We propose a new class of crossover operators for genetic algorithms (CrossNet) which use a network-based (or graph-based) chromosomal representation. We designed Cross-Net with the intent of providing a framework for creating crossover operators that take advantage of domain-specific knowledge for solving problems. Specifically, GA users supply a network which defines the epistatic relationships between genes in the genotype. CrossNet-based crossover uses this information with the goal of improving linkage. We performed two experiments that compared CrossNet-based crossover with one-point and uniform crossover. The first experiment involved the density classification problem for cellular automata (CA), and the second experiment involved fitting two randomly generated hyperplane-defined functions (hdf's). Both of these exploratory experiments support the hypothesis that CrossNet-based crossover can be useful, although performance improvements were modest. We discuss the results and remain hopeful about the successful application of CrossNet to other domains. We conjecture that future work with the CrossNet framework will provide a useful new perspective for investigating linkage and chromosomal representations.