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We prove that the curl operator on closed oriented $3$-manifolds, i.e., thesquare root of the Hodge Laplacian on its coexact spectrum, generically has$1$-dimensional eigenspaces, even along $1$-parameter families of$\mathcal{C}^k$ Riemannian metrics, where $k\geq 2$. We show further that theHodge Laplacian in dimension $3$ has two possible sources for nonsimpleeigenspaces along generic $1$-parameter families of Riemannian metrics: eithereigenvalues coming from positive and from negative eigenvalues of the curloperator cross, or an exact and a coexact eigenvalue cross. We provide examplesfor both of these phenomena. In order to prove our results, we generalize amethod of Teytel \cite{Teytel1999}, allowing us to compute the meagrecodimension of the set of Riemannian metrics for which the curl operator andthe Hodge Laplacian have certain eigenvalue multiplicities. A consequence ofour results is that while the simplicity of the spectrum of the Hodge Laplacianin dimension $3$ is a meagre codimension $1$ property with respect to the$\mathcal{C}^k$ topology as proven by Enciso and Peralta-Salas in\cite{Enciso2012}, it is not a meagre codimension $2$ property.

On spectral simplicity of the Hodge Laplacian and Curl Operator along paths of metrics