Let $F$ be a surface of genus $g$ that is decorated with $g-$many $\alpha$ curves (in red) and $g-$many $\beta$ curves (in blue). If we like, we can take each curve to be non-separating. Now suppose that $\alpha'$ is an entirely different collection of $g-$many non-separating red curves. Is there necessarily a homeomorphism $\phi:F\to F$ which takes $\alpha$ to $\alpha'$? Is it unique?
If so, is there an algorithm to compute what the new blue curves will be? Something like cutting up the original surface by the red curves and keeping track of intersections.
Thanks