Laplace operators with eigenfunctions whose nodal set is a knot

TL;DR

This paper proves that for any knot in a compact 3-manifold, one can find a Riemannian metric making the Laplacian's eigenfunction's nodal set contain that knot, extending to higher dimensions.

Contribution

It establishes the existence of Laplacian eigenfunctions with prescribed knot-shaped nodal sets in 3-manifolds, a novel geometric and spectral construction.

Findings

Existence of eigenfunctions with prescribed knot nodal sets

Construction of Riemannian metrics for desired nodal topology

Extension to higher-dimensional manifolds

Abstract

We prove that, given any knot $\gamma$ 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 $\gamma$. Higher dimensional analogs of this result will also be considered.