What is the homotopy dimension of knot and link complements?

Question

Let $\Sigma$ be a one dimensional subspace of $S^3$ representing a knot or a link. $M = S^3$ \ $\Sigma$ is the knot or link complement I am interested in. If $\Sigma$ represents an unknot, I know $M$ is of the same homotopy type as $S^1$. If $\Sigma$ is a Hopf link, I know $M$ is of the same homotopy type as $T^2$. I want to know if there is more general statement about the homotopy dimension of $M$, in particular if it is $\le 2$.

Answer 1

A triangulated noncompact $n$-manifold always deformation retracts onto a simplicial complex of dimension $\le n-1$. See this answer of Lee Mosher for details. Since your manifold has dimension $3$, it is triangulable, and the result follows.