_3(S^1)

TL;DR

This paper introduces a hyperbolic geometric approach to analyze the configuration spaces of subsets of the circle, providing explicit structures and new proofs for classical topological theorems involving exp_3(S^1).

Contribution

It presents a novel hyperbolic geometric method to explicitly determine the structure of configuration spaces of the circle and offers new proofs of classical theorems about their embeddings.

Findings

Explicit structure of exp_k(S^1) for small k

Reproof of Bott's theorem using hyperbolic geometry

Proof that the embedding of exp_1(S^1) into exp_3(S^1) is the trefoil knot

Abstract

In this paper we investigate a new geometric method of studying exp_k(S^1), the set of all non-empty subsets of the circle of cardinality at most k. By considering the circle as the boundary of the hyperbolic plane we are able to use its group of isometries to determine explicitely the structure of its first few configuration spaces. We then study how these configuration spaces fit together in their union, exp_3(S^1), to reprove an old theorem of Bott as well as to offer a new proof (following that of E. Shchepin) of the fact that the embedding exp_1(S^1) into exp_3(S^1) is the trefoil knot.