Continuum Many Planar Embeddings of the Knaster Accordion

Anusic, Bruin, and Cinc have asked 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, that any HDCC which is the compactification of a ray with an arc likely has this property. Here, we show two methods for constructing $\mathfrak{c}$-many mutually inequivalent planar embeddings of the classic Knaster $V \Lambda$-continuum, $K$, also referred to 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 $K$, each of which preserve the accessibility of all points in the standard embedding.