  In this article, we consider an infinite family of normal surface
singularities with an integral homology sphere link which is related to the
family of space monomial curves with a plane semigroup. These monomial curves
appear as the special fibers of equisingular families of curves whose generic
fibers are a complex plane branch, and the related surface singularities appear
in a proof of the monodromy conjecture for these curves. To investigate whether
the link of a normal surface singularity is an integral homology sphere, one
can use a characterization that depends on the determinant of the intersection
matrix of a (partial) resolution. To study our family, we apply this
characterization with a partial toric resolution of our singularities
constructed as a sequence of weighted blow-ups.
