A Symbolic-Numeric Algorithm for Genus Computation

An Adapted Version of the Bentley-Ottmann Algorithm for Invariants of Plane Curves Singularities

“…[1], [12], the 3-dimensional graph computed as the piecewise linear approximation of an implicitly defined space algebraic curve is called the topology of the curve. We use the Axel [13] free algebraic geometric modeler to compute the 3-dimensional graph as presented in [8], [10]. For the special case of smooth implicitly defined space algebraic curves, Axel uses certified algorithms to compute their topology.…”

“…Furthermore, from the Alexander polynomial we compute other topological invariants for the plane complex algebraic curve, i.e. the Milnor number, the delta-invariant of each singularity and the genus, as described in [10]. Figure 3 …”

“…To use directly this computed 3-dimensional graph and to obtain the essential extra information on each intersection point, we use Axel for the implementation of the adapted Bentley-Ottmann algorithm in a simplified form. Consequently, the adapted Bentley-Ottmann algorithm allows us to design more algorithms for computing the topological properties of each singularity of a plane complex algebraic curve as described in [10]. Another reason for the choice of the implementation system was that we wanted to write our package in one language, and we also needed algebraic functions for surfacesurface intersection, which are provided by Axel's libraries.…”

“…For our purpose, the adapted Bentley-Ottmann algorithm operates on a 3-dimensional graph data structure, which represents the piecewise linear approximation of a closed and smooth space algebraic curve, implicitly defined as the intersection of two algebraic surfaces. We compute this space algebraic curve as the link of the singularity of a plane complex algebraic curve, as described in [10].…”