Plaut and Berestovskii have written some papers that might be of interest. They are not exactly the same thing. They talk about $\epsilon$-chains and homotopy of $\epsilon$-chains, where an $\epsilon$-chain is a sequence of points where each point is within $\epsilon$ of the previous. They have some interesting results.

I can't resist mentioning Neal Stephenson's series of novels "The Baroque Cycle," for an interesting take on both Newton and Leibniz. en.wikipedia.org/wiki/The_Baroque_Cycle

I'd also like to see a database of group cohomology computations for things like the mapping class group and other important groups.

Another example: Take a knot $K$ in $S^3$ and consider the cone on the knot in $B^4$. This is a submanifold of $B^4$ homeomorphic to a disk, but it is not flat at the cone point.

$\mathbb Z/p$ is the worst of both worlds, in that it makes no sense.

This seems highly unlikely to me. My approach to rule it out would be to look at simple knot and link diagrams to constrain what the chord-diagram valued R-matrix could possibly be.

@Tommaso: Alexander duality.

And I should have mentioned $n>2$.

It follows directly from the Nielsen presentation for $Out(F_n)$. Vogtmann has a survey, "Automorphisms of free groups and outer space," which covers this point.

The abelianization is almost trivial: it's $\mathbb Z_2$.

By the way, there are uncountably many different embeddings of infinite genus surfaces in $\mathbb R^3$. Take a long tube, and put a line of tubes tied in knots along this tube. Letting these knots vary gives an uncountable family of surfaces,

You might find Benoit Kloeckner's answer at mathoverflow.net/questions/4155/… illuminating, which gives a classification of all non-compact surfaces.

I suppose you know that $S_\lambda(V)=F_\lambda\otimes_{S_k} V^{\otimes k}$, where $F_\lambda$ is the $\lambda$-irrep of the symmetric group $S_k$.

Yes, in dimension three a closed simply connected 3-manifold is a homotopy sphere. This comes from Poincare duality.