Suppose one has a knot $K$ embedded in $\mathbb{R}^3$;
but view $\mathbb{R}^3$ as a 3-flat in $\mathbb{R}^4$.
Of course $K$ is not a knot in $\mathbb{R}^4$.
I am wondering if there has been any study of how many "moves"
are needed to unravel $K$ using the 4th dimension?


One might make this a sharper question in several ways.
Here is one attempt. Say $K$ is represented as a 3D polygon
of $n$ vertices in $\mathbb{R}^3$.
A *move* consists of rotating a subchain of $C \subset K$
with endpoints $(a,b)$ into 4D
and then back again at some new orientation into the 3-flat containing $K$.
The endpoints $a$ and $b$ remain fixed, 
while $C$ is replaced by $C'$.
Call the result knot $K'$.



> 
> **Q1**.
>  Can every knot $K \subset \mathbb{R}^3$ be unraveled by these moves?
> 
> 
> 


**Answer**: *Yes*, by Ian Agol's convincing argument.



> 
> **Q2**.
>  How many such moves are needed to untangle $K$, as a function of
>  some measure of $K$'s complexity, say, its crossing number?
> 
> 
> 


These are entirely naïve questions, and I am not sure I have formulated
them coherently. If not, apologies for the distraction!



Unknotting knots in 4D
I think your moves suffice. One may prove that your
moves may rotate the polygonal knot into a convex planar polygon by induction.


As a warmup, suppose we have a polygonal knot in the plane. Consider its
convex hull, one gets a convex polygon. If the knot is convex, then it lies
on the boundary of this polygon. Otherwise, there is an edge $e$ in the polygon
which is not an edge of the knot, and whose endpoints are vertices of the knot. 
Take an arc (chain of edges) of the knot bounded by these vertices, and rotate it to the
other side of the line containing the edge $e$. Repeat this process until
you obtain a convex polygon. So this shows that any planar polygon may
be chain-rotated to be convex. 


Now take a 3D polygonal knot $K$ with $n$ segments. If $n=3$, then we are done, so assume $n>3$.
Take the convex hull of the knot, and assume it is nonplanar.
The boundary is a polyhedron, and there is some edge $e$ of the polyhedron 
which is not an edge of the knot. Take the two endpoints of this edge, these must
be vertices of the polygonal knot. The interior of $e$ might meet other
vertices of the polygonal knot, so take two vertices $v\_1,v\_2$ of the knot 
in the edge $e$ which have no other vertices between them. Take a supporting plane
for the polyhedron along the edge $e$, and choose an arc $a$ of the knot with the vertices $v\_1,v\_2$ as endpoints.
Rotate this arc in 4D to lie on the other side of the support plane. Now the 
knot has been reduced to a geometric connect sum: there is a plane meeting
the knot in two vertices of the polygon, with a rotated copy of $a$ lying
on one side, and $K-a$ lying on the other. Take the two knots which this 
is a connect sum of, by taking $a\cup e$ and $K-a\cup e$. 
Each of these knots may be rotated into a convex polygon in the plane by induction, 
so perform
the rotations individually, making sure that they don't interfere with
the other side by genericity. Once we have rotated both knots to be convex planar,
then we may rotate their connect sum to be planar, and therefore convex. So by induction,
any polygonal knot may be chain-rotated to be convex planar. 


