# Is it true that Morse function on non-trivial knot has at least 4 critical points?

I'm actually interested in the continuous case, for a non-trivial knot $S^1\rightarrow \mathbb{R}^3$ is it true that the function $\sin(t)$ can not extend to a continuous function on $\mathbb{R}^3$?

• I don't quite understand: For the title, a morse function on the knot is just a morse function on $\mathbb S^1$. It could have only two critical points. For the question in the body, are you looking for a function $f : \mathbb R^3 \to \mathbb R$ so that $f|_{\text{knot}} = \sin t$?
– user99914
Commented Oct 11, 2015 at 4:13
• @JohnMa yes, I mean Morse function w.r.t the embedding $S^1\rightarrow \mathbb{R}^3$ Commented Oct 11, 2015 at 4:20
• You should clarify precisely what it means to be a morse function with respect to an embedding. I have an idea, but it would be helpful for you to be explicit. Regarding the question in the body, Tietze extension theorem.
– user98602
Commented Oct 11, 2015 at 5:22
• @MikeMiller Yes this is the theorem I need, thanks a lot. Commented Oct 11, 2015 at 8:38