A Symbolic-Numeric Algorithm for Computing the Alexander Polynomial of a Plane Curve Singularity

Abstract

We report on a symbolic-numeric algorithm for computing the Alexander polynomial of each singularity of a plane complex algebraic curve defined by a polynomial with coefficients of limited accuracy, i.e. the coefficients are both exact and inexact data. We base the algorithm on combinatorial methods from knot theory which we combine with computational geometry algorithms in order to compute efficient and accurate results. Nonetheless the problem we are dealing with is ill-posed, in the sense that tiny perturbations in the coefficients of the defining polynomial cause huge errors in the computed results.

A regularization method for computing approximate invariants of plane curves singularities

“…In this way, we can use the Alexander polynomial of the link of a singularity to distinguish the topological type of the singularity itself. In [9] we present a straightforward algorithm to compute the Alexander polynomial attached to the link of a singularity by using combinatorial objects from knot theory such as the diagram of the link, i.e. its arcs and its crossings.…”

“…We base this algorithm on Definition 8 from Subsection 2.2. For more details on this algorithm and an example see [9]. We now present the algorithm APPROXDELTA(Δ , μ, r) for computing the -delta-invariant from the -Alexander polynomial of degree μ and with r variables.…”

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.…”

“…From this diagram and the cycles of the graph, we compute the Alexander polynomial of each singularity of the plane complex algebraic curve as presented in [8]. Furthermore, from the Alexander polynomial we compute other topological invariants for the plane complex algebraic curve, i.e.…”

“…For our purpose, the adapted Bentley-Ottmann algorithm offers essential benefits: it allows us to compute the Alexander polynomial of the singularity of a plane complex algebraic curve as reported in [8]. From the Alexander polynomial we compute other invariants of the singularity, e.g.…”