Suppose $S_g$ is a sorface of genus $g>1$. Let $\gamma_1$ and $\gamma_2$ be two simple closed curves containing points $p_1, p_2$. Suppose $\gamma_1$ and $\gamma_2$ are isotopic. Now there can be many isotopies between them which also takes $p_1$ to $p_2$for example if they bound a cylinder then we can drag one curve to another along that cylinder and then rotate. Now you can rotate the ending curve many ways to take $p_1$ to $p_1$. My question is
Is there any invariant which detects these two different isotopies.
What I mean by that is for circle homeomorphism rotation number is an invariant which detects the difference between two isotopic homeomorphism. Can we define something similar which can differentiate isotopies upto rotation of the initial and ending curve.
PS:1) I have tried with a fixed cover which has a convex fundamental domain and tried to defined it, but I am not sure how to translate this intuitive idea.
PS:2) I want it in topological or smooth level i.e. I don't want to use the facts of hyperbolic geometry e.g. in every homotopoy class there is a unique geodesic.
PS: 3) Even if the invariant can detect the rotations of the curve that will also help.
PS: 4) As I haven;t been able to formulate it properly any kind of suggestions or references will be helpful. Thanks in advance.