- Email: [email protected]
Knot theory and plane algebraic curves
Chaos, Solilions
Pergamon
& Fracrais, Prmted
Vol. 9. No. 415, pp. 779-792. 1998 0 1998 Elsevrer Science Ltd in Great Britain. All rights reserved 0960-0779/9x $19.00 + 0.00
PII: so960-0779(97)00103-3
Knot Theory and Plane Algebraic MICHEL
BOILEAU
and LAURENCE
Curves
FOURRIER
Universite Paul Sabatier, Laboratoire Emile Picard, U.M.R. 5580,118 route de Narbonne, 31062Toulouse Cedex 4, France
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 F c C* passing through the origin, let L, = F fl JB: be the intersection of F with a round ball in C* of radius r>O centered at the origin. When this intersection is transverse, L, is an oriented link in S: = dB;‘. The main purpose of this paper is to present a survey of the results relating the topology of the pair (SJ,L,) to the topology of the pair (B:,F f’ B:). 0 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The study of plane algebraic curves in UC2(i.e. of subsets I = ((x,y) E C2 If(x,y) =0} c (C2, where f: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 (UC2,1).By Hilbert’s Nullstellensatz, with a reduced plane algebraic curve I is associated a well defined reduced polynomial map f:c’+ C. Many results about the topology of the pair (~‘,I) 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 infinity has been introduced by Rudolph [l] (cf. [2, 31) to study plane algebraic curves in the large. This approach has led 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 reflect the authors’ main interests. The starting point of this paper is the data of a reduced plane algebraic curve I c UC2 passing through the origin. Let Bz = ((x,y) E @* : 1x12+ Iy]*sr2} be a round ball in C* centered at the origin, of radius r>O. Except for finitely many values of the radius r> 0, the intersection I II dB1 is transverse, and hence consists of a disjoint union of finitely many smoothly oriented embedded circles in Sz = c?B~,where the orientation is induced by the one of I. We denote by L, the oriented link I fl c?B~and by I, the piece of algebraic curve rn B;. The main purpose of this paper is to present a survey of the results relating the topology of the pair (S;,L,) to the topology of the pair (B:‘,r,). In order to make the discussion precise, we first define two basic invariants of an oriented link in S3,the so-called Seifert and Murasugi Euler characteristic. Seifert has shown [4-6] 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 c S3 is the supremum of the Euler characteristic of the Seifert surfaces for L without any closed components. 779
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Viewing S 3 as the oriented boundary of the ball B 4, Murasugi [7] has considered properly embedded oriented surfaces in B 4 with boundary the oriented link L (see [8]). Then the Murasugi characteristic z4(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 B 4 to become a Murasugi surface, the inequality z3(L) <-z4(L) holds for any oriented link L c S 3. The simplest topological invariant of a reduced plane algebraic curve F c C 2 is its Euler characteristic z(F). When F is singular, we will consider its "corrected" Euler characteristic Zc(F) =z(F) - tz(F), where Iz(F) is the sum of the Milnor's numbers of the isolated singular points of F. A topological definition of the Milnor's number/~x of an isolated singular point x of F is as follows: let Lx be the transverse intersection of F with a small round ball B,(x) centered at x, with sufficiently small e < 1; then/z~ = 1 - z3(Lx) is independent of the radius • < 1 and is called the Milnor's number of the singular point x ~ F. We define =
x E Sing(F)
where Sing (F) is the set of singular points of F. A first step to study the relationship between the topologies of the pairs (S3,L~) and (B~,Fr) is to understand the relationship between the three invariants z3(L~), z4(Lr) and Zc(F~), described above. This is the object of the first 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 remainder of the paper, we focus in Section 3 on the classical case of algebraic links of isolated plane singularities (cf. [9, 10]) and in Section 4 on the more recent case of links at infinity of plane algebraic curves (cf. [3,2,1]).
2. THE G E N E R A L CASE
In 1923, Alexander [11] showed that any oriented link L in S 3 can be isotoped into a closed braid form. This representation of links is strongly related to complex geometry and singularity theory (cf. Sections 3 and 4). 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 [12]. The situation is the one described in the introduction. Let F be a plane algebraic curve in C 2 passing through the origin. For r > 0 (except finitely many values), let Lr be the transverse intersection F A dB~ and let Fr be the piece of curve in the 4-ball B 4. The oriented link Lr c S~ = 0B 4 is transverse to the standard contact structure on S 3 induced by the 2-field of complex lines tangent to S3r. By a transverse isotopy one can always transform Lr into a closed braid form. This is one of the key ingredients of the so-called Bennequin's inequality. Bennequin [13] has proved a beautiful inequality on closed braids which was a crucial step in his search for an exotic contact structure on the 3-sphere. In our context, Bennequin's inequality can be stated as follows:
Theorem (Bennequin). With the notation above, z3(Lr) <-Zc(Fr). A straightforward corollary is the following:
Corollary. With the notation above, z3( Lr) <-Zc(Fr) <-z4( Lr). To prove his result, Bennequin used closed braids and Markov surfaces which are special kinds of braided Seifert surfaces (cf. also [14, 15]). He first shows that z3(Lr) can be realized by a Markov surface and reduces the proof of his inequality to study the topology of a
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Markov surface with respect to the standard contact structure of S3.A complete proof of the inequality can be found in [13] (see also [16]; see [17] for a generalization). Here is now a deep recent result by Kronheimer and Mrowka (the so-called Thorn conjecture, see [S]) which they have proved in full generality using Seiberg-Witten invariants (cf. [l&21]). Th is result shows that the corrected Euler characteristic x#,) of a piece of algebraic curve I?, properly embedded in B:’ is determined by the topology of the pair tssJ4. Theorem (Kronheimer-Mrowka). With the notation above, xc( r,) = x4(Lr). Corollary. The topological type of a non-singular connected piece of algebraic curve r, c B: is determined by the topological type of the pair (S), L,).
This result leads to the following question: Question 1. Does the topology of the pair (Sp, L,) always determine the topological piece of algebraic curve r, when Tr is non-singular?
type of the
Kronheimer and Mrowka’s result implies that the map r H b,(L,) - x4(Lr) is nondecreasing, where b,(L,) is the number of connected components of L,. In order to compare the links L, while r is increasing, let us introduce the following complexity C(L,) of L,: C(L)
= (b,(L)
- x~tL),WJ
- x4Lr))
EN x N3
with the lexicographic order on N X N. Question 2. With the notation above, is the map r H C( L,) non-decreasing?
Kronheimer-Mrowka’s result reduces Question 2 to the following: Question 2’. With the notation above, if b,( L,) - x4( L,) = b,( L,,) - x4( L,,), for r 2 r’ , does the inequality b,( L,) - x3( L,,) 2 b,( L,.) - x3( L,,) hold?
According to Gordon [22], we show that Question 2’ (and hence Question 2) has a positive answer if both L, and L,, are fibred knots. We recall that an oriented link L c S3 is fibred if it is the binding of an open book decomposition of S3. Proposition 1. With the notation above, assume that L, and L,,, for r 2 r’ , are Jibred knots. If x4( L,) = x4( L,,), then 1 - x3( L,) 2 1 -x3( L,,) and equality holds if and only if the pairs (Ss, L,) and (ST,,L,.) are equivalent.
The hypothesis x4( L,) =x4( L,.) and Kronheimer-Mrowka’s theorem imply that r II S3x [r’,r] is an annulus. Therefore r induces a ribbon concordance between the fibred knots L, and L,. Let Y = S3 x [r,r’]\r be the complement of the ribbon concordance, let X, = S;\ L, and X,, = S:(\ L,. be the complements of the knots L, and L,,, and let P, 2, and %,, denote their infinite cyclic coverings. The following argument is due to Gordon [22]. Since the morphism rl(X,) -+ rl(Y) is surjective, the morphism H1(z-$J)+ H,(~;Q) is surjective and then dim H,(F;Q)sdim H,(xr;Q). Moreover cy: H,(dY;Q)-+H,(E;Q) is surjective since 8 = 8, U 8,.. By Milnor’s duality [23], one has the isomorphisms H,(lyl2) = H,(F,dF;Q), H,(F,dF;Q) g H&2), which, combined with the surjectivity of (Y,give the short exact sequence o-,H,(~,d~;~)~H,(d~;~)~H*(P;Q)~o. So dim H,(z’,;Q) + dim H,(a,.;CD) = 2dim H1(P;UJ) 5 2dim H,(z,;Q). To conclude it suffices to remark that if K is an oriented fibred knot in S3, we have the equality dim H,(z;;Q) = 1 -x3(K) (see [6]), where 2 denotes the infinite cyclic covering of the complement of K.
Proof:
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The proof of the equality case is more intricate and we refer to Lemma 3.2 of [22] for it. Moreover a positive answer to Question 2 would show that if Lr is a trivial link then Lr, is a trivial link for r' < r. This special case of Question 2 is still open, and we are only able to give a partial result about it.
Proposition 2. I f Lr is a trivial link f o r some r > O, then (i) the piece o f curve Fr is a disjoint union o f non-singular holomorphic disks, (ii) Fr is unknotted in the ball B 4, that is to say, Fr can be isotoped in OB 4 = S3r to a Seifert surface o f Lr, (iii) f o r 0 < r' < r, Lr, is a trivial link if bo( L~,) <- bo( Lr). Proof'. The first assertion (i) is a direct consequence of Bennequin's inequality.Assertion (ii) follows from Eliashberg's work [24], using the fact that the holomorphic convex hull of Lr contains a union of disjoint holomorphic disks, and thus can be filled up by disjoint families of holomorphic disks (cf. [25]).For the last assertion (iii), we use the fact that if bo(L~,) <- bo(L~), then there is a ribbon concordance W 2 between the link L r, and a sublink Lr of L r. Let F; be the union of disks in B 4 bounding Lr- By assumption F; f3 B 4, = F r f-) B 4, = F~,. Since we are dealing with ribbon concordance, we have the following sequence of surjective morphisms induced by inclusions: 7r,(S3r\Lr) ~
7r,(B4\W 2) -~ 7rl(B4\Fr)
~ O.
From assertion (ii), 7 r l ( B 4 \ r ; ) is a free group of rank bo(L;))=bo(Lr) like crl(S3\Lr). By the hopficity of finitely generated free groups, it follows that the morphism ¢rl(Sr3\ L;) ~ ~ ' 1 ( B 4 \ F r°) is an isomorphism, and hence the morphism 7r1($3\ L r ) ~ 7 r l ( B 4 \ W 2) is an isomorphism. According to [22], the ribbon concordance implies that the morphism induced by the inclusion crl(S3r\L~,)~ 7rl(Br4\W 2) is injective. It follows t h a t 7rl(S3r,\Lr ,) is a free group, and hence Lr, is a trivial link.
Remark. Proposition 2 can be view as a general topological version of the famous Abhyankar-Moh-Suzuki Theorem [26, 27] which asserts that any algebraic embedding of C in C 2 is algebraically unknotted. One may remark that, in general, the topology of the pair (S3,L~) does not determine the topology of the pair (B4,F~) even if r r is non-singular. In [28], Artal-Bartolo constructed two connected non-singular plane algebraic curves in C 2 having the same links at infinity (i.e. the same links L , for r sufficiently large), but whose complements in C 2 have distinct fundamental groups. Thus the topology of the pair (Sr3,L~) does not even determine the group 7rl(B4\F~). We conclude this general discussion by considering the problem of characterizing the possible types of the links L r in S 3, for any r > 0. This is still a wide open question. In his important pioneering work [1,29,14], Rudolph found a large class of links in S 3, called quasipositive links, which can be realized as the transverse intersection of a plane algebraic curve with the boundary of a round ball in C 2 (see also [30]). These links are characterized by the fact that they admit a quasipositive closed braid presentation. A braid word is said quasipositive if it is a product of conjugates of positive words. The quasipositive links form a large family of links, including classical algebraic links, links at infinity of plane algebraic curves, closed positive braids. In contrast to the local case or the case at infinity, Rudolph's construction [31-35] shows that some hyperbolic links (i.e. whose complement in S 3 admits a complete hyperbolic structure) may appear as links L , for some r which is not too small and not too big. Rudolph's construction is so explicit that it allows one to build knotted holomorphic disks in a round ball. Rudolph's conjecture, still open, asserts that any link Lr is a quasipositive link.
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The following result shows that any link L, must verify a specific inequality, already known for quasipositive links [36]. Before stating the result, we need to recall the definition of the 2-variable HOM FLY-polynomial for an oriented link L in S3. We use the notation of [37,36]. To an oriented link L in S3, one can associate a 2-variable Laurent polynomial PI>(W) E qv *I ,Z *I], defined recursively in the following way:
Let L,, L, and L- be three links whose projections differ only at the crossing shown m the diagram. Then define P&v,z) = 1 for the trivial knot 0, and P,, (v,::) = vz PL,(v,z) + v2P, _ (v,z). Let Ord, P, be the valuation of PL in the variable v (i.e. the least degree in v of the polynomial P,(v,z) viewed as a Laurent polynomial in v with coefficients in a=[2* ‘I). Proposition 3 (see [38]): With the notation above, Ord, PL, 2 1 - x4(Lr) 2 1 - b,(L,). Proof: Although this inequality has been explicitly stated only recently in [38], it is an easy consequence of Bennequin’s thesis [13] and of the Franks-Williams [39] and Morton [37] inequality for closed braids. According to [13], after a transverse isotopy to the standard contact structure on S:, L, admits a closed braid presentation with II strands and algebraic length e. An application of the Hopf index theorem to the complex vector field tangent to the piece of algebraic curve I’, c B;’ and orthogonal to the Hopf field shows that x(I’~) =n - e (see [13,401). Then, the Franks-Williams and Morton inequality for closed braids implies that Ord,P,y?e-n+l=l-X,(r,)?l-X,(L,). This inequality is sharp enough to characterize the links L, among 2-bridge links (see [38]). In particular it shows the following: Corollary 1. If L, is a knot and is equivalent to its mirror image then the piece of algebraic curve r, is a non-singular disk and x4( L,) = 1. Proof If L, is equivalent to its mirror image, then by symmetry on the variable v, Ord, P,, is non-positive. So x4(Lr) 2 1 and, if L, is a knot, ,YJ(L,.)= 1. Since by Kronheimer-Mrowka’s theorem x4(Lr) =xJI,), the assertion follows. Corollary 2 (see [4l’]). The figure-eight knot is never the transverse intersection of a plane algebraic curve with the boundary of a round ball in C*. Proof From tabulations (see [42]), one has Ord, P,= - 2 for the figure-eight knot K, contradicting the inequality Ord, P,, L 1 - b,(L,). One also may use the inequality to get an estimate of Milnor’s number ,u(r,) of r, from the topology of the pair (S:,L,). Corollary 3. With the notations above, p(T,) I Ord, P,, + x(Tr) - 1.
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3. ALGEBRAIC LINKS: THE LOCAL CASE IN BRIEF
This section is devoted to presenting some classical results about the topology of isolated singularities of plane algebraic curves. From our point of view, given a plane algebraic curve with an isolated singularity at the origin, we are interested in the embedding of the piece of algebraic curve F, = F f3 B 4 in the round ball B 4 of radius sufficiently small (a so-called Milnor's ball). The link L, = F, N dB 4 is called the link of the singularity of F at the origin. The study of such links (so-called algebraic links) goes back to the works of [43-46] and had a revival from Milnor's seminal work [9], showing that such a link is the binding of an open book decomposition of S 3 and describing the fibration of S 3 \ L ~ over S 1.
Theorem (Milnor's fibration [9]). Let F be a plane algebraic curve in C 2 having an isolated singularity at the origin. Then there exists Eo> 0 such that, for any E <- Eo, (i) the pair (B4,F,) is homeomorphic to the cone on the pair (0B4,L,), (ii) i f F = {f(x,y) = 0] c C 2, for some polynomial map f: C2---~ C, then the argument map 0 = f/I f I : --, S ' is an open book decomposition, (iii) the induced fibration O: S3\IV(L,)---~S l is diffeomorphic to the fibration f: f - 1( dD (0, 7)) A B 4~ 3D (0, ~1), for a radius ~7sufficiently small depending o f e. The proof of this result can be found in Chapters IV and V of [9] (see also [47, 48]). For an explicit description of the monodromy of the fibration see [49] and also [50].A straightforward corollary of this result together with Kronheimer-Mrowka's theorem is the following:
Corollary. In the local case (e < < 1), the topology o f the pair (B4,F,) is determined by the topology o f the pair (S 3,L,). Moreover, the equality z3(L,) = Zc(F,) = x 4 ( L , ) holds. Algebraic links L, are completely characterized by the topology of their complement S 3 \ L , and some special inequalities, called Puiseux inequalities. This is mainly due to works of Brauner [43], K~ihler [51], Burau [44, 45] and Zariski [46] (more details may be found in Reeve's work [52]). For a description of algebraic links L, via Jaco-Shalen and Johannson characteristic family of tori and splicing diagrams, we refer to Eisenbud and Neumann's book [10] (see also [53]).Recall that a Seifert link in S 3 is a union of finitely many fibres of some Seifert fibration of S 3. For example, Seifert knots are the so-called torus knots which may be represented as a simple closed curve on the boundary of a standard unknotted solid torus in S3.An iterated torus link in S 3 is a link obtained from a Seifert link by applying finitely many of the two following constructions: Cabling operation: Starting from a link L0, let L be the union of finitely many parallel simple closed curves on each boundary component of a regular neighborhood N(Lo) of Lo, such that each curve is not null homologous in N(Lo). If some family of parallel curves is not homologous to a component of L0 in N(Lo), we say that L is a non-trivial cable around L0. Connected sum operation: Let L1 and L2 be oriented links in $3; then we say that L is a connected sum of L1 and L2 if L is obtained from L1 and L2 by identifying a small oriented unknotted arc of a component of L1 with a small oriented unknotted arc of a component of Lz by an orientation reversing homeomorphism. For a proof of the following theorem, we refer to [10] in the general case (see also [53]) and to [43] (see also [52,47]) in the case of knots.
Theorem (characterization of algebraic links): An oriented link L in S 3 is an algebraic link L, if and only if L, is an iterated torus link obtained only by cabling operations satisfying the Puiseux inequalities. In particular L, is a positive braid.
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For example, if the singularity of I is analytically irreducible at the origin, the Puiseux expansion theorem gives the following parametrization of I near the origin: y = x41/Pl(ul+ x421PlP2 (a2 + ... + (a, _ l + u$q’pI-+y ...). with pi,qr > 0 and gcd (pi,qi) = 1, for i = l,..., Y. To describe the knot L,, we consider the intersection of I with the bidisk [(x, y) E @’ : 1x ] % E,1y 1I e) which g’Ives a knot equivalent to L,. The knot L, then can be built step by step using successive truncations of the parametrization of I. Let K, be the knot given in the solid torus V= ((x,y) E C* : ]x I= E,Iy ) 5 l } by the equation y = ~~~~~~~~~ It is easy to see that Kr =O(q,,p,) is a torus knot of type (ql,pl): it means that K, is equivalent to the simple closed curve on dV homologous to q,(meridian) +p,(longitude]. Then the knot K2 corresponding to the equation y = xql’pl(al + u2xq2/p~2)is a (n,,m,)-cable around K1 with m2=p2 and n2 = q2+plp2ql. Therefore K2 =B((q,,p,),(n,,m,)) is isotopic in a regular neighborhood N(K,) of Kr to a simple closed curve on dN(K,) homologous to n2 meridian (K,) + m2 longitude (K,). By induction (see [10,54,47]), one obtains that L, is the iterated torus knot ~((nl,ml>,..., (nm,)), with ml = ply ~zr=ql and, for i? 1, mitl =pitl, r~,+~=qi+l +piPi+lni. Moreover, these numbers (ni,mj) must satisfy some inequalities, called Puiseux inequalities, coming from the fact that qi/pi > 0, for 15 i 5 r. If the singularity of I at the origin is not analytically irreducible, the link L, is no longer connected. The number of connected components b,(L,) corresponds to the number of irreducible branches at the origin of I. So L, is an iterated torus link built from the Puiseux expansions of each irreducible branch. In particular, each connected component of L, is an iterated torus knot as above. The way these components are linked together can be determined by comparing pair-wise the Puiseux expansions of the branches. Important are the first terms where the two expansions differ (see [10,53]). In fact a well-known result of Zariski and Jalabert-Lejeune [55] asserts that an algebraic link L, is completely determined by the topological types of its components and their linking numbers. For an iterated torus link L, the operations of connected sums and of non-trivial tablings correspond to embedded incompressible non boundary parallel tori in the exterior X(L) = P\fi(L) (’i.e. r,-injective tori in X(L) which are not isotopic to dX(L)). One can extract from this family of tori a minimal family T-L of embedded, pairwise disjoint and not parallel tori such that the closure of each component of X(L)\TL is the exterior of a Seifert link in S3 (see [56-581). Th en one says that L is obtained by splicing together all these Seifert companions links (see [10,59]). The splicing tree TL is the dual tree to the family FL, whose vertices correspond to the connected components of X(L)\FL and the edges to the tori in 5-L. Moreover, we add an arrowed edge for each component of L. We associate with each vertex a weight corresponding to the linking number of L with a generic fibre of the Seifert fibration of the component of X(L)\F, corresponding to this vertex. For an algebraic link L, all these weights are strictly positive. This is no longer true, in general, for iterated torus links at infinity and it is an important feature of this theory at infinity, as we shall see in the next section. 4. LINKS AT INFINITY
In this section, we give some applications of knot theory to the study of affine algebraic curves in C2. As in the introduction, let I = {(x,y) E C2 : f(x,y) = O] be a reduced algebraic curve where f: C* + @is a reduced polynomial.
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For R sufficiently large, the intersection Ln= c~B~n F is always transverse and the topological type of the pair (3B4,LR) is independent of the radius R. We call this oriented link Ln, for R sufficiently large, the link at infinity of F and denote it by L~(F) (see [1-3]). A major difference compared with the local case is that the Milnor's argument map f~ I f 1: S ~ \ L ~ ( F ) ~ S ~ is not in general an open book fibration. However there still exists a kind of "Milnor's fibration at infinity" given by the following result: Theorem (Milnor's fibration at infinity [2,60]): Let F - - { f ( x , y ) = O ] c C 2 be a reduced algebraic curve as above. Then there exist a disk o f radius s centered at the origin D (O,s ) and a real number Ro > 0 such that, for any R >-Ro, (i) No = f - l ( D ( O , s ) ) n S 3 is a union o f solid tori, (ii) O = f / l f [ • S 3n \ f - I(D(0,s)) ""-> S 1 is a locally trivial fibration called Milnor's fibration at infinity, (iii) this fibration is diffeomorphic to the fibration f: f 1( 3D (0,s)) O B4--> OD (0,s). Statement (i) is a very nice application of the maximum principle due to Rudolph [3]. The remaining statements follow by extending Milnor's methods at infinity. The proof of statement (ii) can be found in [60] and the proof of statement (iii) in [3,2,60].In the Milnor's fibration at infinity appear the following links naturally associated with the polynomial map f." C2---> C: the multilink at infinity and the regular link at infinity. The multilink at infinity: Let Lo be the link whose components are the cores of the solid tori of No. The Milnor's fibration 0 = f~ I f I • X(Lo) ~ S 1 on the exterior of Lo allows us to consider Lo as a fibred multilink [3].A multilink L(m~ ..... mr) is an oriented link with r components each of whose components is weighted by an integer mi (the multiplicity of the ith component)• Such a multilink L(ml ..... mr) is fibred if there is a fibration of its exterior p: X ( L ) ~ S 1 whose restriction along any meridian curve of the ith component of L(ml,...,mr) has degree mi. It follows that a fibred multilink L(ml,...,mr) defines an open book decomposition of S 3 if and only if mi = + 1, for i = 1 ..... r. For more details about multilinks, we refer to [10]. Coming back to Lo, the argument map O = f / [ f l " S 3 \ L ~ ( F ) - - > S ~ restricted to X(Lo) = S 3 \ b l o defines the Milnor's fibration of the multilink Lo(ml ..... mr). Assuming that the value 1 ~ S 1 is a regular one for 0, then 0-1(1)=2~ is a Seifert surface for the oriented link L~(F). The intersection Eo = E n X(Lo) is a fibre of the fibred multilink Lo(ml ..... mr). In particular, 3Eo=E N 3No is a family of disjoint closed curves on the tori c~No which is homologous to L~(F) in No. It follows that the multiplicities mi, i=1 ..... r, of the fibred multilink Lo(ml,...,m~) are given by the homology class of L~(F) in No: [L~(F)] = [3Eo] = (ml ..... m~) ~ H1(No;2~) ~- 7/r. So there naturally appears a second link: The regular link at infinity: For a value t ~ c~D(0,s), let L~(F,) = 3Eo be the link at infinity of the generic fibre F, = {f(x,y) = t} c C 2. From the discussion above, it follows that any Seifert surface of the link Lo~(F) intersects transversely 3No along a link isotopic to L~(F,) on 3No. In particular, one has the following inequalities: z3(L~(F)) -< Z.~(L~(F,)) <- z3(Lo). If the pairs (S3,L~(F)) and (S3,L~(F,)) are equivalent, we say that F is regular at infinity• Otherwise, F is said to be irregular at infinity• In particular, F is regular at infinity if and only if, outside a sufficiently large ball, it looks like its nearby fibres F,, I t] < e. Using Section 2, the discussion above and results by Hh and L6 [61] and Suzuki [27] we obtain the following: Proposition 4: Let F be a reduced algebraic curve in C 2. Then (i) the embedding of F in C 2 determines the embedding of any generic fibre F, in C 2, (ii) the curve F is regular at infinity if and only if z4( L~(F)) = z3(L~(F)). Proof'. Assertion (i) is a consequence of the following facts:
Knot theory and plane algebraic curves
The topological type of the pair (,$&,(I,)) is determined by the topological type of the pair (Si,L,(I’)). This follows from the previous discussion and the fact that the Milnor’s fibration at infinity is determined up to isotopy by the multiplicities (m,,...,m,) (see [62, 631). The topological type of the pair (S&L,(I,)) determines the topological type of the pair (C”,I’,) by assertion (iii) of Milnor’s fibration theorem at infinity. To prove assertion (ii), we need the following result of 1611and [27]: For any generic fibre I,, x(I) rX(I’J, with equality if and only if I is regular at infinity. Moreover, assertion (iii) of Milnor’s fibration theorem at infinity implies that X(IJ =X3(La(IJ). Therefore by Kronheimer-Mrowka’s theorem m4r)) = im wLmu2) ~~4ur)). So x4(Lcs(I)) =X3(Lm(I)) implies that xc(I) =&It). So I is regular at infinity by [61, 271 and the pairs ($&,(I)) and ($&(I,)) are equivalent. Conversely if I is regular at infinity, then idv3) = ~3(w2) = m) = im = m4w This proposition shows that there is a natural and well defined notion of irregular link at infinity. Like algebraic links in the local case (see Section 3), the link at infinity L,(I) is an iterated torus link but, in this case, connected sum operations may appear in the construction of L,(I). Let us recall briefly the construction of L,(I) (see [3,2]). Let P2@= C2 U PL be a projective compactification of C2 and let r be the projective compactification of I in P’C. Then Lp( I) = r n JN( Pk) where the regular neighborhood N( IFPL)of the line at infinity is a disk bundle 9: N( [FDL) + lu’: with Euler class 1. If l? n ln)L= (x1,... ,.r,J, let D c pi be a disk containing l? n R& such that r n aN( p:) = r fl dq -l(D). Then f n dq -l(D) in dq -l(D) is a closed braid b which is a disjoint union of the local algebraic links associated with each point xi of r n iFDk. To pass from the embedding of L,(I) in dq-l(D) to the embedding of L,(I) in JN( [FDL),one must perform a positive Dehn twist around the axis of the braid p and change the ambient orientation. It follows easily that L,(I) in aN( IFpi)is an iterated torus link which must satisfy some reversed Puiseux inequalities (see [3]). However, there is not yet a complete characterization of links at infinity, except in the case of knots [64]. So, with the topological pair ($&,(I)) one can associate a weighted splicing tree as in Section 3 for the local case.This allows one to give a simple characterization of regular link at infinity: Corollary (see [3,65, 661). A link at infinity L,(T) only non-negative weights.
is regular if and only if its splicing tree has
This corollary follows from Proposition 4 and the formula computing xJL&)) and x4(L&)) from the weighted splicing tree of L,(I) (see [3]). Milnor’s fibration theorem at infinity shows that the embedding in C2 of a non-singular curve I which is regular at infinity is determined by the topological type of the pair (&L,(T)). As we pointed out in Section 2, this is no longer true if I is irregular at infinity (see [28]). To obtain more precise results on this problem, one has to take into account the other irregular fibres of the polynomial map defining I. In other words, the topology of the pair (@‘,I’), where I = cf(x,y) = 0}, is intimately connected with the topology of the polynomial map f: C2+ C. Without loss of generality, we shall always assume from now on that all the generic fibres off are connected (i.e. f is a primitive polynomial). By the Stein factorization theorem, any non-primitive polynomial g E C[x,y] is equal to h of, for a primitive polynomial f E C[x,y] and h E @[t] (see [67]).
788
M. BOILEAU and L. FOURRIER
For a polynomial map f." C 2 ~ C, there exists a finite set of values A I = {cl ..... cn} such that the restriction fl: C 2 \ f - X ( A f ) ~ C \ A f is a C~-locally trivial fibration (see [68, 69,61,70, 71]). The set A s of these so-called irregular values of f can be characterized by using the resultant between the equation f(x,y) - c = 0 and a generic polar equation (see [69,72])
~f
--
~f
(x,y) +
( x , y ) = 0.
According to [73], we define the link at infinity L r of a polynomial f as the link at infinity of the algebraic curve F I given by the equation II~= o(f(x,y) - ci) = 0, where Co is a regular value of f and ci s A~ for i=l,...,n. Since F r = IA ~=0f l(ci), the link Lyis the union of the links at infinity of all the irregular fibres of f and of a regular one. A polynomial f e C[x,y] is said to be good (at infinity) if its link at infinity L I is regular (i.e. if all the fibres of f are regular at infinity). A consequence of the Milnor's fibration theorem at infinity is that the polynomial f is good if and only if the link at infinity of a generic fibre of f is fibred, i.e. isotopic to Lo (see [3,2]). Together with the Proposition 4, this leads to the following:
Proposition 5. Let F = {f(x,y) = 0} c C 2 be an algebraic curve. The polynomial f is good if and only if the splicing tree TL~(V) o f L~(F) has strictly positive weights. In particular, if F is connected at infinity, f is good. Define/x¢ =/xr~ to be the total Milnor number of the polynomial f. Then we can prove the following: Proposition 6. For a primitive polynomial f • C [x ,y], 1 - x 4 ( L f ) = t z f - nz4(L~(Fco)), where Fco = {f(x,y) = Col is a generic fibre and n is the number o f irregular fibres o f f.
Proof: Since the restriction f: C 2 \ f - 1 ( A t ) ~ C \ A s defines a locally trivial fibration, whose fibre is dif f eomorphic to Fc,,, one obtains n
Z(C2\f
- ~(Af)) = 1 -
~
z(F,,) = (1 - n)z(F,,,, ).
i=l
Moreover, Kronheimer-Mrowka's theorem implies that z4(Cf) = ~
Xc(Fq) + X(I'c0)
i=l
and ,;];'4(L~(F,,o) ) = X(['co)" The proposition then follows easily from these equalities. A consequence of this proposition is to give an upperbound for/x I and for the number of irregular fibres at infinity of )~
Proposition 7. Let F = [ f ( x , y ) = 0} c C 2 be an algebraic curve. I f p is the number o f irregular fibres at infinity of the polynomial f, then t x f + p -< 2 - z4(L~(F)) -< 1 + Ord~ PLy(r). Proof'. The equality given in Proposition 6 is equivalent to the following one: n
1 - z4(L~(F,o)) = / x I + ~
(z4(L~(F,,)) - z4(L~(F~o))).
i=l
Since f is a primitive polynomial, by Kronheimer-Mrowka's theorem and H~-Le and Suzuki's results [61,27], z4(L~(F~.~))>-z4(L~(Fc.)), for i= 1,...,n, with equality if and only if
Knot theory and plane algebraic curves
789
I,, is regular at infinity. The proof is then straightforward by considering two cases according to whether I is regular at infinity or not. Remark: In the case where I is regular at infinity, the proof gives the better estimate (see [741): A direct consequence of this proposition is the following: Corollary (see [67]). Let IY = {f(x,y) = 0) c C2 b e an algebraic curve. Zf x(r) the only possible irregular fibre of the polynomial f.
= 1 then r is
We present now a topological version of a classical result by Zaidenberg and Lin [67] (see also [2]). Two algebraic plane curves I and I’ in C2 are said to be topologically equivalent if there is a homeomorphism h:C2 -+ C* such that h(lY)=I”. We say that a reduced curve I = cf(x,y) = 0) is quasihomogeneous (resp. nodal) if the polynomial f is quasihomogeneous (resp. of the type yg(x), where g E C[X]). Theorem. Let r = (f(x,y) = 0) c C2 b e an algebraic reduced curve which is regular at infinity. Zf x(T) = 1 then the curve r is topologically equivalent to a quasihomogeneous curve or to a nodal curve. Remark. This is a weak form of Zaidenberg-Lin’s
theorem in the sense that it only gives a topological equivalence instead of an algebraic equivalence. Proof By the corollary of Proposition 7, I is the only possible irregular fibre of the polynomial $ Since I is regular at infinity, the polynomial f is good. In particular Pi= pi-. Therefore the restriction of f: C2\I’+ C\(O) induces a locally trivial fibration, whose monodromy is of finite order by Corollary 12 of [75] (see also Lemma 3 of [67]). Since the polynomial f is good, by Milnor’s fibration theorem at infinity, L,(T) is a fibred link with a monodromy of finite order. So L, (I’)is a Seifert link [76]. Moreover, by Theorem 2.7 of [2], L-(I) is the connected sum of the local algebraic links of the singular points of I. The classification of essential annuli in the complements of Seifert links shows that either L,(I) is a prime link or a connected sum of Hopf links (i.e. a key-ring). In the second case, all singular points of I are ordinary double points (i.e. nodes). Using methods from [73], one can show that the curve I is topologically equivalent to a nodal curve. In the first case, if I? is non-singular, then L-(r) is a trivial knot. By Proposition 2 of Section 2, I is an algebraic embedding of @ in C2 which is topologically unknotted, hence equivalent to a coordinate embedding of C in C2. Otherwise, P is a singular curve and admits only one singular point. Without loss of generality, we may assume that the origin is the only singular point of I. Moreover the algebraic link L, of this singularity (E < < 1) is equivalent to L,(r). Since L,(T) = L, is a Seifert link, the germ of curve P, = I rl B:’ is topologically equivalent to a quasihomogeneous one. Since I is the only irregular fibre of f, the fact that the curve I itself is globally equivalent to a quasihomogeneous curve follows from the methods of [73], which in fact give a stronger result (see theorem and corollary below). Remark. In the case where the curve I admits an injective polynomial parametrization by 16, similar methods allow one to obtain a classification of I up to algebraic automorphism of C2 (see [l-3]). This leads to new proofs of Abhyankar-Moh-Suzuki’s Theorem and ZaidenbergLin’s Theorem. Artal’s examples [28] show that the link at infinity Lf of f does not determine the topological embedding in C2 of the algebraic curve I, However, by considering the singular foliation 9 of C2 given by the connected components of the fibres of J the second author proved the following:
790
M. BOILEAU and L. FOURRIER
Theorem (see[73]). The topological type o f the pair ( S 3 , L f ) , for R sufficiently large, determines up to homeomorphism the foliation ~ at infinity (i.e. ,~ A C 2 \ B R ) and in a neighborhood o f each singular fibre o f f. C r u c i a l for u n d e r s t a n d i n g t h e t o p o l o g i c a l e m b e d d i n g in C 2 o f t h e c u r v e F I is t h e g l o b a l m o n o d r o m y of t h e i n d u c e d f i b r a t i o n f." C Z \ f - 1 (A;)--+ C \ A ~ I n p a r t i c u l a r , t h e c o n n e c t i o n b e t w e e n t h e local m o n o d r o m i e s a r o u n d t h e i r r e g u l a r fibres a n d t h e m o n o d r o m y at infinity (i.e. of t h e M i l n o r ' s f i b r a t i o n at infinity) is still l a r g e l y u n k n o w n . If F = {f(x,y) -- 0} is t h e o n l y i r r e g u l a r fibre of f, t h e n t h e g l o b a l m o n o d r o m y o f t h e f i b r a t i o n f." C 2 \ F - - - + C \ { 0 } c o i n c i d e s w i t h t h e m o n o d r o m y at infinity. H e n c e we h a v e t h e f o l l o w i n g :
Corollary. I f F = { f ( x , y ) = 0} is the only irregular fibre o f f, then L:~(F) determines the singular foliation ~ o f C 2 up to homeomorphism. In particular, L ~ ( F ) determines the topological embedding o f F in C 2. I n view of this result, t h e f o l l o w i n g q u e s t i o n s s e e m n a t u r a l :
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Pergamon
& Fracrais, Prmted
Vol. 9. No. 415, pp. 779-792. 1998 0 1998 Elsevrer Science Ltd in Great Britain. All rights reserved 0960-0779/9x $19.00 + 0.00
PII: so960-0779(97)00103-3
Knot Theory and Plane Algebraic MICHEL
BOILEAU
and LAURENCE
Curves
FOURRIER
Universite Paul Sabatier, Laboratoire Emile Picard, U.M.R. 5580,118 route de Narbonne, 31062Toulouse Cedex 4, France
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 F c C* passing through the origin, let L, = F fl JB: be the intersection of F with a round ball in C* of radius r>O centered at the origin. When this intersection is transverse, L, is an oriented link in S: = dB;‘. The main purpose of this paper is to present a survey of the results relating the topology of the pair (SJ,L,) to the topology of the pair (B:,F f’ B:). 0 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The study of plane algebraic curves in UC2(i.e. of subsets I = ((x,y) E C2 If(x,y) =0} c (C2, where f: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 (UC2,1).By Hilbert’s Nullstellensatz, with a reduced plane algebraic curve I is associated a well defined reduced polynomial map f:c’+ C. Many results about the topology of the pair (~‘,I) 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 infinity has been introduced by Rudolph [l] (cf. [2, 31) to study plane algebraic curves in the large. This approach has led 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 reflect the authors’ main interests. The starting point of this paper is the data of a reduced plane algebraic curve I c UC2 passing through the origin. Let Bz = ((x,y) E @* : 1x12+ Iy]*sr2} be a round ball in C* centered at the origin, of radius r>O. Except for finitely many values of the radius r> 0, the intersection I II dB1 is transverse, and hence consists of a disjoint union of finitely many smoothly oriented embedded circles in Sz = c?B~,where the orientation is induced by the one of I. We denote by L, the oriented link I fl c?B~and by I, the piece of algebraic curve rn B;. The main purpose of this paper is to present a survey of the results relating the topology of the pair (S;,L,) to the topology of the pair (B:‘,r,). In order to make the discussion precise, we first define two basic invariants of an oriented link in S3,the so-called Seifert and Murasugi Euler characteristic. Seifert has shown [4-6] 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 c S3 is the supremum of the Euler characteristic of the Seifert surfaces for L without any closed components. 779
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Viewing S 3 as the oriented boundary of the ball B 4, Murasugi [7] has considered properly embedded oriented surfaces in B 4 with boundary the oriented link L (see [8]). Then the Murasugi characteristic z4(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 B 4 to become a Murasugi surface, the inequality z3(L) <-z4(L) holds for any oriented link L c S 3. The simplest topological invariant of a reduced plane algebraic curve F c C 2 is its Euler characteristic z(F). When F is singular, we will consider its "corrected" Euler characteristic Zc(F) =z(F) - tz(F), where Iz(F) is the sum of the Milnor's numbers of the isolated singular points of F. A topological definition of the Milnor's number/~x of an isolated singular point x of F is as follows: let Lx be the transverse intersection of F with a small round ball B,(x) centered at x, with sufficiently small e < 1; then/z~ = 1 - z3(Lx) is independent of the radius • < 1 and is called the Milnor's number of the singular point x ~ F. We define =
x E Sing(F)
where Sing (F) is the set of singular points of F. A first step to study the relationship between the topologies of the pairs (S3,L~) and (B~,Fr) is to understand the relationship between the three invariants z3(L~), z4(Lr) and Zc(F~), described above. This is the object of the first 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 remainder of the paper, we focus in Section 3 on the classical case of algebraic links of isolated plane singularities (cf. [9, 10]) and in Section 4 on the more recent case of links at infinity of plane algebraic curves (cf. [3,2,1]).
2. THE G E N E R A L CASE
In 1923, Alexander [11] showed that any oriented link L in S 3 can be isotoped into a closed braid form. This representation of links is strongly related to complex geometry and singularity theory (cf. Sections 3 and 4). 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 [12]. The situation is the one described in the introduction. Let F be a plane algebraic curve in C 2 passing through the origin. For r > 0 (except finitely many values), let Lr be the transverse intersection F A dB~ and let Fr be the piece of curve in the 4-ball B 4. The oriented link Lr c S~ = 0B 4 is transverse to the standard contact structure on S 3 induced by the 2-field of complex lines tangent to S3r. By a transverse isotopy one can always transform Lr into a closed braid form. This is one of the key ingredients of the so-called Bennequin's inequality. Bennequin [13] has proved a beautiful inequality on closed braids which was a crucial step in his search for an exotic contact structure on the 3-sphere. In our context, Bennequin's inequality can be stated as follows:
Theorem (Bennequin). With the notation above, z3(Lr) <-Zc(Fr). A straightforward corollary is the following:
Corollary. With the notation above, z3( Lr) <-Zc(Fr) <-z4( Lr). To prove his result, Bennequin used closed braids and Markov surfaces which are special kinds of braided Seifert surfaces (cf. also [14, 15]). He first shows that z3(Lr) can be realized by a Markov surface and reduces the proof of his inequality to study the topology of a
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Markov surface with respect to the standard contact structure of S3.A complete proof of the inequality can be found in [13] (see also [16]; see [17] for a generalization). Here is now a deep recent result by Kronheimer and Mrowka (the so-called Thorn conjecture, see [S]) which they have proved in full generality using Seiberg-Witten invariants (cf. [l&21]). Th is result shows that the corrected Euler characteristic x#,) of a piece of algebraic curve I?, properly embedded in B:’ is determined by the topology of the pair tssJ4. Theorem (Kronheimer-Mrowka). With the notation above, xc( r,) = x4(Lr). Corollary. The topological type of a non-singular connected piece of algebraic curve r, c B: is determined by the topological type of the pair (S), L,).
This result leads to the following question: Question 1. Does the topology of the pair (Sp, L,) always determine the topological piece of algebraic curve r, when Tr is non-singular?
type of the
Kronheimer and Mrowka’s result implies that the map r H b,(L,) - x4(Lr) is nondecreasing, where b,(L,) is the number of connected components of L,. In order to compare the links L, while r is increasing, let us introduce the following complexity C(L,) of L,: C(L)
= (b,(L)
- x~tL),WJ
- x4Lr))
EN x N3
with the lexicographic order on N X N. Question 2. With the notation above, is the map r H C( L,) non-decreasing?
Kronheimer-Mrowka’s result reduces Question 2 to the following: Question 2’. With the notation above, if b,( L,) - x4( L,) = b,( L,,) - x4( L,,), for r 2 r’ , does the inequality b,( L,) - x3( L,,) 2 b,( L,.) - x3( L,,) hold?
According to Gordon [22], we show that Question 2’ (and hence Question 2) has a positive answer if both L, and L,, are fibred knots. We recall that an oriented link L c S3 is fibred if it is the binding of an open book decomposition of S3. Proposition 1. With the notation above, assume that L, and L,,, for r 2 r’ , are Jibred knots. If x4( L,) = x4( L,,), then 1 - x3( L,) 2 1 -x3( L,,) and equality holds if and only if the pairs (Ss, L,) and (ST,,L,.) are equivalent.
The hypothesis x4( L,) =x4( L,.) and Kronheimer-Mrowka’s theorem imply that r II S3x [r’,r] is an annulus. Therefore r induces a ribbon concordance between the fibred knots L, and L,. Let Y = S3 x [r,r’]\r be the complement of the ribbon concordance, let X, = S;\ L, and X,, = S:(\ L,. be the complements of the knots L, and L,,, and let P, 2, and %,, denote their infinite cyclic coverings. The following argument is due to Gordon [22]. Since the morphism rl(X,) -+ rl(Y) is surjective, the morphism H1(z-$J)+ H,(~;Q) is surjective and then dim H,(F;Q)sdim H,(xr;Q). Moreover cy: H,(dY;Q)-+H,(E;Q) is surjective since 8 = 8, U 8,.. By Milnor’s duality [23], one has the isomorphisms H,(lyl2) = H,(F,dF;Q), H,(F,dF;Q) g H&2), which, combined with the surjectivity of (Y,give the short exact sequence o-,H,(~,d~;~)~H,(d~;~)~H*(P;Q)~o. So dim H,(z’,;Q) + dim H,(a,.;CD) = 2dim H1(P;UJ) 5 2dim H,(z,;Q). To conclude it suffices to remark that if K is an oriented fibred knot in S3, we have the equality dim H,(z;;Q) = 1 -x3(K) (see [6]), where 2 denotes the infinite cyclic covering of the complement of K.
Proof:
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The proof of the equality case is more intricate and we refer to Lemma 3.2 of [22] for it. Moreover a positive answer to Question 2 would show that if Lr is a trivial link then Lr, is a trivial link for r' < r. This special case of Question 2 is still open, and we are only able to give a partial result about it.
Proposition 2. I f Lr is a trivial link f o r some r > O, then (i) the piece o f curve Fr is a disjoint union o f non-singular holomorphic disks, (ii) Fr is unknotted in the ball B 4, that is to say, Fr can be isotoped in OB 4 = S3r to a Seifert surface o f Lr, (iii) f o r 0 < r' < r, Lr, is a trivial link if bo( L~,) <- bo( Lr). Proof'. The first assertion (i) is a direct consequence of Bennequin's inequality.Assertion (ii) follows from Eliashberg's work [24], using the fact that the holomorphic convex hull of Lr contains a union of disjoint holomorphic disks, and thus can be filled up by disjoint families of holomorphic disks (cf. [25]).For the last assertion (iii), we use the fact that if bo(L~,) <- bo(L~), then there is a ribbon concordance W 2 between the link L r, and a sublink Lr of L r. Let F; be the union of disks in B 4 bounding Lr- By assumption F; f3 B 4, = F r f-) B 4, = F~,. Since we are dealing with ribbon concordance, we have the following sequence of surjective morphisms induced by inclusions: 7r,(S3r\Lr) ~
7r,(B4\W 2) -~ 7rl(B4\Fr)
~ O.
From assertion (ii), 7 r l ( B 4 \ r ; ) is a free group of rank bo(L;))=bo(Lr) like crl(S3\Lr). By the hopficity of finitely generated free groups, it follows that the morphism ¢rl(Sr3\ L;) ~ ~ ' 1 ( B 4 \ F r°) is an isomorphism, and hence the morphism 7r1($3\ L r ) ~ 7 r l ( B 4 \ W 2) is an isomorphism. According to [22], the ribbon concordance implies that the morphism induced by the inclusion crl(S3r\L~,)~ 7rl(Br4\W 2) is injective. It follows t h a t 7rl(S3r,\Lr ,) is a free group, and hence Lr, is a trivial link.
Remark. Proposition 2 can be view as a general topological version of the famous Abhyankar-Moh-Suzuki Theorem [26, 27] which asserts that any algebraic embedding of C in C 2 is algebraically unknotted. One may remark that, in general, the topology of the pair (S3,L~) does not determine the topology of the pair (B4,F~) even if r r is non-singular. In [28], Artal-Bartolo constructed two connected non-singular plane algebraic curves in C 2 having the same links at infinity (i.e. the same links L , for r sufficiently large), but whose complements in C 2 have distinct fundamental groups. Thus the topology of the pair (Sr3,L~) does not even determine the group 7rl(B4\F~). We conclude this general discussion by considering the problem of characterizing the possible types of the links L r in S 3, for any r > 0. This is still a wide open question. In his important pioneering work [1,29,14], Rudolph found a large class of links in S 3, called quasipositive links, which can be realized as the transverse intersection of a plane algebraic curve with the boundary of a round ball in C 2 (see also [30]). These links are characterized by the fact that they admit a quasipositive closed braid presentation. A braid word is said quasipositive if it is a product of conjugates of positive words. The quasipositive links form a large family of links, including classical algebraic links, links at infinity of plane algebraic curves, closed positive braids. In contrast to the local case or the case at infinity, Rudolph's construction [31-35] shows that some hyperbolic links (i.e. whose complement in S 3 admits a complete hyperbolic structure) may appear as links L , for some r which is not too small and not too big. Rudolph's construction is so explicit that it allows one to build knotted holomorphic disks in a round ball. Rudolph's conjecture, still open, asserts that any link Lr is a quasipositive link.
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The following result shows that any link L, must verify a specific inequality, already known for quasipositive links [36]. Before stating the result, we need to recall the definition of the 2-variable HOM FLY-polynomial for an oriented link L in S3. We use the notation of [37,36]. To an oriented link L in S3, one can associate a 2-variable Laurent polynomial PI>(W) E qv *I ,Z *I], defined recursively in the following way:
Let L,, L, and L- be three links whose projections differ only at the crossing shown m the diagram. Then define P&v,z) = 1 for the trivial knot 0, and P,, (v,::) = vz PL,(v,z) + v2P, _ (v,z). Let Ord, P, be the valuation of PL in the variable v (i.e. the least degree in v of the polynomial P,(v,z) viewed as a Laurent polynomial in v with coefficients in a=[2* ‘I). Proposition 3 (see [38]): With the notation above, Ord, PL, 2 1 - x4(Lr) 2 1 - b,(L,). Proof: Although this inequality has been explicitly stated only recently in [38], it is an easy consequence of Bennequin’s thesis [13] and of the Franks-Williams [39] and Morton [37] inequality for closed braids. According to [13], after a transverse isotopy to the standard contact structure on S:, L, admits a closed braid presentation with II strands and algebraic length e. An application of the Hopf index theorem to the complex vector field tangent to the piece of algebraic curve I’, c B;’ and orthogonal to the Hopf field shows that x(I’~) =n - e (see [13,401). Then, the Franks-Williams and Morton inequality for closed braids implies that Ord,P,y?e-n+l=l-X,(r,)?l-X,(L,). This inequality is sharp enough to characterize the links L, among 2-bridge links (see [38]). In particular it shows the following: Corollary 1. If L, is a knot and is equivalent to its mirror image then the piece of algebraic curve r, is a non-singular disk and x4( L,) = 1. Proof If L, is equivalent to its mirror image, then by symmetry on the variable v, Ord, P,, is non-positive. So x4(Lr) 2 1 and, if L, is a knot, ,YJ(L,.)= 1. Since by Kronheimer-Mrowka’s theorem x4(Lr) =xJI,), the assertion follows. Corollary 2 (see [4l’]). The figure-eight knot is never the transverse intersection of a plane algebraic curve with the boundary of a round ball in C*. Proof From tabulations (see [42]), one has Ord, P,= - 2 for the figure-eight knot K, contradicting the inequality Ord, P,, L 1 - b,(L,). One also may use the inequality to get an estimate of Milnor’s number ,u(r,) of r, from the topology of the pair (S:,L,). Corollary 3. With the notations above, p(T,) I Ord, P,, + x(Tr) - 1.
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3. ALGEBRAIC LINKS: THE LOCAL CASE IN BRIEF
This section is devoted to presenting some classical results about the topology of isolated singularities of plane algebraic curves. From our point of view, given a plane algebraic curve with an isolated singularity at the origin, we are interested in the embedding of the piece of algebraic curve F, = F f3 B 4 in the round ball B 4 of radius sufficiently small (a so-called Milnor's ball). The link L, = F, N dB 4 is called the link of the singularity of F at the origin. The study of such links (so-called algebraic links) goes back to the works of [43-46] and had a revival from Milnor's seminal work [9], showing that such a link is the binding of an open book decomposition of S 3 and describing the fibration of S 3 \ L ~ over S 1.
Theorem (Milnor's fibration [9]). Let F be a plane algebraic curve in C 2 having an isolated singularity at the origin. Then there exists Eo> 0 such that, for any E <- Eo, (i) the pair (B4,F,) is homeomorphic to the cone on the pair (0B4,L,), (ii) i f F = {f(x,y) = 0] c C 2, for some polynomial map f: C2---~ C, then the argument map 0 = f/I f I : --, S ' is an open book decomposition, (iii) the induced fibration O: S3\IV(L,)---~S l is diffeomorphic to the fibration f: f - 1( dD (0, 7)) A B 4~ 3D (0, ~1), for a radius ~7sufficiently small depending o f e. The proof of this result can be found in Chapters IV and V of [9] (see also [47, 48]). For an explicit description of the monodromy of the fibration see [49] and also [50].A straightforward corollary of this result together with Kronheimer-Mrowka's theorem is the following:
Corollary. In the local case (e < < 1), the topology o f the pair (B4,F,) is determined by the topology o f the pair (S 3,L,). Moreover, the equality z3(L,) = Zc(F,) = x 4 ( L , ) holds. Algebraic links L, are completely characterized by the topology of their complement S 3 \ L , and some special inequalities, called Puiseux inequalities. This is mainly due to works of Brauner [43], K~ihler [51], Burau [44, 45] and Zariski [46] (more details may be found in Reeve's work [52]). For a description of algebraic links L, via Jaco-Shalen and Johannson characteristic family of tori and splicing diagrams, we refer to Eisenbud and Neumann's book [10] (see also [53]).Recall that a Seifert link in S 3 is a union of finitely many fibres of some Seifert fibration of S 3. For example, Seifert knots are the so-called torus knots which may be represented as a simple closed curve on the boundary of a standard unknotted solid torus in S3.An iterated torus link in S 3 is a link obtained from a Seifert link by applying finitely many of the two following constructions: Cabling operation: Starting from a link L0, let L be the union of finitely many parallel simple closed curves on each boundary component of a regular neighborhood N(Lo) of Lo, such that each curve is not null homologous in N(Lo). If some family of parallel curves is not homologous to a component of L0 in N(Lo), we say that L is a non-trivial cable around L0. Connected sum operation: Let L1 and L2 be oriented links in $3; then we say that L is a connected sum of L1 and L2 if L is obtained from L1 and L2 by identifying a small oriented unknotted arc of a component of L1 with a small oriented unknotted arc of a component of Lz by an orientation reversing homeomorphism. For a proof of the following theorem, we refer to [10] in the general case (see also [53]) and to [43] (see also [52,47]) in the case of knots.
Theorem (characterization of algebraic links): An oriented link L in S 3 is an algebraic link L, if and only if L, is an iterated torus link obtained only by cabling operations satisfying the Puiseux inequalities. In particular L, is a positive braid.
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For example, if the singularity of I is analytically irreducible at the origin, the Puiseux expansion theorem gives the following parametrization of I near the origin: y = x41/Pl(ul+ x421PlP2 (a2 + ... + (a, _ l + u$q’pI-+y ...). with pi,qr > 0 and gcd (pi,qi) = 1, for i = l,..., Y. To describe the knot L,, we consider the intersection of I with the bidisk [(x, y) E @’ : 1x ] % E,1y 1I e) which g’Ives a knot equivalent to L,. The knot L, then can be built step by step using successive truncations of the parametrization of I. Let K, be the knot given in the solid torus V= ((x,y) E C* : ]x I= E,Iy ) 5 l } by the equation y = ~~~~~~~~~ It is easy to see that Kr =O(q,,p,) is a torus knot of type (ql,pl): it means that K, is equivalent to the simple closed curve on dV homologous to q,(meridian) +p,(longitude]. Then the knot K2 corresponding to the equation y = xql’pl(al + u2xq2/p~2)is a (n,,m,)-cable around K1 with m2=p2 and n2 = q2+plp2ql. Therefore K2 =B((q,,p,),(n,,m,)) is isotopic in a regular neighborhood N(K,) of Kr to a simple closed curve on dN(K,) homologous to n2 meridian (K,) + m2 longitude (K,). By induction (see [10,54,47]), one obtains that L, is the iterated torus knot ~((nl,ml>,..., (nm,)), with ml = ply ~zr=ql and, for i? 1, mitl =pitl, r~,+~=qi+l +piPi+lni. Moreover, these numbers (ni,mj) must satisfy some inequalities, called Puiseux inequalities, coming from the fact that qi/pi > 0, for 15 i 5 r. If the singularity of I at the origin is not analytically irreducible, the link L, is no longer connected. The number of connected components b,(L,) corresponds to the number of irreducible branches at the origin of I. So L, is an iterated torus link built from the Puiseux expansions of each irreducible branch. In particular, each connected component of L, is an iterated torus knot as above. The way these components are linked together can be determined by comparing pair-wise the Puiseux expansions of the branches. Important are the first terms where the two expansions differ (see [10,53]). In fact a well-known result of Zariski and Jalabert-Lejeune [55] asserts that an algebraic link L, is completely determined by the topological types of its components and their linking numbers. For an iterated torus link L, the operations of connected sums and of non-trivial tablings correspond to embedded incompressible non boundary parallel tori in the exterior X(L) = P\fi(L) (’i.e. r,-injective tori in X(L) which are not isotopic to dX(L)). One can extract from this family of tori a minimal family T-L of embedded, pairwise disjoint and not parallel tori such that the closure of each component of X(L)\TL is the exterior of a Seifert link in S3 (see [56-581). Th en one says that L is obtained by splicing together all these Seifert companions links (see [10,59]). The splicing tree TL is the dual tree to the family FL, whose vertices correspond to the connected components of X(L)\FL and the edges to the tori in 5-L. Moreover, we add an arrowed edge for each component of L. We associate with each vertex a weight corresponding to the linking number of L with a generic fibre of the Seifert fibration of the component of X(L)\F, corresponding to this vertex. For an algebraic link L, all these weights are strictly positive. This is no longer true, in general, for iterated torus links at infinity and it is an important feature of this theory at infinity, as we shall see in the next section. 4. LINKS AT INFINITY
In this section, we give some applications of knot theory to the study of affine algebraic curves in C2. As in the introduction, let I = {(x,y) E C2 : f(x,y) = O] be a reduced algebraic curve where f: C* + @is a reduced polynomial.
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For R sufficiently large, the intersection Ln= c~B~n F is always transverse and the topological type of the pair (3B4,LR) is independent of the radius R. We call this oriented link Ln, for R sufficiently large, the link at infinity of F and denote it by L~(F) (see [1-3]). A major difference compared with the local case is that the Milnor's argument map f~ I f 1: S ~ \ L ~ ( F ) ~ S ~ is not in general an open book fibration. However there still exists a kind of "Milnor's fibration at infinity" given by the following result: Theorem (Milnor's fibration at infinity [2,60]): Let F - - { f ( x , y ) = O ] c C 2 be a reduced algebraic curve as above. Then there exist a disk o f radius s centered at the origin D (O,s ) and a real number Ro > 0 such that, for any R >-Ro, (i) No = f - l ( D ( O , s ) ) n S 3 is a union o f solid tori, (ii) O = f / l f [ • S 3n \ f - I(D(0,s)) ""-> S 1 is a locally trivial fibration called Milnor's fibration at infinity, (iii) this fibration is diffeomorphic to the fibration f: f 1( 3D (0,s)) O B4--> OD (0,s). Statement (i) is a very nice application of the maximum principle due to Rudolph [3]. The remaining statements follow by extending Milnor's methods at infinity. The proof of statement (ii) can be found in [60] and the proof of statement (iii) in [3,2,60].In the Milnor's fibration at infinity appear the following links naturally associated with the polynomial map f." C2---> C: the multilink at infinity and the regular link at infinity. The multilink at infinity: Let Lo be the link whose components are the cores of the solid tori of No. The Milnor's fibration 0 = f~ I f I • X(Lo) ~ S 1 on the exterior of Lo allows us to consider Lo as a fibred multilink [3].A multilink L(m~ ..... mr) is an oriented link with r components each of whose components is weighted by an integer mi (the multiplicity of the ith component)• Such a multilink L(ml ..... mr) is fibred if there is a fibration of its exterior p: X ( L ) ~ S 1 whose restriction along any meridian curve of the ith component of L(ml,...,mr) has degree mi. It follows that a fibred multilink L(ml,...,mr) defines an open book decomposition of S 3 if and only if mi = + 1, for i = 1 ..... r. For more details about multilinks, we refer to [10]. Coming back to Lo, the argument map O = f / [ f l " S 3 \ L ~ ( F ) - - > S ~ restricted to X(Lo) = S 3 \ b l o defines the Milnor's fibration of the multilink Lo(ml ..... mr). Assuming that the value 1 ~ S 1 is a regular one for 0, then 0-1(1)=2~ is a Seifert surface for the oriented link L~(F). The intersection Eo = E n X(Lo) is a fibre of the fibred multilink Lo(ml ..... mr). In particular, 3Eo=E N 3No is a family of disjoint closed curves on the tori c~No which is homologous to L~(F) in No. It follows that the multiplicities mi, i=1 ..... r, of the fibred multilink Lo(ml,...,m~) are given by the homology class of L~(F) in No: [L~(F)] = [3Eo] = (ml ..... m~) ~ H1(No;2~) ~- 7/r. So there naturally appears a second link: The regular link at infinity: For a value t ~ c~D(0,s), let L~(F,) = 3Eo be the link at infinity of the generic fibre F, = {f(x,y) = t} c C 2. From the discussion above, it follows that any Seifert surface of the link Lo~(F) intersects transversely 3No along a link isotopic to L~(F,) on 3No. In particular, one has the following inequalities: z3(L~(F)) -< Z.~(L~(F,)) <- z3(Lo). If the pairs (S3,L~(F)) and (S3,L~(F,)) are equivalent, we say that F is regular at infinity• Otherwise, F is said to be irregular at infinity• In particular, F is regular at infinity if and only if, outside a sufficiently large ball, it looks like its nearby fibres F,, I t] < e. Using Section 2, the discussion above and results by Hh and L6 [61] and Suzuki [27] we obtain the following: Proposition 4: Let F be a reduced algebraic curve in C 2. Then (i) the embedding of F in C 2 determines the embedding of any generic fibre F, in C 2, (ii) the curve F is regular at infinity if and only if z4( L~(F)) = z3(L~(F)). Proof'. Assertion (i) is a consequence of the following facts:
Knot theory and plane algebraic curves
The topological type of the pair (,$&,(I,)) is determined by the topological type of the pair (Si,L,(I’)). This follows from the previous discussion and the fact that the Milnor’s fibration at infinity is determined up to isotopy by the multiplicities (m,,...,m,) (see [62, 631). The topological type of the pair (S&L,(I,)) determines the topological type of the pair (C”,I’,) by assertion (iii) of Milnor’s fibration theorem at infinity. To prove assertion (ii), we need the following result of 1611and [27]: For any generic fibre I,, x(I) rX(I’J, with equality if and only if I is regular at infinity. Moreover, assertion (iii) of Milnor’s fibration theorem at infinity implies that X(IJ =X3(La(IJ). Therefore by Kronheimer-Mrowka’s theorem m4r)) = im wLmu2) ~~4ur)). So x4(Lcs(I)) =X3(Lm(I)) implies that xc(I) =&It). So I is regular at infinity by [61, 271 and the pairs ($&,(I)) and ($&(I,)) are equivalent. Conversely if I is regular at infinity, then idv3) = ~3(w2) = m) = im = m4w This proposition shows that there is a natural and well defined notion of irregular link at infinity. Like algebraic links in the local case (see Section 3), the link at infinity L,(I) is an iterated torus link but, in this case, connected sum operations may appear in the construction of L,(I). Let us recall briefly the construction of L,(I) (see [3,2]). Let P2@= C2 U PL be a projective compactification of C2 and let r be the projective compactification of I in P’C. Then Lp( I) = r n JN( Pk) where the regular neighborhood N( IFPL)of the line at infinity is a disk bundle 9: N( [FDL) + lu’: with Euler class 1. If l? n ln)L= (x1,... ,.r,J, let D c pi be a disk containing l? n R& such that r n aN( p:) = r fl dq -l(D). Then f n dq -l(D) in dq -l(D) is a closed braid b which is a disjoint union of the local algebraic links associated with each point xi of r n iFDk. To pass from the embedding of L,(I) in dq-l(D) to the embedding of L,(I) in JN( [FDL),one must perform a positive Dehn twist around the axis of the braid p and change the ambient orientation. It follows easily that L,(I) in aN( IFpi)is an iterated torus link which must satisfy some reversed Puiseux inequalities (see [3]). However, there is not yet a complete characterization of links at infinity, except in the case of knots [64]. So, with the topological pair ($&,(I)) one can associate a weighted splicing tree as in Section 3 for the local case.This allows one to give a simple characterization of regular link at infinity: Corollary (see [3,65, 661). A link at infinity L,(T) only non-negative weights.
is regular if and only if its splicing tree has
This corollary follows from Proposition 4 and the formula computing xJL&)) and x4(L&)) from the weighted splicing tree of L,(I) (see [3]). Milnor’s fibration theorem at infinity shows that the embedding in C2 of a non-singular curve I which is regular at infinity is determined by the topological type of the pair (&L,(T)). As we pointed out in Section 2, this is no longer true if I is irregular at infinity (see [28]). To obtain more precise results on this problem, one has to take into account the other irregular fibres of the polynomial map defining I. In other words, the topology of the pair (@‘,I’), where I = cf(x,y) = 0}, is intimately connected with the topology of the polynomial map f: C2+ C. Without loss of generality, we shall always assume from now on that all the generic fibres off are connected (i.e. f is a primitive polynomial). By the Stein factorization theorem, any non-primitive polynomial g E C[x,y] is equal to h of, for a primitive polynomial f E C[x,y] and h E @[t] (see [67]).
788
M. BOILEAU and L. FOURRIER
For a polynomial map f." C 2 ~ C, there exists a finite set of values A I = {cl ..... cn} such that the restriction fl: C 2 \ f - X ( A f ) ~ C \ A f is a C~-locally trivial fibration (see [68, 69,61,70, 71]). The set A s of these so-called irregular values of f can be characterized by using the resultant between the equation f(x,y) - c = 0 and a generic polar equation (see [69,72])
~f
--
~f
(x,y) +
( x , y ) = 0.
According to [73], we define the link at infinity L r of a polynomial f as the link at infinity of the algebraic curve F I given by the equation II~= o(f(x,y) - ci) = 0, where Co is a regular value of f and ci s A~ for i=l,...,n. Since F r = IA ~=0f l(ci), the link Lyis the union of the links at infinity of all the irregular fibres of f and of a regular one. A polynomial f e C[x,y] is said to be good (at infinity) if its link at infinity L I is regular (i.e. if all the fibres of f are regular at infinity). A consequence of the Milnor's fibration theorem at infinity is that the polynomial f is good if and only if the link at infinity of a generic fibre of f is fibred, i.e. isotopic to Lo (see [3,2]). Together with the Proposition 4, this leads to the following:
Proposition 5. Let F = {f(x,y) = 0} c C 2 be an algebraic curve. The polynomial f is good if and only if the splicing tree TL~(V) o f L~(F) has strictly positive weights. In particular, if F is connected at infinity, f is good. Define/x¢ =/xr~ to be the total Milnor number of the polynomial f. Then we can prove the following: Proposition 6. For a primitive polynomial f • C [x ,y], 1 - x 4 ( L f ) = t z f - nz4(L~(Fco)), where Fco = {f(x,y) = Col is a generic fibre and n is the number o f irregular fibres o f f.
Proof: Since the restriction f: C 2 \ f - 1 ( A t ) ~ C \ A s defines a locally trivial fibration, whose fibre is dif f eomorphic to Fc,,, one obtains n
Z(C2\f
- ~(Af)) = 1 -
~
z(F,,) = (1 - n)z(F,,,, ).
i=l
Moreover, Kronheimer-Mrowka's theorem implies that z4(Cf) = ~
Xc(Fq) + X(I'c0)
i=l
and ,;];'4(L~(F,,o) ) = X(['co)" The proposition then follows easily from these equalities. A consequence of this proposition is to give an upperbound for/x I and for the number of irregular fibres at infinity of )~
Proposition 7. Let F = [ f ( x , y ) = 0} c C 2 be an algebraic curve. I f p is the number o f irregular fibres at infinity of the polynomial f, then t x f + p -< 2 - z4(L~(F)) -< 1 + Ord~ PLy(r). Proof'. The equality given in Proposition 6 is equivalent to the following one: n
1 - z4(L~(F,o)) = / x I + ~
(z4(L~(F,,)) - z4(L~(F~o))).
i=l
Since f is a primitive polynomial, by Kronheimer-Mrowka's theorem and H~-Le and Suzuki's results [61,27], z4(L~(F~.~))>-z4(L~(Fc.)), for i= 1,...,n, with equality if and only if
Knot theory and plane algebraic curves
789
I,, is regular at infinity. The proof is then straightforward by considering two cases according to whether I is regular at infinity or not. Remark: In the case where I is regular at infinity, the proof gives the better estimate (see [741): A direct consequence of this proposition is the following: Corollary (see [67]). Let IY = {f(x,y) = 0) c C2 b e an algebraic curve. Zf x(r) the only possible irregular fibre of the polynomial f.
= 1 then r is
We present now a topological version of a classical result by Zaidenberg and Lin [67] (see also [2]). Two algebraic plane curves I and I’ in C2 are said to be topologically equivalent if there is a homeomorphism h:C2 -+ C* such that h(lY)=I”. We say that a reduced curve I = cf(x,y) = 0) is quasihomogeneous (resp. nodal) if the polynomial f is quasihomogeneous (resp. of the type yg(x), where g E C[X]). Theorem. Let r = (f(x,y) = 0) c C2 b e an algebraic reduced curve which is regular at infinity. Zf x(T) = 1 then the curve r is topologically equivalent to a quasihomogeneous curve or to a nodal curve. Remark. This is a weak form of Zaidenberg-Lin’s
theorem in the sense that it only gives a topological equivalence instead of an algebraic equivalence. Proof By the corollary of Proposition 7, I is the only possible irregular fibre of the polynomial $ Since I is regular at infinity, the polynomial f is good. In particular Pi= pi-. Therefore the restriction of f: C2\I’+ C\(O) induces a locally trivial fibration, whose monodromy is of finite order by Corollary 12 of [75] (see also Lemma 3 of [67]). Since the polynomial f is good, by Milnor’s fibration theorem at infinity, L,(T) is a fibred link with a monodromy of finite order. So L, (I’)is a Seifert link [76]. Moreover, by Theorem 2.7 of [2], L-(I) is the connected sum of the local algebraic links of the singular points of I. The classification of essential annuli in the complements of Seifert links shows that either L,(I) is a prime link or a connected sum of Hopf links (i.e. a key-ring). In the second case, all singular points of I are ordinary double points (i.e. nodes). Using methods from [73], one can show that the curve I is topologically equivalent to a nodal curve. In the first case, if I? is non-singular, then L-(r) is a trivial knot. By Proposition 2 of Section 2, I is an algebraic embedding of @ in C2 which is topologically unknotted, hence equivalent to a coordinate embedding of C in C2. Otherwise, P is a singular curve and admits only one singular point. Without loss of generality, we may assume that the origin is the only singular point of I. Moreover the algebraic link L, of this singularity (E < < 1) is equivalent to L,(r). Since L,(T) = L, is a Seifert link, the germ of curve P, = I rl B:’ is topologically equivalent to a quasihomogeneous one. Since I is the only irregular fibre of f, the fact that the curve I itself is globally equivalent to a quasihomogeneous curve follows from the methods of [73], which in fact give a stronger result (see theorem and corollary below). Remark. In the case where the curve I admits an injective polynomial parametrization by 16, similar methods allow one to obtain a classification of I up to algebraic automorphism of C2 (see [l-3]). This leads to new proofs of Abhyankar-Moh-Suzuki’s Theorem and ZaidenbergLin’s Theorem. Artal’s examples [28] show that the link at infinity Lf of f does not determine the topological embedding in C2 of the algebraic curve I, However, by considering the singular foliation 9 of C2 given by the connected components of the fibres of J the second author proved the following:
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M. BOILEAU and L. FOURRIER
Theorem (see[73]). The topological type o f the pair ( S 3 , L f ) , for R sufficiently large, determines up to homeomorphism the foliation ~ at infinity (i.e. ,~ A C 2 \ B R ) and in a neighborhood o f each singular fibre o f f. C r u c i a l for u n d e r s t a n d i n g t h e t o p o l o g i c a l e m b e d d i n g in C 2 o f t h e c u r v e F I is t h e g l o b a l m o n o d r o m y of t h e i n d u c e d f i b r a t i o n f." C Z \ f - 1 (A;)--+ C \ A ~ I n p a r t i c u l a r , t h e c o n n e c t i o n b e t w e e n t h e local m o n o d r o m i e s a r o u n d t h e i r r e g u l a r fibres a n d t h e m o n o d r o m y at infinity (i.e. of t h e M i l n o r ' s f i b r a t i o n at infinity) is still l a r g e l y u n k n o w n . If F = {f(x,y) -- 0} is t h e o n l y i r r e g u l a r fibre of f, t h e n t h e g l o b a l m o n o d r o m y o f t h e f i b r a t i o n f." C 2 \ F - - - + C \ { 0 } c o i n c i d e s w i t h t h e m o n o d r o m y at infinity. H e n c e we h a v e t h e f o l l o w i n g :
Corollary. I f F = { f ( x , y ) = 0} is the only irregular fibre o f f, then L:~(F) determines the singular foliation ~ o f C 2 up to homeomorphism. In particular, L ~ ( F ) determines the topological embedding o f F in C 2. I n view of this result, t h e f o l l o w i n g q u e s t i o n s s e e m n a t u r a l :
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