# Finding my mistake in a "proof" that there is no non-trivial knot

I am working on a presentation about the complexity of determining whether a knot is trivial, and while reading about some topological stuff I somehow managed to "show" that there are no non-trivial knots. Obviously, there has to be something wrong with my "proof", but I can't see where I did something wrong, or maybe which of the things I read about are inaccurate.

Here are the definitions and propositions I've picked up from various papers.

Definition 1: An embedding $$N\hookrightarrow M$$ is proper, if $$\partial N=N\cap\partial M$$. And a properly embedded hypersurface $$S\subset M$$ is essential in $$M$$, if it cannot be homotoped into $$\partial M$$ while keeping $$\partial S$$ fixed in place, and if the fundamental group $$\pi_1(S)$$ of $$S$$ injects into $$\pi_1(M)$$.

Proposition 2: Let $$K$$ be a tame knot embedded in $$\mathbb S^3$$, and let $$T_K\subseteq\mathbb S^3$$ be an open tubular neighborhood of $$K$$. The knot $$K$$ is trivial, if and only if the knot complement $$M_K:=\mathbb S^3\setminus T_K$$ of $$K$$ contains an essential disk (whose boundary certainly is a longitude of $$\overline{T_K}$$).

(as far as I know this is a corollary from the fact that a knot embedded in $$\mathbb R^3$$ is trivial, if and only if it has a spanning disk, that is, a disk embedded in $$\mathbb R^3$$ whose boundary is the knot traversed once.)

Proposition 3: The complement $$T'$$ of the interior of a solid torus $$T$$ in $$\mathbb S^3$$ is homeomorphic to a solid torus, the longitudes of $$T$$ are the meridians of $$T'$$ and the meridians of $$T$$ are the longitudes of $$T'$$.

Proposition 4: Every solid torus $$T$$ contains an essential disk, and every such disk has a meridian of $$T$$ as boundary.

And here is my corollary from those propositions.

My corollary: Every tame knot is trivial. Proof. Let $$K\subset\mathbb S^3$$ be any tame knot. Since $$K$$ is an embedding of the circle $$\mathbb S^1$$ into $$\mathbb S^3$$, any open tubular neighborhood $$T_K\subseteq\mathbb S^3$$ of $$K$$ is homeomorphic to $$\mathbb S^1\times(D^2)^\circ$$, i.e. the interior of a solid torus. By proposition 3 the knot complement $$M_K=\mathbb S^3\setminus T_K$$ is homeomorphic to a solid torus which, by proposition 4, contains an essential disk (with a meridian of $$M_K$$ or a longitude of $$\overline{T_K}$$ as boundary). Hence, $$K$$ is trivial by proposition 1.

Can you see where I've made a mistake or which propositions are inaccurate?