The following question is motivated by Anton Petrunin's https://mathoverflow.net/q/257420/36904 :

> **Question**: Does every sphere topologically embedded in
> $\mathbb{R}^3$ necessarily contain a ***planar*** arc?

An *arc* is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is *planar* if it is contained in some plane. A negative answer to this question would immediately answer Anton's question in the negative as well.