Let $K:\mathbb{S}^1\times I\to \mathbb{S}^3$ be an isotopy of the knots $K_0 = K(\cdot, 0) $ and $K(\cdot, 1).$ Fix $v \in \mathbb{R}^3\subset \mathbb{S}^3$.
I have seen in my knot theory course that we can perturb slightly $K$ in such a way that the projection into the orthogonal of $v$, $$p = \pi_{v^{\perp}} \circ K:\mathbb{S}^1\times I\to v^{\perp}\simeq \mathbb{R}^2$$ satisfies: 1) $\frac{\partial}{\partial z}p(z,t) = 0$ only for a finite number of $t\in I$ and isolated $z$ in $\mathbb{S}^1$
2) $p$ can have double points (also non-tranversal)
3) if $p(z)$ is a triple point then it is a transversal triple point
$p$ has at most double or triple point (i.e. the cardinality of the fiber is at most 3).
This fact is very important in knot theory as is used to introduce Reidemeister moves but I cannot find a proof of it.