The main goal of this REU was the exploration the cubiquity obstruction due to Greene-Jabuka and Greene-Owens. In short, if an alternating link L in the 3-sphere bounds a smooth properly embedded surface with Euler characteristic 1 and no closed components embedded in the 4-ball, then there exists an associated sublattice of the standard integer lattice in Euclidean space with the property that every unit cube in the integer lattice contains a point of the sublattice. The main question that the participants of this REU chose to explore is the following: "Which sublattices are cubiquitous?"

Although this seems like an innocuous question, it is very difficult to answer, in general. Upon restricting our view to sublattices with orthogonal bases, we were able to produce a classification such sub lattices that are cubiquitous and provide applications to topology. A paper is forthcoming.