The third symmetric potency of the circle and the Barnette sphere

TL;DR

This paper provides an elementary proof that the third symmetric potency of a circle is homeomorphic to a 3-sphere and contains a trefoil knot, also explicitly decomposing it as the Barnette sphere.

Contribution

It offers a new elementary proof of known topological results and explicitly constructs a simplicial decomposition of the space.

Findings

The third symmetric potency of the circle is homeomorphic to a 3-sphere.

The inclusion of one-element subsets forms a trefoil knot.

The space is isomorphic to the Barnette sphere.

Abstract

We give an elementary (not cut just paste) proof of results of Bott and Shchepin: the space of non-empty subsets of a circle of cardinality at most 3, which is called the third symmetric potency of the circle, is homeomorphic to a 3-sphere and the inclusion of the space of one element subsets is a trefoil knot. Moreover, we give an explicit simplicial decomposition of the third symmetric potency of the circle which is isomorphic to the Barnette sphere.