One can define a knot in two ways:
(1) A knot is a closed polygonal curve in $\mathbb R^3$
(2) A knot is an equivalence class of embeddings $S^1 \hookrightarrow \mathbb R^3$
And perhaps also:
(3) A knot is an equivalence class of smooth $1$-dimensional submanifolds of $S^3$
Question 1: Can I replace $S^3$ in (3) with $\mathbb R^3$?
Question 2: I would like to define what a regular projection of a knot is. Unfortunately, it depends on whether I use (1), (2) or (3). I would like to use (2) and I have the definition using (1), which goes as follows:
(Definition) A knot projection is called a regular if no three points on the knot project to the same point, and no vertex projects to the same point as any other point on the knot.
How can I define regular projection without (1), that is, how can I define it for embeddings instead of polygonal curves?
Thanks a lot!