Is there any way to represent every element of the mapping class group of a surface as an arc on that surface?



Arcs and elements of the mapping class group
The natural action (of mapping classes act on isotopy classes of arcs) has large stabilisers. So the "correct" answer to your question is "no". 


Now, the mapping class group is countable. The set of isotopy classes of arcs is also countable. With a bit of work, you can construct a bijection between them. So in that sense the answer to your question is "yes". (In fact, you will learn a lot from constructing such a bijection.) But making the bijection will involve many choices; so it cannot be used to prove anything...


