(Apologies in advance for any imprecision in the following; I am a computer scientist and regret never having taken an actual course on topology.)

One way to define the pure braid group $P_n$ is as follows: consider a pure braid to be a set of $n$ non-intersecting arcs in $x,y,t$-space which are monotone in the $t$ direction, such that the $i$th arc connects $(i,0,0)$ to $(i,0,1)$. Sets of arcs which can be deformed continuously into each other without any arcs intersecting are considered equivalent pure braids.

Consider a generalization where the endpoints are not integer positions on two lines, but arbitrary fixed positions in the $t = 0$ and $t = 1$ planes. It seems clear to me that this does not change the topology of the space [1], so I will call such a collection of arcs an *embedded braid*. I'm looking at characterizing "optimal" embedded braids that minimize a certain functional $F$. For concreteness, let $F$ be the total length of the arcs (the actual functional I need to use is slightly different, but this should be close enough to carry over the results).

If the requirement that arcs do not intersect is removed, the space of embedded braids can be given an affine structure over which $F$ is convex, and it is immediately apparent that there is a single local minimum which is the global minimum: connect each pair of endpoints with a straight line. What can we say about the local minima of $F$ if we retain the non-intersection property? My intuition says that each topologically distinct braiding (corresponding to a particular element of $P_n$) forms a connected component with a unique local minimum, but I cannot tell how to begin proving this. It would be true if one could show that the connected components are convex subsets of the space; this does not hold under the "obvious" affine structure where we treat each arc and each $t$ independently, but that leaves the possibility of some other choice of affine structure which works. Are there some nice proof techniques that would help proving uniqueness of local minima in this context? Would any ideas or analogies from the theory of minimal surfaces help?

As Andrew Stacey mentions, the connected components are *open* subspaces; I believe existence of local minima can be guaranteed by considering the *closure* of the component seen as a subset of the space of embeddings that allow intersection. (Following Kevin Walker's comment, I realize this is called compactification.) This would include embeddings "on the boundary", whose arcs can intersect but which are only an infinitesimal displacement away from non-intersecting embeddings with the right topology. As a concrete example, the following braid,

\ / \ / \ / \ \ / \ / \ / \

on being pulled tight, approaches a configuration where the arcs are infinitesimally close together in the middle; the minimal length is attained by the intersecting embedding where the two arcs share a common point.

[1] Since the endpoints are fixed, we can associate each such generalized braid uniquely with a pure braid by shrinking the $t$ dimension slightly and composing each arc $i$ with a fixed arc from its endpoints to $(i,0,0)$ and $(i,0,1)$ respectively.