  The relationship between the shape of a fitness landscape and the underlying
gene interactions, or epistasis, has been extensively studied in the two-locus
case. Gene interactions among multiple loci are usually reduced to two-way
interactions. We present a geometric theory of shapes of fitness landscapes for
multiple loci. A central concept is the genotope, which is the convex hull of
all possible allele frequencies in populations. Triangulations of the genotope
correspond to different shapes of fitness landscapes and reveal all the gene
interactions. The theory is applied to fitness data from HIV and Drosophila
melanogaster. In both cases, our findings refine earlier analyses and reveal
previously undetected gene interactions.
