Topology of Hypersurface Singularities

Abstract

Kähler's paperÜber die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle" offered a more perceptual view of the link of a complex plane curve singularity than that provided shortly before by Brauner. Kähler's innovation of using a "square sphere" became standard in the toolkit of later researchers on singularities. We describe his contribution and survey developments since then, including a brief discussion of the topology of isolated hypersurface singularities in higher dimension.Key words and phrases. plane curve singularity, link of a singularity, splice diagram.

Stable pairs and the HOMFLY polynomial

“…Given a plane curve C = {f (x, y) = 0} ⊂ C 2 with singularity at p = (0, 0), the algebraic link L C,p can be described via an iterated satellite construction using the diagrams listed above. A more detailed discussion can be found in Appendix A of [12] as well as the first few sections of [29].…”

“…This convention is explained in the references (see, e.g. footnote 1 in [29]). Second, in some of the references (e.g.…”

“…In this section, we recall from [12], [29], and [39], the presentation of algebraic links in terms of the constructions from the last section. We also discuss the lowest degree behavior of their HOMFLY polynomials, which will be the most important feature for our application, as well as how they change with respect to blowup.…”

“…The link component for this branch is constructed as above; to explain how these components are linked, we use the splice diagram S n m from Section 3.3. As before, we can assume each Puiseux series is finite by truncating after a sufficiently large number of steps, without affecting the topology of the link (see, e.g., Theorem 1.1 in [29]), and we can construct the link inductively.…”

Knot Theory of Complex Plane Curves

“…There is a huge literature devoted to studying the algebraic and geometric topology L, and especially of its fibration (which, at the geometric level, completely determines L and all its invariants-though not necessarily in a perspicuous fashion). Some starting points for the dedicated reader are Eisenbud and Neumann (1985); Durfee (1999);Neumann (2001);Lê Dũng Tráng (2003). Here it may simply be noted that manythough certainly not all-of the interesting topological features of the fibration of L can profitably be explored using one or another of the representations of L alluded to in earlier sections.…”