gt.geometric topology - Minimal-length embeddings of braids into R^3 with fixed endpoints

UPDATE.

I revisited the question and realized that verifying the local CAT(0) property is not that easy. When I wrote the original answer, I was under impression that removing any collection of codimension 2 subspaces (more precisely, their tubular neighborhoods) from $mathbb R^n$ leaves one with a locally CAT(0) space. This is true in $mathbb R^3$ but there are counterexamples in $mathbb R^4$. This particular subset might satisfy the condition but this does not seem to follow from any "generic" argument.

Further, there are some discouraging examples. First, approximating by arcs by "ropes" with fixed-size square sections does not actually work: in this case there are non-unique minimal configurations. So one really needs to deal with zero-width ones. Second, if you consider 3 arcs and allow two of them intersect while deforming the braid (but still disallow intersections with the 3rd one), then again, you can have two distinct minimal configurations in the same equivalence class.

Below is the original answer (and I suggest that it is unaccepted).

---

I can prove uniqueness of a local minimum for another length-like functional
(similar but not equal to the sum of lengths). I believe that it should work the same way
for the sum of lengths, but unfortunately the underlying geometric theory does not
seem to exist (yet?).

First let me reformulate the problem. Let $X$ denote the set of possible horisontal cross-sections of braids.
This is the set of $n$-tuples of distinct points in $mathbb R^2$. Geometrically this is $(mathbb R^2)^n$
minus a collection of codimension 2 subspaces corresponding to positions where some two points coincide.

Actually I prefer another formalization: the arcs forming the braid are ropes of nonzero width
and with square cross-sections.
More precisely, the horizontal section of every rope is an $varepsilontimesvarepsilon$ square (with sides parallel
to the coordinate axes), and these sections should not overlap. So $X$ is $mathbb R^{2n}$ minus a union of polyhedral
heighborhoods of codimension 2 subspaces. This formalization makes the local structure simpler,
and the original one is the limit as $varepsilonto 0$.

An embedded braid is a path $f:[0,1]to X$. We want to minimize the functional
$$
L = int_0^1 left(sqrt{(df_1/dt)^2+1}+dots+sqrt{(df_n/dt)^2+1}right) dt,
$$
over all paths between two given positions $a,bin X$, in a given connected component
of the space of such paths. Here $f_i=f_i(t)$ are 2-dimensional coordinates.
Another functional,
$$
L' = int_0^1 sqrt{(df_1/dt)^2+dots+(df_n/dt)^2+1} dt,
$$
is easier to deal with, because this is the Euclidean length of the corresponding
path $(f(t),t)$ in $Xtimesmathbb Rsubset mathbb R^{2n+1}$. Let us work with $L'$.

The connected component of the set of paths is the homotopy class. Fixing the homotopy class of a path is the same as
fixing endpoints of its lift to the universal cover of the space.
(In the case of zero-width ropes, you first take the universal cover and then the completion in order to add "boundary cases".)
And local length minimizers are called geodesics. So we want to show that, in the universal cover of $Xtimesmathbb R$,
every pair of points is connected by a unique geodesic.

Our space $X$ (and hence $Xtimesmathbb R$) is a locally CAT(0) space.
This can be shown using standard curvature tests for polyhedral spaces.
A globalization theorem says that a complete, simply connected, locally CAT(0) space is globally CAT(0),
in other words, it is a Hadamard space. So the universal cover is a Hadamard space.
Hadamard spaces feature uniqueness of geodesics, hence the result.

Now what about the original functional $L$? It is also a length functional, but for a non-Euclidean norm on $mathbb R^{2n+1}$.
So, to carry over the proof, one needs to develop an analog of CAT(0) spaces modelled after Finsler spaces rather than Riemannian. I think it should be possible - after all, all this CAT(0) business is about convexity of distance, and this convexity is there in all normed vector spaces.
But I have not heard of such generalizations (maybe this is just my ignorance).