Knot theory and plane algebraic curves

Abstract

Knot theory has been known for a long time to be a powerful tool for the study of the topology of local isolated singular points of a plane algebraic curve. However it is rather recently that knot theory has been used to study plane algebraic curves in the large. Given a reduced plane algebraic curve Γ − ℂ 2 passing through the origin, let L r =Γ∩ ∂ B r 4 be the intersection of Γ with a round ball inC 2 of radius r > 0 centered at the origin. When this intersection is transverse, L r is an oriented link in S r 3 = ∂ B r 4. The main purpose of this paper is to present a survey of the results relating the topology of the pair ( S r 3, L r ) to the topology of the pair ( B r 4,Γ∩ B r 4).