I will add that Dror Bar-Natan in Toronto teaches a whole course on computing Alexander modules and Khovanov homology in shameless Wolfram Mathematica.

For reference, consider Gromov's "Sign and Geometric Meaning of Curvature", pp. 16, especially "when $\epsilon$ becomes equal to $-\lambda_i^{-1}(0) \ldots"$. Whenever the principle curvatures are negative with respect to the exterior, then there will be nonunique argmins at a sufficiently far distance. Another reference is the fact that Blum's medial axis $M(A)$ of an open set $A$ is nonempty, not to mention homotopy-equivalent to $A$.

I think the stable packings constructed by Werner Fischer (see the above figures) are effectively counter examples to Guth's Sponge Problem. Stable packings do not have any strictly local deformations -- if any sphere is moved, it is moved by a global deformation. And this prevents any expanding embedding from deforming the packing into a smaller domain.

@Bence Racsko: Doesn't the topological singular chain theory, like defined by Eilenberg-Greenberg, and reviewed in Gromov's "VBC", have everything you need?

Sounds like you're talking about Federer's geometric integration, and the problem of representing currents via subsets. I remember finding this paper pretty interesting in this direction : Sullivan, Dennis. "Cycles for the dynamical study of foliated manifolds and complex manifolds." Inventiones mathematicae 36.1 (1976): 225-255. You might also consider Gromov's "Volume and Bounded Cohomology" and integrating cochains over "averaged chains".

Following the answer given by Sam, I have computed exactly what i've been looking for with Bell's curver program. The action of $MCG$ on curve complex is essentially equivalent to conjugacy action of $MCG$ on Dehn twists. And Bell's curver program is effective enough at this.

Mumford's Tata Lectures, especially volumes 1 and 3 were excellent references. Mumford also had a little red book, with alot of characteristic p, which i never used. But the Tata lectures had the most detail concerning the different linear algebraic descriptions of "abelian variety". I studied them as symplectic lattices, flat 2g-dimensional real tori with symplectic and complex structures parameterized by Riemann Conditions and Siegel Upper Half space.

Thank you yes, I think you made the point. It's not a great question because after your answer it's obvious from Birman exact sequence. And that means all the hard work's been done in curver, and i just need work with curver code.

Your answer suggests that Bell's curver program can be used to compute the action of pure mapping classes on the curve complex. I think I'll ask Mark directly.

To your point about conjugacy, yes, i wish i could compute the action of MCG directly on geodesics in the hyperbolic universal cover. I would find that basically computable in symbolic python or wolfram. W.r.t. simplicial maps, maybe it's possible, and maybe thats what Mark Bell's curver does for pointed groups. I've never seen that in a paper on MCG.