  We show that higher order epistasis has a substantial impact on evolutionary
dynamics by analyzing peaks in the fitness landscapes. There are 193,270,310
fitness graphs, or cube orientations, for 4-locus systems, distributed on 511,
863 isomorphism classes. We identify all fitness graphs with 6 or more peaks.
81 percent of them imply 4-way epistasis, whereas 9 percent of all 4-locus
fitness graphs imply 4-way epistasis. Fitness graphs are useful in that they
reflect the entire collection of fitness landscapes rather than focusing on a
particular model. Our results depend on a characterization of fitness graphs
that imply $n$-way epistasis. The characterization is expressed in terms of a
partition property that can be derived from Hall's marriage theorem for
bipartite graphs. A similar partition condition holds for any partial order.
The result answers an open problem posed at a conference on interactions
between algebra and the sciences at the Max Planck institute.
