The following question is motivated by Anton Petrunin's Disc bounded by a plane curve :

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.