Knot Theory and Plane Algebraic Curves

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 C2 passing through the origin, let Lr = \ @B4 r be the intersection of with a round ball in C2 of radius r > 0 centered at the origin. When this intersection is transverse, Lr is an oriented link in S3 r = @B4 r . The main purpose of this paper is to present a survey of the results relating the topology of the pair (S3 r ; Lr) to the topology of the pair (B4 r ; \B4 r ). 0. Introduction The study of plane algebraic curves in C2 (i. e. of subsets = f(x; y) 2 C2 j f(x; y) = 0g C2, where f : C2 ! C is a polynomial map) is a very old subject which has lead to many developments in algebraic geometry. Surprisingly few results are known about the topology of plane algebraic curves and of the pair (C2; ). By the Hilbert's Nullstellensatz, to a reduced plane algebraic curve is associated a well de ned reduced polynomial map f : C2 ! C. Many results about the topology of the pair (C2; ) can be interpreted as results about the topology of the polynomial map f . It is one of the points of view that we will present here. 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 at in nity has been introduced by L. Rudolph [Ru1] (cf [N-R], [N2]) to study plane algebraic curves in the large. This approach has lead to very elegant and geometrical proofs of some classical results as well as to new important results about the topology of plane algebraic curves. We will present here some of these recent developments. This survey article does not intend to be complete but rather to re ect the authors' main interests. The starting point of this paper is the data of a reduced plane algebraic curve C2 passing through the origin. Let B4 r = f(x; y) 2 C2n j x j2 + j y j2 r2g be a round ball in C2 centered at the origin, of radius r > 0. Except for nitely many values of the radius r > 0, the intersection \ @B4 r is transverse, and hence consists of a disjoint union of nitely many smoothly oriented embedded circles in S3 r = @B4 r , where the orientation is induced by the one of . We denote by Lr the oriented link \ @B4 r and by r the piece of algebraic curve \B4 r . The main purpose of this paper is to present a survey of the results relating the topology of the pair (S3 r ; Lr) to the topology of the pair (B4 r ; r). In order to make the discussion precise, we rst de ne two basic invariants of an oriented link in S3, the so-called "Seifert" and "Murasugi" Euler characteristic. H. Seifert has shown ([Se], [B-Z], [Ro]) that any oriented link in the 3-sphere S3 is the boundary of a smooth embedded oriented surface in S3. The Seifert (Euler) characteristic X3(L) of an oriented link L S3 is the supremum of the Euler characteristic of the Seifert surfaces for L without any closed components. Viewing S3 as the oriented boundary of the ball B4, K. Murasugi ([Mu]) has considered properly embedded oriented surfaces in B4 with boundary the oriented link L (see [B-W]). Then the Murasugi characteristic X4(L) of the oriented link L is the supremum of the Euler characteristic of the Murasugi surfaces for L without any closed components. Since any Seifert surface for L can be pushed, relatively to L, into the interior of B4 to become a Murasugi surface, the inequality X3(L) X4(L) holds for any oriented link L S3. The simplest topological invariant of a reduced plane algebraic curve C2 is its Euler characteristic X ( ). When is singular, we will consider its "corrected" Euler characteristic Xc( ) = X ( ) ( ), where ( ) is the sum of the Milnor's numbers of the isolated singular points of . A topological de nition of the Milnor's number x of an isolated singular point x of is as follows : let Lx be the transverse intersection of with a small round ball B"(x) centered at x, with su ciently small " << 1; then x = 1 X3(Lx) is independent of the radius " << 1 and is called the Milnor's number of the singular point x 2 . We de ne ( ) = X x2Sing( ) x; where Sing( ) is the set of singular points of . A rst step to study the relationship between the topologies of the pairs (S3 r ; Lr) and (B4 r ; r) is to understand the relationship between the three invariants X3(Lr), X4(Lr) and Xc( r), described above. This is the object of the rst section, in which we give also some results about the types of links which may appear as the transverse intersection of a plane algebraic curve with the boundary of a round ball. In the two remaining sections we focus on the classical case of algebraic links of isolated plane singularities (cf section 2. and [Mi1], [E-N]) and on the more recent case of links at in nity of plane algebraic curves (cf section 3. and [N2], [N-R], [Ru1]). 1. The general case In 1923, Alexander [Al] has shown that any oriented link L in S3 can be isotoped into a closed braid form. This representation of links is strongly related to complex geometry and singularity theory (cf section 2. and section 3.). In this section, we will not distinguish between links and closed braids and use both notions interchangeably. For more details on braid theory, we refer the reader to [Bi].