Suppose $K^1 \subset S^3$ is a (connected) knot. First, I am wondering is there a Morse function $f$ on $S^3$ such that $f$ restricted to a neighborhood $D^2 \times S^1$ of $K^1$ have standard form, i.e the standard Morse function on $S^1$ with two critical points plus $r^2$, where $r$ is the radial coordinate on $D^2$. In particular, $f$ restricted to $D^2 \times S^1$ has two critical points of index 0,1 and $\nabla f$ is outward pointing along $\partial (D^2 \times S^1)$.
I am also wondering what the genus of a knot has to do with the number of critical points of such a Morse function. If the knot genus is $k$, is it true that there are at least $k+1$ critical points of $f$ of index $1$? If $K^1$ is the unknot, then $f$ can be a Morse function with 4 critical points of index $0,1,2,3$. Then by topology, there must be at least $k+1$ critical points of index 2 to cancel these critical points of index 1.
Finally, I am trying to understand the index 3 critical points, which are the most confusing to me. I think, in general, there need to be many index 3 critical points but I am not sure how they appear from the perspective of the genus/Seifert surface. It would be helpful if someone could explain or provide a reference to a concrete example, say the $(p,q)$-torus knot (which has genus $(p-1)(q-1)$). Maybe this has to do with Heegaard splittings of $S^3$. Essentially I want a concrete description of the knot complements in terms of Morse theory or cell decompositions.