$\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are studying the well known group action $\Gamma \times V \to V$, yet we find ourselves utterly incapable of computing and translating elements $([\phi], [\alpha]) \mapsto [\phi(\alpha)]$ where $\alpha$ represents a "simple closed curve" in $V$. In otherwords it appears that the action of the MCG on the curve complex is incomputable for closed surfaces.

I am aware of Mark C. Bell's curver program. It's documentation indicates that it uses results of Saul Schleimer relating paths in a "flip graph" with words in the pointed mapping class groups. I am not aware of any flip graph analogy for the closed mapping class groups.

**My question:** What is best computational approach to being able to solve the following problem for the group action $\Gamma \times V \to V$:

``*Given $[\phi] \in \Gamma, [\alpha] \in V, [\beta] \in V$, determine whether $[\phi(\alpha)]=[\beta]$ is True or False in $V$.*"

Evidently the fact that we have no linear representations of $\Gamma$ is one computational obstruction, for otherwise all the quantities could essentially be represented as matrices.

In my research I have interest in constructing finite subsets $I$ for which a chain sum $\sum_{[\phi] \in I} \sum_{[\alpha] \in B}[\phi(\alpha)]=0$ mod 2. Here $B$ is a finite subset of $V$ based on Nathan Broaddus' homology spheres. (e.g. $B$ is the vertex set if the two dimensional spheres in genus $g=2$ discovered by Broaddus).

(**) I'll accept Sam Nead's answer since it suggests that Birman's exact sequence tells us that forgetful map from pointed mapping classes to unpointed mapping classes is onto, with kernel equal to the image of $\pi_1(S, pt)$ under the Push map. So it's obvious from Birman's sequence that Bell-Webb's curver program can be used to compute pure mapping class actions on the curve complex. Although this raises the question of whether the membership problem for image of $\pi_1(S, pt)$ by the push map is computable in the pointed groups. Assuming it is, then to solve the subset-sum problem I'll start with the pointed groups in curver.