Darwinian evolution can be illustrated as an uphill walk in a landscape, where the
surface consists of genotypes, the height coordinates represent fitness, and each
step corresponds to a point mutation. Epistasis, roughly defined as the dependence
between the fitness effects of mutations, is a key concept in the theory of adaptation.
Important recent approaches depend on graphs and polytopes. Fitness graphs are useful
for describing coarse properties of a landscape, such as mutational trajectories and
the number of peaks. The graphs have been used for relating global and local properties
of fitness landscapes. The geometric theory of gene interaction, or the shape theory,
is the most fine-scaled approach to epistasis. Shapes, defined as triangulations of
polytopes for any number of loci, replace the well established concepts of positive
and negative epistasis for two mutations. From the shape one can identify the fittest
populations, i.e., populations where allele shuffling (recombination) will not increase
the mean fitness. Shapes and graphs provide complementary information. The approaches
make no structural assumptions about the underlying fitness landscapes, which make
them well suited for empirical work.