Cylindrical contact homology of links of simple singularities

We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to $S^3\mathrm{/}G$ for finite subgroups $G\subset \text{SU}(2)$. We perturb the degenerate contact form on $S^3\mathrm{/}G$ with a Morse function, invariant under the corresponding $H\subset \text{SO}(3)$ action on $S^2$, to achieve nondegeneracy up to an action threshold. The cylindrical contact homology is recovered by taking a direct limit of the action filtered homology groups. The ranks of this homology are given in terms of $\mathrm{\mid }\text{Conj}(G)\mathrm{\mid }$, demonstrating a form of the McKay correspondence.