  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.
