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.

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.

The following questions are motivated by Anton Petrunin's Disc bounded by a plane curve :

Question 1. Does every topologically embedded sphere in $\mathbb{R}^3$ necessarily contain a planar arc?

A negative answer to this question would immediately answer Anton's question in the negative as well.

Question 2. Does every topologically embedded disk in $\mathbb{R}^3$ necessarily contain a planar arc?

Remark. Obviously, a positive answer to Question 2 would imply the same for Question 1, but the converse is not obvious, perhaps not even true.