# gt.geometric topology – Harmonic functions on knot complements

In Axler’s Harmonic Function Theory, he and his coauthors develop the theory of harmonic polynomials on $$mathbb{R}^n$$ by considering the restrictions of arbitrary polynomials on the sphere $$S^{n-1} = {x in mathbb{R}^n : ||x||^2 = 1 }$$ and taking the Poisson integral to get a harmonic polynomial in the interior ball. One can then take the Kelvin transform to get a harmonic polynomial on the exterior of the sphere. This process yields a canonical projection $$mathscr{P}(mathbb{R}^n) to mathscr{H}(mathbb{R}^n)$$, from the space of polynomials to the space of harmonic polynomials, factoring through the restriction map to $$L^2(S^{n-1})$$. By taking polynomial approximations via the Stone-Weierstrass theorem, and verifying some series convergence, I believe one can expand the domain and codomain of the projection to larger, more general function spaces. I have not verified this myself, so I could be wrong.

Does this theory generalize to knot complements? Say we have a knot $$K subseteq mathbb{R}^3$$, and we take a small tubular neighborhood $$V$$ around $$K$$, whose boundary is topologically a torus $$T$$. Given a function on the knot complement, one could restrict to $$T$$ and then solve the Dirichlet problem on the knot complement to get a projection like the one above. However, in the sphere case, there are many nice properties of the projection; namely it comes with an efficient algorithm for computation which involves repeatedly differentiating the function $$f(x) = |x|^{2-n}$$.

Is anyone aware of any theory along this vein? Are there any obstacles to generalizing what happens in the sphere case?