The main objective of this paper is to construct smooth 1-parameter families of embedded minimal surfaces in euclidean space that are invariant under screw motions and are asymptotic to the helicoid. Some of these families are significant because they generalize the screw motion invariant helicoid with handles and thus suggest a pathway to the construction of higher genus helicoids. As a byproduct, we are able to relate limits of minimal surface families with the zero sets of Hermite polynomials.