Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent? Suppose we are given embeddings $f_1,f_2:[0,1]\to\mathbb R^3$.
Does there exist a homeomorphism $g:\mathbb R^3\to\mathbb R^3$ such that $g\circ f_1=f_2$?
This question seems to be classical eighty years old result and looks like the Jordan curve theorem. I think I can deduce this for smooth mappings (using the Tubular neighborhood theorem), but in continuous case I'm a bit confused...
$\bullet$ On the one hand, in the Knot theory embeddings $S^1\to\mathbb R^3$ are considered up to ambient isotopy. Since the Reidemeister theorem holds, there are no wild pathologies.
$\bullet$ On the other hand, one can construct a curve like Alexander horned sphere (e.g. for the "horned map" $h:S^2\to\mathbb R^3$ it is sufficient to take a path on $S^2$ which passes through the Cantor set of "self-touches", and then take the image of the path under $h$).
The complement of such curve has to be not simply connected, I think.
Which strategy is wrong?
How can we classify embeddings $[0,1]\to\mathbb R^3$ up to topological equivalence?
 A: As igorf pointed out in the comment, the answer to the first question is 'no'. A quick counterexample is by looking into the complement of Fox-Artin arc, which is not simply connected. See figure which is from Wikipedia (which is also from Fox-Artin original paper).

It seems a complete classification would be nearly impossible. Even for the arcs which are wild at one point is hard. Note the first figure has two wild points. The second figure is wild at one end point p (quoted from Daverman-Venema's book Embeddings in Manifolds P. 76)

This is what I called the basic Fox-Artin arc in the comments. In the literature, arcs which are wild at one end point is called nearly polyhedral arc. I think there is no classification even for nearly polyhedral arcs. Only some invariants exist. For example, J. M. McPherson in Pacific J. Math. 45(2): 585-598 (1973) developed some invariant which can show when two Fox-Artin type arcs are equivalent. As a consequence, there exists an uncountably family of locally noninvertible Fox-Artin type arcs in $R^3$, and an uncountable family of nonamphicherial such arcs.
However, there does exist some classification for special arcs. The most successful one would be R. H. Fox and O. G. Harrold's complete classifications of the Wilder arcs of penetration index 2 (denoted P(A, x) =2: an arc $A$ at a point $x \in A$ to be the smallest cardinal number $n$ such that there are arbitrarily small 2-spheres
enclosing $x$ and containing no more than $n$ points of $A$) in "The Wilder arcs", Topology of 3-manifolds and related topics, Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 184-187. ) Later, Alford and Ball (Some almost polyhedral wild arcs, Duke J. Math. 30 (1963),
33-38) showed that the penetration index can distinguish a large class of wild arcs.
Using Brody's technique, S. J. Lomonaco (An algebraic theory of local knottedness, I, Trans. Amer. Math. Soc.
129 (1967), 322-343 and the reference therein) developed an algebraic theory to distinguish wild arcs with one wild point in the interior. The following picture is a connected sum of a Wilder arc and a tame arc. Arcs like this is called mildly wild arc.

For topics like arcs with one wild point there are many theories. Not to mention arcs with more wild points. One can also discuss things like moving one arc off of itself using ambient $\epsilon$-homeomorphism. For instance, see a construction in J. W. Cannon and D. G. Wright (Slippery Cantor set in $E^n$, Fundamenta Mathematicae 106.2 (1980): 89-98). To detect local wildness in general, I would recommend Daverman-Venema's book Embeddings in Manifolds, Chapter 2 for instance, they gave a thorough explanation for notions like 1-alg (local homotopically unknottedness) for codimension 2 embeddings, and 1-LCC for codimension 3 embeddings, which are powerful tools.
By the way, if one is willing to relax the condition from isotopy to pseudo-isotopy, Keldys has shown that for any locally unknotted arc $A$ in $E^3$ there exists a pseudo-isotopy of $E^3$ translating a straight line segment homeomorphically onto $A$. See Topological imbeddings of simple arcs and closed curves in E3, Soviet Math. Dokl. 10 (1969), pp. 376-379.
