On Planar Embeddings of the Knaster V Lambda-Continuum

Abstract

A\v nusi\'c, Bruin, and \v Cin\v c have asked in \cite{Anusic2} which hereditarily decomposable chainable continua (HDCC) have uncountably many mutually inequivalent planar embeddings. It was noted, as per the embedding technique of John C. Mayer with the $\sin(1/x)$-curve \cite{Mayer}, that any HDCC which is the compactification of a ray with an arc likely has this property. We show here two methods for constructing $\mathfrak{c}$-many mutually inequivalent planar embeddings of the classic Knaster $V \Lambda$-continuum, $K$, also referred to here as the Knaster accordion. The first of these two methods produces $\mathfrak{c}$-many planar embeddings of $K$, all of whose images have a different set of accessible points from the image of the standard embedding of $K$, while the second method produces $\mathfrak{c}$-many embeddings of $K$ which preserve the set of accessible points of the standard embedding.