I am trying to prove that every tame knot is isotopic to the unknot. Here is my attempt. $\newcommand{\sone}{\mathbf{S}^1}$ $\newcommand{\rthree}{\mathbf{R}^3}$
A (tame) knot is a locally flat embedding of $\sone$ in $\rthree$ and $\sone \subset \rthree$. Let $f \colon \sone \rightarrow K$ be an embedding of a knot $K$. Consider a locally flat neighbourhood $U_p \subset \rthree$ of a point $p \in K$, and a locally flat neighbourhood of $W_q \subset \rthree$ of a point $q \in \sone$. Now, $$f^{-1}|_{U_p \cap K} \colon U_p \cap K \rightarrow W_p \cap \sone$$ and $$f^{-1}|_{K \setminus (U_p \cap K)} \colon K \setminus (U_p \cap K) \rightarrow \sone \setminus (W_p \cap \sone)$$ are homeomorphisms.
So, $K \setminus (U_p \cap K)$ contains all the 'knotted' part and $f$ maps $\sone \setminus (W_p \cap \sone)$ to the 'knotted' part.
Let $\sim$ be the equivalence relation on $\sone$ defined by identifying all points of $\sone \setminus (W_p \cap \sone)$ to one single point. $\sone$ is isotopic to $\sone \setminus \!\sim$. Let $i \colon [0, 1] \times \sone \rightarrow \mathbf{R}^3$ be that isotopy, with $i_0(\sone) = \sone$ and $i_1(\sone) = \sone\setminus\!\sim$.
Now, is it correct that if one proves that $i \circ f^{-1}$ is an isotopy, then we are done?
I am not able to prove that $i \circ f^{-1}$ is an isotopy. Composition of two continuous and bijective functions is continuous and bijective. Thus, each $i_t \circ f^{-1}$ is continuous and bijective, for all $t \in [0,1]$.
By local flatness, I mean the following. A point $p \in K$ is said to be locally flat if there exists a neighbourhood $U_p \ni p$ such that the ordered pair $(U_p, U_p \cap K)$ is homeomorphic to $(B, d)$, where $B \subset \mathbf{R}^3$ is the unit ball around origin and $d \subset B$ is the set of points along a diameter of the ball. A knot $K$ is said to be locally flat if all points are locally flat.
By isotopy, I mean a homotopy $i$ such that all the individual $i_t$s are bijective. I distinguish between isotopy and ambient isotopy, although people usually mean ambient isotopy when they mean isotopy in a knot theory context. By ambient isotopy, I mean $a \colon [0, 1] \times \mathbf{R}^3 \rightarrow \mathbf{R}^3$. The domain in the ambient isotopy is not $\mathbf{S}^1$ but rather $\mathbf{R}^3$. My terminology follows Knots and Links by Cromwell.
$i \circ f^{-1}$ is defined as follows. $i \circ f^{-1} \colon [0, 1] \times K \rightarrow \rthree$, $(i \circ f^{-1}) (t, p) = i_t(f^{-1}(p))$, for all $t \in [0, 1]$ and $p \in K$.