I know that the higher homotopy groups of spheres are extremely challenging to compute and have driven a lot of research in algebraic topology. I was under the impression that many such computaions are beyond the reach of available techniques, even in principal. However,
this mathoverflow post (challenging to parse as a nonexpert) seems to suggests there are algorithm(s) which can compute $\pi_k S^n$ for arbitrary $k$ and $n$? If it is already known how to do all such computations in principal, with the only limit being computational power, why is there still significant interest (what's the current focus) in studying how to compute higher homotopy groups of spheres.
Edit to focus question: Given that there exists an algorithm; albeit impractical, for computing arbitrary higher homotopy groups of spheres, what are some current trends in studying such computations? Specifically, is the focus at this point primarily on algorithmic complexity; or, is there some consensus that major gaps in conceptual understanding (apart from complexity issues) still exist.
Best Answer
The actual value of these groups are never important. (Well, sometimes other parts of mathematics / physics need to take one off the shelf for their calculations, but that's beside the point.) What we care about is the deep structure in these homotopy groups. Usually discovering a structure helps us come up with calculation methods to reach further, so filling up this table is a decent gauge of our progress. Also, most of the state-of-the-art calculation is not algorithmic calculation. It's the sort of calculation that requires you to "notice" stuff and use that to progress. So it cannot be written as a robust algorithm.
Let's give an elementary example, say we know that $\pi_n (\mathbb S^n) = \mathbb Z$ from previous work. We notice that the Hopf fibration $\mathbb S^1 \hookrightarrow \mathbb S^3 \to \mathbb S^2$ gives a homotopy exact sequence, which means that $\pi_n (\mathbb S^3) \cong \pi_n (\mathbb S^2)$ for all $n > 2$. So this in particular determines $\pi_3 (\mathbb S^2) = \mathbb Z$. Filling up this spot at $\pi_3 (\mathbb S^2)$ means that we uncovered an interesting structure that emerges from the impeccably simple concept of spheres wrapping around each other.
Further more, since the Hopf fibration can be constructed through the algebraic structure of the complex numbers, we naturally try to push further using quaternions. This gives us another fibration $\mathbb S^3 \hookrightarrow \mathbb S^7 \to \mathbb S^4$. So we can use that to fill some more spots in the table, like a sudoku puzzle. Next we try octonions, which isn't even associative, but you get a fibration $\mathbb S^7 \hookrightarrow \mathbb S^{15} \to \mathbb S^8$ anyway. Reflecting on why this works reveals some exceptional structures about octonions, such as the $G_2$ group. This pattern finally breaks and cannot continue to the next one, i.e. the fibration $\mathbb S^{15} \hookrightarrow \mathbb S^{32} \to \mathbb S^{16}$. These sequences will tell you a lot about $\mathbb R$-algebras, and shows that the breakdown of algebraic properties in the sequence $\mathbb R, \mathbb C, \mathbb H, \mathbb O, \dots$ has a topological significance.
There is no end of fascinating work in this direction. To be honest, most homotopy theorists probably aren't even thinking about algorithms when they study the sphere homotopy groups.