Show that a curve which is perpendicular to a vector exactly twice is the unknot

Question

If I know that the tangent vector $T(s)$ of $\gamma(s)$, a smooth closed curve in $\mathbb{R}^3$, is perpendicular to some vector $\vec{v}$ exactly twice, say at $s_0$ and $s_1$, how can I show that $\gamma$ isn't knotted? I'm not allowed to use Fary-Milnor. I can see the picture, though.

Vector $\vec{v}$ gives us a family of planes. If one of these planes meets $\gamma$, then the intersection will contain one or two points. The plane will contain exactly one point of $\gamma$ when $T$ is perpendicular to $\vec{v}$ at either $s_0$ or $s_1$. Otherwise, as can be seen if we move the plane slightly away from $\gamma(s_0)$, $\gamma$ punches through the plane twice.

Answer 1

This is actually part of one of the possible proofs of Fáry-Milnor. As you pass planes orthogonal to $\vec v$ from the low point to the high point of the curve, you can have only two points of intersection (by a Rolle's Theorem argument). Joining those two points by a line segment in each of those planes will fill out a spanning disk for the curve, so it is unknotted.

Answer 2

Let $P_0$ be the plane intersecting $\gamma$ between $\gamma(s_0)$ and $\gamma(s_1)$.

Suppose the curve $\gamma$ intersects $P_0$ an odd number of times, say $2n+1$ times. Then $\gamma$ is not closed. Contradiction.

Suppose $\gamma$ intersects $P_0$ an even number of times, $2n$ many times. Since $\gamma$ is closed, each intersection point is paired with another such that $\gamma$ enters in one point and exits through the other. Then by Rolle's Theorem, there should be $n$ distinct places where $\gamma$ is tangent to the plane. But there is only one place where this happens on either side of the intersecting plane, so $n=1$.

Hence each intersecting plane contains either one or two points, one being only if the plane is tangent to $\gamma$ at $s_0$ or $s_1$. If we connect each pair of points with a line segment for each plane that intersects $\gamma$ twice, we fill out a "pringle". It is easy to see that the boundary of a pringle can be continuously deformed to a unit cirlce. Unknotted.