  This paper studies the global structure of algebraic curves defined by
generalized unitarity cut of four-dimensional three-loop diagrams with eleven
propagators. The global structure is a topological invariant that is
characterized by the geometric genus of the algebraic curve. We use the
Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with
the help of techniques involving convex hull polytopes and numerical algebraic
geometry. Some interesting properties of genus for arbitrary loop orders are
also explored where computing the genus serves as an initial step for integral
or integrand reduction of three-loop amplitudes via an algebraic geometric
approach.
