# Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

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.