Abstract

The main purpose of this course is to study isolated singularities of plane algebraic curves by knot theoretical methods.
In the first lecture, we explain the dichotomy between simple and non-simple singularities, which is based on the difference between the topological and analytical equivalence of singularities. We will prove that all singularities of multiplicity two are simple, and give an easy example of a non-simple singularity of multiplicity four.
In the second lecture, we discuss links associated with isolated singularities of plane curves, and, more generally, positive braid links. As we will see, the links associated with simple singularities can be characterized as prime positive braid links which admit a positive definite Seifert form.
The third lecture focuses on the canonical fibre surface of positive braid links. In particular, we will see how these surfaces can be constructed by an operation called positive Hopf plumbing. This provides a simple description of their monodromies.
The fourth lecture will be a research talk by Marius Huber on the Floer homology of positive fibred Pretzel knots. These admit an even simpler Hopf plumbing structure than positive braid knots. We describe a pair of mutant knots of this type that cannot be distinguished by the hat version of Floer homology. Moreover, we give a conjectural picture on the Floer homology that implies mutation invariance for this class of knots.



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