Is a non-geometrically essential knot in a standard solid torus in $\mathbb{R}^3$ trivial?

Question

Let $V=S^1 \times D^2$ be a standard solid torus in $\mathbb{R}^3$, where $S^1$ and $D^2$ denote a standard circle and a 2-disk, respectively. Let $K$ be a knot that is embedded inside $V$ so that $K$ is contained within a 3-ball in $V$ ($K$ is not geometrically essential in $V$). In other words, there is a meridian disk $D_1^2$ such that $K \cap D_1^2=\emptyset $. Can we conclude that K is trivial?

Answer 1

I think you mean that the intersection with every meridian disc is non-empty.

Take the trefoil knot and embed it in the torus in such a way that the hole of the torus passes through the center of the trefoil as in the picture in the link.

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The non-geometrically essential is also not related to the knot not being trivial.

Embed the same trefoil in a ball and put the ball (isometrically) inside the torus.

It is, however, a trivial loop, in the sense that you can contract it to a point.