Laplace operators with eigenfunctions whose nodal set is a knot
We prove that, given any knot γ in a compact 3-manifold M, there exists a Riemannian metric on M such that there is a complex-valued eigenfunction u of the Laplacian, corresponding to the first nontrivial eigenvalue, whose nodal set u^(-1)(0) has a connected component given by γ. Higher dimensional analogs of this result will also be considered.
Enciso, A., Hartley, D. James. & Peralta-Salas, D. (2016). Laplace operators with eigenfunctions whose nodal set is a knot. Journal of Functional Analysis, 271 (1), 182-200.