Change of coordinates principle for mapping class groups

Question

So I am reading Farb and Margalit's text, and his statement for the change of coordinates principle doesn't seem complete. Namely, he states that there is an orientation preserving homeomorphism of a genus g surface sending one collection of curves to another if they have the same intersection pattern. However I think we also need to stipulate that the complements of the curve have the same type of connected components. Am I mistaken? Please correct me if I'm wrong. I just want to make sure my understanding of the principle is correct or not.

Answer 1

Your intuition that cutting along these curves should not result in different surfaces is completely correct. The following two pairs of curves, $(\alpha,\beta)$ and $(a,b)$, cannot be sent to one another by any homeomorphism, for the connected components they cut off of the surface are not homeomorphic.

However, this is somewhat of a special case, as I chose a bounding pair of curves that have different genus. Note that the Primer mentions that the given statement of the change of coordinates principle is "rough." Pages 39-40 give examples of different types of pairs of curves, and mentions a few where only the intersection number is needed, e.g. when it is equal to one.