TITLE: Simultaneously minimizing intersections
QUESTION [5 upvotes]: This may be a standard problem in homotopy theory, but I don't know a good reference.
Let $\Sigma$ be a smooth, oriented surface, and let $X_1,X_2$ and $X_3$ be three smoothly embedded curves in $\Sigma$.  Can I find smoothly embedded curves $Y_1,Y_2$ and $Y_3$ such that...

$Y_i$ is homotopic to $X_i$ through smooth embeddings,
each intersection in $Y_1\cup Y_2\cup Y_3$ is a transverse double intersection, and
the number of intersections between $Y_i$ and $Y_j$ is minimal among all pairs homotopic to $X_i$ and $X_j$ through smooth embeddings?

Roughly speaking, I have three curves, and I want to use homotopy to simultaneously minimize the number of intersections between each pair of curves.

REPLY [5 votes]: Firstly, "homotopic through smooth embeddings" is known as "isotopic", and there is a well known paper of David Epstein titled "Curves on 2-manifolds and isotopies" where he shows that homotopy is equivalent to isotopy in this setting.
Secondly, if you put a hyperbolic structure on your surface, then the curve shortening flow will isotope your curves into geodesics, which will automatically minimize intersection numbers (this can be seen by going to the hyperbolic plane as in the Klein model, and seeing what intersection numbers mean in terms of lifts and cross-ratios), and will also minimize self-intersection numbers, so will stay embedded if your curves are to begin with.
Thirdly, the geodesics, as above, might have triple intersections, but it is clear (by transversality, if you like) that a small local perturbation will remove the triple points (this would be less obvious if your curves $X_i$ were not embedded, but they are.)