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mathematicae Inventiones manuscript No. (will be inserted by the editor) Knots, Links, and 4-Manifolds Ronald Fintushel1 , Ronald J. Stern2 1 2 Department of Mathematics, Michigan State University East Lansing, MI 48824 e-mail: ron"nt@math.msu.edu Department of Mathematics, University of California Irvine, CA 92697 e-mail: rstern@math.uci.edu January 15, 1998 1 Introduction In this paper we investigate the relationship between isotopy classes of knots and links in S 3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 surface. In order to state our theorems we need to view the Seiberg-Witten invariant of a smooth 4-manifold as a multivariable (Laurent) polynomial. To do this, recall that the Seiberg-Witten invariant of a smooth closed oriented 4-manifold X with b+ (X) > 1 is an integer valued function which is de2 "ned on the set of spin c structures over X, (cf. [W], [KM],[Ko1],[T1]). In case H1 (X, Z) has no 2-torsion (which will be the situation in this paper) there is a natural identi"cation of the spin c structures of X with the characteristic elements of H 2 (X, Z). In this case we view the Seiberg-Witten invariant as SWX : {k H 2 (X, Z)|k w2 (T X) (mod 2))} Z. The Seiberg-Witten invariant SWX is a diffeomorphism invariant whose sign depends on an orientation of 2 H 0 (X, R) det H+ (X, R) det H 1 (X, R). The "rst author was partially supported NSF Grants DMS9401032 and DMS9704927 and the second author by NSF Grant DMS9626330 Knots, Links, and 4-Manifolds 3 If SWX ( ) = 0, then we call a basic class of X. It is a fundamental fact that the set of basic classes is "nite. Furthermore, if is a basic class, then so is with SWX ( ) = ( 1)(e+sign)(X)/4 SWX ( ) where e(X) is the Euler number and sign(X) is the signature of X. Now let { 1 , . . . , n } be the set of nonzero basic classes for X. Consider variables t = exp( ) for each H 2 (X; Z) which satisfy the relations t + = ta tb . For the purposes of this paper we de"ne the SeibergWitten invariant of X to be the Laurent polynomial n SW X = b0 + j=1 bj (t j + ( 1)(e+sign)(X)/4 t 1 ) j where b0 = SWX (0) and bj = SWX ( j ). For example, for the simply connected minimally elliptic surface E(n) with holomorphic Euler characteristic n and no multiple "bers, we have SW E(n) = (t t 1 )n 2 where t = tF for F the cohomology class Poincar dual to the "ber class. e n 2 I.e. SWE(n) ((n 2m)F ) = ( 1)m 1 m 1 for m = 1, . . . , n 1 and SWE(n) ( ) = 0 for any other . We refer to any Laurent polynomial n P (t) = a0 + j=1 aj (tj + t j ) of one variable with coef"cient sum n a0 + 2 j=1 aj = 1 as an A-polynomial. If, in addition, an = 1 we call P (t) a monic Apolynomial. Let X be any simply connected smooth 4-manifold with b+ > 1. We de"ne a cusp in X to be a PL embedded 2-sphere of self-intersection 0 with a single nonlocally #at point whose neighborhood is the cone on the right-hand trefoil knot. (This agrees with the notion of a cusp "ber in an elliptic surface.) The regular neighborhood N of a cusp in a 4-manifold is a cusp neighborhood; it is the manifold obtained by performing 0-framed surgery on a trefoil knot in the boundary of the 4-ball. Since the trefoil knot is a "bered knot with a genus 1 "ber, N is "bered by smooth tori with one singular "ber, the cusp. If T is a smoothly embedded torus representing a nontrivial homology class [T ], we say that T is c-embedded if T is a smooth "ber in a cusp neighborhood N ; equivalently, T has two vanishing cycles. Note that a c-embedded torus has self-intersection 0. We can now state our "rst theorem. 4 Ronald Fintushel, Ronald J. Stern Theorem 1.1 Let X be a simply connected smooth 4-manifold with b+ > 1. Suppose that X contains a smoothly c-embedded torus T such that 1 (X \ T ) = 1. Then for any A-polynomial P (t), there is a smooth 4manifold XP which is homeomorphic to X and has Seiberg-Witten invariant SW XP = SW X P (t) where t = exp(2[T ]). The basic classes of X were de"ned above to be elements of H 2 (X). To make sense of the statement of the theorem, we need to replace [T ] by its Poincar dual. Throughout this paper we allow ourselves to pass freely e between H 2 (X) and H2 (X) without further comment. As a corollary to the construction of XP we shall show: Corollary 1.2 Suppose further that X is symplectic and that T is symplectically embedded. If P (t) is a monic A-polynomial, then XP can be constructed as a symplectic manifold. Using the work of Taubes [T1 T5] concerning the nature of the SeibergWitten invariants of symplectic manifolds, we shall also deduce: Corollary 1.3 If P (t) is not monic, then XP does not admit a symplectic structure. Furthermore, if X contains a surface g of genus g disjoint from T with 0 = [ g ] H2 (X; Z) and with [ g ]2 < 2 2g if g > 0, or [ g ]2 0 if g = 0, then XP with the opposite orientation does not admit a symplectic structure. As a corollary, we have many interesting new homotopy K3 surfaces (i.e. manifolds homeomorphic to the K3 surface). In particular, since SW K3 = 1 we have: Corollary 1.4 Any A-polynomial P (t) can occur as the Seiberg- Witten invariant of an irreducible homotopy K3 surface. If P (t) is not monic, then the homotopy K3 surface does not admit a symplectic structure with either orientation. Furthermore, any monic A-polynomial can occur as the Seiberg-Witten invariant of a symplectic homotopy K3 surface. In fact, there are three disjoint c-embedded tori T1 , T2 , T3 in the K3 surface representing distinct homology classes [Tj ], j = 1, 2, 3 (cf. [GM]). Theorem 1.1 then implies that the product of any three A-polynomials Pj (tj ), j = 1, 2, 3, can occur as the Seiberg-Witten invariant of a homotopy K3 surface K3P1 P2 P3 with tj = exp(2[Tj ]), SW K3P1 P2 P3 (t1 , t2 , t3 ) = P1 (t1 )P2 (t2 )P3 (t3 ). Knots, Links, and 4-Manifolds 5 Furthermore, if all three of the polynomials are monic, the resulting homotopy K3 surface can be constructed as a symplectic manifold. If any one of the A-polynomials Pj (tj ) is not monic, then the resulting homotopy K3 surface admits no symplectic structure. A common method for constructing exotic manifolds is to perform log transforms on c-embedded tori. One might ask whether these new homotopy K3 surfaces K3P1 P2 P3 can be constructed in this fashion. However, it is shown in [FS1] that if X is the result of performing log transforms of multiplicities pi on c-embedded tori Ti , i = 1, . . . , n, in the K3 surface, then n SW X = 1 (si (pi 1) + si (pi 3) + + si (pi 1) ) where si = exp(Ti /pi ). In particular, the sum SWX ( ) taken over all characteristic classes in H 2 (X; Z) is p1 pn . (See also [GM].) However, the same sum for K3P1 P2 P3 is SW K3P1 P2 P3 (1, 1, 1) = P1 (1)P2 (1)P3 (1) = 1; so K3P1 P2 P3 cannot be built using log transforms in this way. The A-polynomials appearing in Theorem 1.1 are familiar to knot theorists. It is known that any A-polynomial occurs as the Alexander polynomial K (t) of some knot K in S 3 . Conversely, the Alexander polynomial of a knot K is an A-polynomial. Furthermore, if the A-polynomial is monic then the knot can be constructed as a "bered knot, and if K is "bered, then K (t) is a monic A-polynomial. Indeed, it is a knot K in S 3 which we use to construct XP . Consider a knot K in S 3 , and let m denote a meridional circle to K. Let MK be the 3-manifold obtained by performing 0-framed surgery on K. Then m can also be viewed as a circle in MK . In MK S 1 we have the smooth torus Tm = m S 1 of self-intersection 0. Since a neighborhood of m has a canonical framing in MK , a neighborhood of the torus Tm in MK S 1 has a canonical identi"cation with Tm D2 . Let XK denote the "ber sum XK = X#T =Tm (MK S 1 ) = [X \ (T D2 )] [(MK S 1 ) \ (Tm D2 )] where T D2 is a tubular neighborhood of the torus T in the manifold X (with 1 (X) = 1 (X \ T ) = 0). The two pieces are glued together so as to preserve the homology class [pt D2 ]. This latter condition does not, in general, completely determine the diffeomorphism type of XK (cf. [G]). We take XK to be any manifold constructed in this fashion. Because MK 6 Ronald Fintushel, Ronald J. Stern has the homology of S 2 S 1 with the class of m generating H1 , the complement (MK S 1 )\(T D2 ) has the homology of T 2 D2 . Thus XK has the same homology (and intersection pairing) as X. Furthermore, the class of m normally generates 1 (MK ); so 1 (MK S 1 ) is normally generated by the image of 1 (T ). Since 1 (X \F ) = 1, it follows from Van Kampen's Theorem that XK is simply connected. Thus XK is homotopy equivalent to X. It is conceptually helpful to note that XK is obtained from X by removing a neighborhood of a torus and replacing it with the complement of the knot K in S 3 crossed with S 1 . Also, in order to de"ne Seiberg-Witten invariants, the oriented 4-manifold X must also be equipped with an orien2 tation of H+ (X; R). The manifold XK inherits an orientation as well as an 2 orientation of H+ (XK ; R) from X. Let [T ] be the class in H2 (XK ; Z) induced by the torus T in X, and let t = exp(2[T ]). Our "rst main theorem, from which Theorem 1.1 is an immediate corollary, is: Theorem 1.5 If the torus T is c-embedded, then the Seiberg-Witten invariant of XK is SW XK = SW X K (t). Let E(1) be the rational elliptic surface with elliptic "ber F . If T is a smoothly embedded self-intersection 0 torus in X, then E(1)#F =T XK = (E(1)#F =T X)#T =Tm (MK S 1 ), and in the manifold E(1)#F =T X, the torus T = F is c-embedded. Thus we have a slightly more general result: Corollary 1.6 Let X be any simply connected smooth 4-manifold with b+ > 1. Suppose that X contains a smoothly embedded torus T of selfintersection 0 with 1 (X \ T ) = 1 and representing a nontrivial homology class [T ]. Then SW E(1)#F =T XK = SW E(1)#F =T X K (t). More can be said if K is a "bered knot. Consider the normalized Alexander polynomial AK (t) = td K (t), where d is the degree of K (t). If K is a "bered knot in S 3 with a punctured genus g surface as "ber, then MK "bers over the circle with a closed genus g surface g as "ber. Thus MK S 1 "bers over S 1 S 1 with g as "ber and with Tm = m S 1 as section. It is a theorem of Thurston [Th] that such a 4-manifold has a symplectic structure with symplectic section Tm . Thus, if X is a symplectic 4 manifold with a symplectically embedded torus with self-intersection 0, then XK = X#T =Tm (MK S 1 ) is symplectic since it can be constructed as a symplectic "ber sum [G]. In a fashion similar to the treatment of the Knots, Links, and 4-Manifolds 7 Seiberg-Witten invariant as a Laurent polynomial, one can view the Gromov invariant of a symplectic 4-manifold X as a polynomial GrX = GrX ( ) t where GrX ( ) is the usual Gromov invariant of . As a corollary to Theorem 1.5 and the theorems of Taubes relating the Seiberg-Witten and Gromov invariants of a symplectic 4 manifold [T3,T5] we have: Corollary 1.7 Let X be a symplectic 4-manifold with b+ > 1 containing a symplectic c-embedded torus T . If K is a "bered knot, then XK is a symplectic 4-manifold whose Gromov invariant is GrXK = GrX AK ( ) where = exp([T ]). Proof The homology H2 (MK S 1 ) is generated by the classes of the symplectic curves Tm and g ; so the canonical class of MK S 1 has the form MK S 1 = a[Tm ] + b[ g ]. Applying the adjunction formula (and using [Tm ]2 = [ g ]2 = 0 and [Tm ] [ g ] = 1) gives b = 0 and a = 2g 2. d But note that the degree of K ( ) = a0 + n=1 an ( n + n ) is d = g. Hence MK S 1 = (2d 2)[Tm ]. This means that the canonical class of the symplectic structure on XK is XK = X + MK S 1 +2[T ] = X +2d[T ]. Taubes' theorem now implies that for any H2 (XK ), the coef"cient of t = exp( ) in GrXK is: GrXK ( ) = SWXK (2 XK = SWXK (2 X 2d[T ]) d = n= d d an SWX (2 X 2(d + n)[T ]) an GrX ( (d + n)[T ]), n= d = and this is the coef"cient of exp( ) in GrX AK ( ). Corollary 1.7 also follows from the work of Ionel and Parker [IP]. They prove a general formula for the Gromov invariants of symplectic manifolds constructed as `symplectic mapping cylinders'. Their formula is valid in all dimensions, and in particular gives the above corollary in dimension 4. Of course, if X is simply connected and 1 (X \ T ) = 1, then XK is homeomorphic to X. This implies Corollary 1.2. As corollary of the initial work of Taubes [T1,T2] on the Seiberg-Witten invariants of symplectic manifolds and the adjunction inequality, we have the following corollary which also implies Corollary 1.3. 8 Ronald Fintushel, Ronald J. Stern Corollary 1.8 If K (t) is not monic, then XK does not admit a symplectic structure. Furthermore, if X contains a surface g of genus g disjoint from T with 0 = [ g ] H2 (X; Z) and with [ g ]2 < 2 2g if g > 0 or [ g ]2 < 0 if g = 0, then XK with the opposite orientation does not admit a symplectic structure. Proof Suppose that XK admits a symplectic structure with symplectic form and canonical class . Taubes has shown that SWXK ( ) = 1 and that if k is any other basic class then |k | < . Since SW XK = SW X K (t) and since K (t) = a0 + d n=1 an (tn + t n ) is a polynomial in one variable t = exp(2[T ]), any nontrivial basic class, and in particular, the canonical class, is of the form = + 2n[T ] where SWX ( ) = 0 and |n| d. Let m be the maximum integer satisfying SWX ( + m[T ]) = 0. Note that m 0. Set = + m[T ]. Because of the maximality of m, we have SWXK ( + 2d[T ]) = ad SWX ( ) = 0. Replacing [T ] with [T ] allows us to assume that [T ] 0. First assume that [T ] > 0. Then ( + 2d[T ]) = + (m + 2(d n))[T ] because m 0 and d n 0, and equality occurs only if m = 0 and n = d. But a strict inequality contradicts Taubes' theorem, thus = and n = d; so = +2d[T ]. This means that 1 = SWXK ( ) = ad SWX ( ); so ad = 1, i.e. K is monic. If [T ] = 0, ( + 2d[T ]) = ( + 2n[T ]) = , which means that = + 2d[T ], and again we see that ad = 1. Finally, if any manifold Y contains a homologically nontrivial surface g of genus g with [ g ]2 > 2g 2, then, if g > 0, it follows from the adjunction inequality [KM,MST] that the Seiberg-Witten invariants of Y vanish, and hence Y does not admit a symplectic structure [T1]. If g = 0 and [ g ]2 0 one can also show that the Seiberg-Witten invariants of Y must vanish [FS2,Ko2]. The authors are unaware of any simply connected smooth oriented 4 manifold X with b+ > 1 and SW X = 0 which does not contain either a sphere with self-intersection 2 or a torus with self-intersection 1. The techniques used in proving Theorem 1.5 generalize to the more general setting of links. Let L = {K1 , . . . , Kn } be an (n 2)-component ordered link in S 3 and suppose that Xj , j = 1, . . . , n, are simply connected smooth 4-manifolds with b+ 1 and each containing a smoothly embedded torus Tj of self-intersection 0 with 1 (Xj \ Tj ) = 1 and representing a Knots, Links, and 4-Manifolds 9 nontrivial homology class [Tj ]. If ( j , mj ) denotes the standard longitudemeridian pair for the knot Kj , let L : 1 (S 3 \ L) Z denote the homomorphism which is characterized by the property that L (mj ) = 1 for each j = 1, . . . , n. Now, mimicking the knot case above, let ML be the 3manifold obtained by performing L ( j ) surgery on each component Kj of L. (The surgery curves form the boundary of a Seifert surface for the link.) Then, in ML S 1 we have smooth tori Tmj = mj S 1 of self-intersection 0 and we can construct the n fold "ber sum n X(X1 , . . . Xn ; L) = (ML S 1 )#Tj =Tmj j=1 Xj Here, the "ber sum is performed using the natural framings Tmj D2 of the neighborhoods of Tmj = mj S 1 in ML S 1 and the neighborhoods Tj D2 in each Xj and glued together so as to preserve the homology classes [pt D2 ]. As in the knot case, it follows from Van Kampen's Theorem that X(X1 , . . . Xn ; L) is simply connected. Furthermore, the signature and Euler characteristic (i.e. the rational homotopy type) of X(X1 , . . . Xn ; L) depend only on the rational homotopy type of the Xi and the number of components in the link L. In the special case that all the Xj are the same manifold X we denote the resulting construction by XL . It is conceptually helpful to note that X(X1 , . . . Xn ; L) is obtained from the disjoint union of the Xj by removing a neighborhood of the tori Tj and replacing them with the complement of the link L in S 3 crossed with S 1 . If L (t1 , . . . , tn ) denotes the symmetric multivariable Alexander polynomial of the n 2 component link L, and E(1) denotes the rational elliptic surface, our second theorem is: Theorem 1.9 The Seiberg-Witten invariant of E(1)L is SW E(1)L = L (t1 , . . . , tn ) where tj = exp(2[Tj ]). As a corollary we shall show: Corollary 1.10 The Seiberg-Witten invariant of X(X1 , . . . Xn ; L) is n SW X(X1 ,...Xn ;L) = L (t1 , . . . , tn ) j=1 SW E(1)#F =T j Xj where tj = exp(2[Tj ]). 10 Ronald Fintushel, Ronald J. Stern As before, the work of Taubes [T3,T5] implies that if L is a "bered link, and each (Xj , Tj ) is a symplectic pair, then X(X1 , . . . Xn ; L) is a symplectic 4-manifold whose Gromov invariant is n GrX(X1 ,...Xn ;L) = AK (t1 , . . . , tn ) j=1 GrE(1)#F =T j Xj . Two words of caution are in order here. First, there may be relations in X(X1 , . . . Xn ; L) among the homology classes [Tj ] represented by the tori Tj = Tmj . These relations are determined by the linking matrix of the link L. In particular, if all the linking numbers are zero, then the [Tj ] are linearly independent. At the other extreme, if the n-component link is obtained from the Hopf link by pushing off one component (n 2) times, then all the [Tj ] are equal. Second, the ordering of the components of the link can affect these relations. If L is a two component link with odd linking number, then E(1)L is a homotopy K3-surface and there are many interesting new polynomials which are not products of A-polynomials (cf. [Hi]) that can occur in this way as the Seiberg-Witten invariants of a homotopy K3-surface. The "rst examples of nonsymplectic simply connected irreducible smooth 4-manifolds were constructed by Z. Szabo [S1]. These manifolds X(k) (k Z, k = 0, 1) can be shown to be diffeomorphic to E(1)W (k) , where W (k) is the 2-component k-twisted Whitehead link (see Figure 1) with 1/2 1/2 1/2 1/2 Alexander polynomial k(t1 t1 )(t2 t2 ). By Theorem 1.9 (and by the computation "rst given in [S1]) SW X(k) = k(t1 1/2 t1 1/2 )(t2 1/2 t2 1/2 ). Thus by Taubes' Theorem [T1], X(k) does not admit a symplectic structure with either orientation (since X(k) contains spheres with self-intersection 2). Note also that for k = 1 the k twisted Whitehead link is "bered; so X( 1) is, in fact, symplectic. The "rst examples of nonsymplectic homotopy K3-surfaces were constructed by the authors. These manifolds Y (k) can be shown to be diffeomorphic to K3T (k) where T (k) is the k-twist knot (see Figure 1) with Alexander polynomial kt (2k + 1) + kt 1 . By Theorem 1.5 SW Y (k) = kt (2k + 1) + kt 1 and, by Corollary 1.8, if k = 0, 1, Y (k) does not admit a symplectic structure with either orientation. Again, for k = 1 the k twist knot is "bered; so Y ( 1) is symplectic. Knots, Links, and 4-Manifolds 11 9 6 ' $ & RHk 8twists % 7 9 ' & RHk 8twists 6 $ % 7 W (k) = k-twisted Whitehead link T (k) = k-twist knot Figure 1 Our next task is to prove Theorem 1.5 and Theorem 1.9. The proof of Theorem 1.5 is constructive and gives an algorithm which relates the Seiberg-Witten invariants of XK with those of X by performing a series of topological log transforms on nullhomologous tori in XK which reduce it to X. This turns out to be the same algorithm used to compute the Alexander polynomial K (t). This proof relies upon important analytical work of Morgan, Mrowka, and Szabo [MMS] (cf. [S1,S2]) and Taubes [T6]and [MMST] regarding the effect on the Seiberg-Witten invariants of removing neighborhoods of tori and sewing in manifolds with nonnegative scalar curvature. These we present in Section 2 in the form of gluing theorems. The proof of Theorem 1.9 can take one of two routes. The "rst is to utilize the algorithm provided by Conway [C] to compute the Alexander polynomial (more precisely the potential function) of a link. The proof then proceeds in a (tedious) manner, similar to the proof of Theorem 1.5. However, we choose to present a more direct proof by showing that the Seiberg-Witten invariants for the manifolds E(1)L satisfy the axioms for the Alexander polynomial of a link as provided by Turaev [T]. The advantage of this proof, aside from its brevity, is that it isolates the required gluing properties of the Seiberg-Witten invariants and perhaps lays the foundation for determining the axioms for an appropriate gauge theory which may expose the other, more sophisticated, knot and link invariants. In the "nal section we discuss examples with b+ = 1 which are given by our construction. It was Meng and Taubes [MT] who "rst discovered the relationship between Seiberg-Witten type invariants and the Alexander polynomial. In [MT] they de"ned an invariant of 3-manifolds by dimension-reducing the Seiberg-Witten invariants, and they showed that this 3-manifold invariant was related to the Milnor torsion. We fell upon Theorems 1.5 and 1.9 by attempting to understand our above mentioned constructions of nonsymplectic homotopy K3-surfaces. We end this introduction with three items. First, we conjecture that if K and K are two distinct knots (or n-component links) then for X = K3, the manifolds XK and XK are diffeomorphic if and only if K is isotopic 12 Ronald Fintushel, Ronald J. Stern to K . Second, we wish to thank Jim Bryan, Bob Gompf, Elly Ionel, Dieter Kotschick, Wladek Lorek, Dusa McDuff, Terry Lawson, Tom Parker, and Cliff Taubes for useful conversations. Finally, we wish to make it clear that the contributions of the present paper are of a purely topological nature. The gauge theoretic input to our theorems is due to Morgan, Mrowka, Szabo, and Taubes. 2 Background for the proofs of Theorems 1.5 and 1.9 In this section we shall survey recent gluing theorems of Morgan, Mrowka, and Szabo [MMS] (cf. [S1,S2]) and Taubes [T6] (cf.[MT]) and their joint work [MMST] which are used in the proof of Theorem 1.5. Also we shall review some of the work of W. Brakes and J. Hoste on `sewn-up link exteriors' which will be used in our constructions. The context for the "rst of the gluing results is as follows. We are given a smooth 4-manifold X with b+ > 1 and with an embedded torus X T which represents a nontrivial homology class [T ] of self-intersection 0 in H2 (X; Z). Any Seiberg-Witten basic class H2 (X; Z) (i.e. any with SW X ( ) = 0) must be orthogonal to the homology class [T ] since the adjunction inequality states that 0 [T ]2 + | [T ]|. The relative SeibergWitten invariant SW (X;T ) is formally de"ned to be SW (X;T ) = SW X#T =F E(1) where E(1) is the rational elliptic surface with smooth elliptic "ber F . It is a consequence of the gluing theorems of Morgan, Mrowka, Szabo, and Taubes [MMST] (cf. [MMS] and [T6]) that Theorem 2.1 Suppose that b+ > 1 and the torus T is c-embedded. Then X SW (X;T ) = SW X (t1/2 t 1/2 ) where t = exp(2[T ]). Note that E(1) has a metric of positive scalar curvature. A much more general (and dif"cult to prove) gluing theorem is: Theorem 2.2 ([MMST]) In the situation above SW X1 #T1 =T2 X2 = SW (X1 ;T1 ) SW (X2 ;T2 ) . To make sense of the homology classes in this statement, one argues as in the paragraph below Theorem 2.3. Knots, Links, and 4-Manifolds 13 The fact that Corollary 1.10 follows from Theorem 1.9 is now an easy consequence of the gluing theorems Theorem 2.1 and Theorem 2.2; for note that X(X1 , . . . , Xn ; L) = X(X2 , . . . , Xn ; L)#Tm1 =T1 X1 . Thus Theorem 2.1 implies that SW X(X1 ,...,Xn ;L) = SW X(E(1),X2 ,...,Xn ;L) SW X1 #T1 =F E(1) , and continuing inductively completes the argument. This is the only time we shall need to use the general gluing theorem (2.2). For our proof of Theorem 1.5 we will form an `internal "ber sum'. For this construction suppose that we have a pair of c-embedded tori T1 , T2 . In our manifold X with b+ > 1 we formally de"ne the relative Seiberg-Witten X invariant SW (X;T1 ,T2 ) to be 1 1 SW (X;T1 ,T2 ) = SW X ( 1 1 ) ( 2 2 ). where j = exp([Tj ]). We construct the internal "ber sum XT1 ,T2 by identifying the boundaries of neighborhoods Ti D2 of the Ti , again preserving the homology classes [pt D2 ]. The "rst gluing theorem we need for the proof of Theorem 1.5 is: Theorem 2.3 ([MMST],[MT]) The Seiberg-Witten of the internal "ber sum XT1 ,T2 is SW XT1 ,T2 = SW (X;T1 ,T2 ) | 1 = 2 . To make sense of Theorem 2.3, we need to describe how to choose lifts in H2 (XT1 ,T2 ; Z) of classes H2 (X; Z) satisfying SWX ( ) = 0. Let T0 denote the image of Ti pt in XT1 ,T2 . Each with SWX ( ) = 0 has a lift represented by a surface in XT1 ,T2 \(T0 D2 ) = X \(T1 D2 T2 D2 ). Let ai , bi be simple closed curves in Ti which generate H1 (Ti ; Z) and such that the curves ai pt are identi"ed to a curve a0 in T0 , and the same for the bi . An arbitrary lift of has the form + mA + nB where A and B are represented by the tori a0 D2 and b0 D2 . In X, the ai and bi bound -1 disks (vanishing cycles) in their respective cusp neighborhoods. This means that, in XT1 ,T2 there are spheres Sa and Sb of self-intersection 2 satisfying [Sa ] B = 1 and [Sb ] A = 1. One can then smooth the singular surfaces Sa + T0 and Sb + T0 to obtain tori Ta , Tb of self-intersection 0 which satisfy [Ta ] B = 1, [Tb ] A = 1 and [Ta ] = 0, [Tb ] = 0. The adjunction inequality then implies that if SWXT1 ,T2 ( + mA + nB) = 0 then 0 = [Ta ] + mA + nB = n, and similarly, m = 0; so the correct choice for the lift in (2.3) is alone. 14 Ronald Fintushel, Ronald J. Stern The other gluing result we need concerns generalized log transforms on nullhomologous tori. Let p and q be relatively prime nonzero integers (or (1, 0) or (0, 1)). If T is any embedded self-intersection 0 torus in a 4manifold Y with tubular neighborhood N = T D2 = S 1 S 1 D2 , let = p,q be the diffeomorphism S 1 S 1 D2 N given by (x, y, z) = (x, y s z q , y r z p ), The manifold Y (p/q) = (Y \ N ) (S 1 S 1 D2 ) is called the (generalized) (p/q)-log transform of Y along T . Our notation and terminology are incomplete since the splitting T = S 1 S 1 is necessary information. Throughout this paper, whenever a log transform is performed on a torus T , there will be a natural identi"cation T = S 1 S 1 and we always perform the transform with p,q in the coordinates S 1 S 1 D2 as above. We shall need to study the situation where a log transform is performed on a nullhomologous torus in Y . If T is such a torus, then in Y (0/1) there appears a new 2-dimensional homology class which is represented by the torus T0 = S 1 S 1 pt S 1 S 1 D2 (X \ N ) (S 1 S 1 D2 ) = Y (0/1). (The old torus pushed to (Y \ N ) is now (S 1 pt D2 ).) Notice that each homology class H2 (Y ; Z) may be viewed as a homology class in each Y (p/q). The theorem of Morgan, Mrowka, and Szabo and Taubes is Theorem 2.4 ([MMS], [T6], [MT])) Let Y be a smooth 4-manifold with b+ 3, and suppose that Y contains a nullhomologous torus T with tubular neighborhood N = T D2 = S 1 S 1 D2 . Let be the homology class of T0 in Y (0/1). Then for each characteristic homology class H2 (Y ; Z), det pq rs = 1. SWY (p/q) ( ) = pSWY ( ) + q i= SWY (0/1) ( + 2i ). The sum in the above formula reduces to a single term in all the situations which are encountered in the proofs of Theorem 1.5 and Theorem 1.9. Speci"cally, Knots, Links, and 4-Manifolds 15 2.5 In Theorem 2.4 suppose that there is a torus in Y (0/1) which is disjoint from T and which represents a homology class of self-intersection 0 whose intersection number with in Y (0/1) is 1. Suppose furthermore that = 0 for all H2 (Y ) H2 (Y (0/1)). Then SW Y (p/q) = pSW Y + qSW Y (0/1) . Proof Note that H2 (Y (0/1)) = H2 (Y ) H( , ) where H( , ) is a hyperbolic pair. If H2 (Y ; Z) satis"es SWY (0/1) ( + 2i ) = 0, the adjunction inequality implies that 0 2 + |( + 2i ) | = |2i|. Thus i = 0, and SWY (p/q) ( ) = pSWY ( ) + qSWY (0/1) ( ). Since for p = 0, each H2 (Y (p/q)) arises from a class in H2 (Y ), the lemma follows. We next wish to describe a method for constructing 3-manifolds which was "rst studied by W. Brakes [B] and extended by J. Hoste [Ho]. Let L be a link in S 3 with two oriented components C1 and C2 . Fix tubular neighborhoods Ni S 1 D2 of Ci with S 1 (pt on D2 ) a longitude of = Ci , i.e. nullhomologous in S 3 \ Ci . For any A GL(2; Z) with det A = 1, we the get a 3-manifold s(L; A) = (S 3 \ int(N1 N2 ))/A called a sewn-up link exterior by identifying N1 with N2 via a diffeomorphism inducing A in homology. For n Z, let An = 1 0 . n1 (In [Ho] this matrix is denoted A n .) A simple calculation shows that H1 (s(L; An ); Z) = Z Z2 n where is the linking number in S 3 of the two components C1 , C2 , of L. (See [B].) The second summand is generated by the meridian to either component. J. Hoste [Ho, p.357] has given a recipe for producing Kirby calculus diagrams for s(L; An ). First we review the notion of a `band sum' on an oriented knot or link L. Consider a portion of L consisting of a pair of strands, oriented in opposite directions, and separated by a band B. We identify B with an embedding of I 2 = [0, 1] [0, 1] in S 3 such that B L = ({0} I) ({1} I). Let K be the (oriented) knot or link obtained by trading the segments B L = {0, 1} I of B for the complementary oriented segments I {0} and (I {1}). The process of exchanging L for K in this fashion is called a band sum. Associated with the band move are two unknots: U , an unknotted circle which bounds a disk whose interior meets B = I 2 in the arc { 1 } I and is disjoint from L, and Uo , 2 which spans a disk whose interior meets B in I { 1 } and is disjoint from 2 K. (See Figure 2.) 16 Ronald Fintushel, Ronald J. Stern ' B C1 T C2 T $ U c & c L Uo % K Figure 2 Proposition 2.6 ([Ho]) Let L = C1 C2 be an oriented link in S 3 . Consider a portion of L consisting of a pair of strands, one from each component, oriented in opposite directions, and separated by a band B. The band sum of C1 and C2 is a knot K, and U links K twice geometrically and 0 times algebraically. The sewn-up link exterior s(L; An ) is obtained from surgery on the the two component link K U in S 3 with surgery coef"cient 0 on U and 2 n on K, where is the linking number of C1 and C2 . Next consider a related situation. Let Z = s(L; An ) where n = 2 ; so Z has H1 (Z; Z) = Z Z. In Z, let T0 denote the torus which is the image of Ni (i = 1, 2). Suppose that B is a band in S 3 meeting L as in Proposition 2.6, with the circle Uo , which links L twice geometrically and 0 times algebraically. Then Uo gives rise to a loop, Uo in Z. To get a Kirby calculus picture of this situation, apply Hoste's formula. We obtain a two component framed link K U with 0-framing on each. In S 3 , Uo bounds a disk which is disjoint from K and meets a disk spanning U in two points with opposite orientations. Thus Uo bounds a punctured torus in S 3 \ (K U ); and so Uo is nullhomologous in Z. This means that o has a naturally de"ned longitude in Z; so p/q- Dehn surgery on Uo is U well-de"ned. Let Z0 denote the result of 0-surgery on Uo in Z. In Z, let m denote the meridian circle to the torus T0 . Then in Z S 1 we have the torus Tm = m S 1 of self-intersection 0. Form the "ber sum X#T (Z S 1 ) of X with Z S 1 by identifying Tm with the torus T of X. (In the name of brevity, we shall sometimes make, as in this case, a mild change in our notation for "ber sum.) Similarly we can form the "ber sum X#T (Z0 S 1 ), which is seen to be the result of performing a (0/1)-log transform on the torus Uo S 1 in X#T (Z S 1 ). We wish to compute the Seiberg-Witten invariant of this 4-manifold. By cutting open Z = s(L; An ) along the torus T0 , we obtain the exterior of the link L in S 3 , and if this is done after performing 0-surgery on Uo , we obtain a link exterior in S 2 S 1 . Now perform the corresponding task in X#T (Z0 S 1 ), removing T0 S 1 . We obtain X#T (S 2 S 1 S 1 ) with a pair of tubular neighborhoods of self-intersection 0 tori removed. Call this manifold Q(L). To reiterate Q(L) is obtained by performing 0-surgery Knots, Links, and 4-Manifolds 17 on Uo in S 3 \ L, crossing with S 1 , and then "ber-summing with X along Tm . The boundary components of Q(L) have a natural framing , , , coming from the longitude and meridian of the link components in S 3 and the S 1 in the last coordinate. We can re-obtain X#T (Z0 S 1 ) by sewing up the boundary 3-dimensional tori of Q(L) using the matrix An (1). Instead, let us "ll in each of the boundary components of Q(L) with a copy of S 1 D2 S 1 . This can be done in many ways. We wish to do it so that, using the framings obtained from our copies of S 1 D2 S 1 , we obtain X#T (Z0 S 1 ) by sewing up the boundary of the resultant manifold with a neighborhood of the (new) link (S 1 pt S 1 ) (S 1 pt S 1 ) removed using the matrix A0 (1). Using the obvious framing for S 1 D2 S 1 , we claim that this is done by gluing each S 1 D2 S 1 to a component of 01 . Q(L) by a diffeomorphism with matrix B (1) where B = 1 We shall denote by W (L) the manifold formed using B (1) to sew in the neighborhoods of the tori Ci S 1 = S 1 pt S 1 . Let V (L) be the result of "lling in the exterior of L in S 3 via the diffeomorphism B on each component. This is just the result of -framed surgery on each component of L. Now if we sew up the link complement V (L) \ L via A0 , we get the result of sewing up S 3 \ L using the diffeomorphism B ( A0 )B 1 = A2 . Thus s(L; An ) = s(L ; A0 ) where L is the link C1 , C2 . Denote by V0 (L) the result of performing 0-surgery on Uo in V (L) \ L = S 3 \ L. Then W (L) = X#T (V0 (L) S 1 ) and X#T (Z0 S 1 ) = W (L)T1 =T2 where Ti = Ci S 1 . Proposition 2.7 Suppose that T is a c-embedded torus in X. Then with the above notation, SW X#T (Z0 S 1 ) = (t1/2 t 1/2 )2 SW XK as Laurent polynomials, where K is the band sum of C1 and C2 using the band B, and t = exp(2[T ]). Proof Since the matrix A0 (1) identi"es the tori Ci S 1 in the boundary components of Q(L), Theorem 2.3 tells us that SW X#T (Z0 S 1 ) = SW W (L)T1 =T2 is obtained from the relative invariant SW (W (L);T1 ,T2 ) by identifying the homology classes in Q(L) represented by the tori Ti . The gluing diffeomorphism B identi"es the homology class of a longitude of Ci with the meridian of Ci in S 3 \ L. Thus Theorem 2.3 implies that SW X#T (Z0 S 1 ) = SW W (L) ( 1 )2 , 18 Ronald Fintushel, Ronald J. Stern where = exp([m S 1 ]). It remains to identify W (L) as XK . By construction, W (L) is obtained from the 3-component link C1 C2 Uo in S 3 by performing -framed surgery on C1 and C2 and 0-framed surgery on Uo , crossing with S 1 , and "ber summing to X along Tm and T . The result of framed surgery on the 3component link is, by sliding C1 over C2 , seen to be the same as 0-framed surgery on K. Thus W (L) = XK , and the handle slide carries the meridian m to a meridian of K. Letting t = exp(2[T ]) = 2 , we get the calculation as claimed. 3 The proof of Theorem 1.5 We "rst recall a standard technique for calculating the (symmetrized) Alexander polynomial of a knot. This uses the skein relation K+ (t) = K (t) + (t1/2 t 1/2 ) K0 (t) (3.1) where K+ is an oriented knot or link, K is the result of changing a single oriented positive (right-handed) crossing in K+ to a negative (left-handed) crossing, and K0 is the result of resolving the crossing as shown in Figure 3. Note that if K+ is a knot, the so is K , and K0 is a 2-component link. If K+ is a 2-component link, then so is K , and K0 is a knot. u e e e ! u e e ! y gg # e e e e e e e e g e g g g g K+ K K0 Figure 3 The point of using (3.1) to calculate K is that K can be simpli"ed to an unknot via a sequence of crossing changes of a projection of the oriented knot or link to the plane. To describe this well-known technique, consider such a projection and choose a basepoint on each component. In the case of a link, order the components. Say that such a projection is descending, if starting at the basepoint of the "rst component and traveling along the component, then from the basepoint of the second component and traveling along it, etc., the "rst time that each crossing is met, it is met by an overcrossing. Clearly a link with a descending projection is an unlinked collection of unknots. Our goal is to start with a knot K and perform skein Knots, Links, and 4-Manifolds 19 moves so as to build a tree starting from K and at each stage adding the bifurcation of Figure 4, where each K+ , K , K0 is a knot or 2-component link, and so that at the bottom of the tree, we obtain only unknots, and split links. Then, because for an unknot U we have U (t) = 1, and for a split link S (of more than one component) we have S (t) = 0, we can work backwards using (3.1) to calculate K (t). K+ K Figure 4 K0 The recipe for constructing the tree is, in the case of a knot, to change the crossing of the "rst `bad' crossing encountered on the traverse described above. In this case, the result of changing crossing the is still a knot, and the result of resolving the crossing is a 2-component link. In the case of a 2-component link, one changes the "rst `bad' crossing between the two components which is encountered on the traverse. The result of changing the crossing is still a 2-component link, and the result of resolving the crossing is a knot. In this way we obtain a tree whose top vertex is the given knot K and which branches downward as in the "gure above. We shall call this tree a resolution tree for the knot K. We claim that the tree can be extended until each bottom vertex represents an unknot or a split link. For the projection of an oriented, based knot K, let c(K) be the number of crossings and b(K) be the number of bad crossings encountered on a traverse starting at the basepoint. The complexity of the projection is de"ned to be the ordered pair (c(K), b(K)). For the projection of an oriented, based 2-component link L, let c(L) be the total number of crossings and let b(L) be the number of bad crossings between the two components. Again the complexity is de"ned to be (c(L), b(L)). Consider a vertex which represents a knot or 2-component link A. Note that c(A ) = c(A+ ), b(A ) < b(A+ ), and c(A0 ) < c(A+ ). Thus in the lexicographic ordering, (c(A ), b(A )) < (c(A+ ), b(A+ )) and (c(A0 ), b(A0 )) < (c(A+ ), b(A+ )). Now a knot K1 with c(K1 ) = 0 or with b(K1 ) = 0 is the unknot, and a link L with c(L) = 0 is the unlink and with b(L) = 0, it is at least a split link. This completes the proof that we can construct the resolution tree as described. (We remark that for the sake of simplicity we have considered only the case where we have changed a positive to a negative crossing in the skein move. Of course, we may as well have to change a negative to a positive crossing in order to lower b at various steps, but this does not change the proof.) 20 Ronald Fintushel, Ronald J. Stern Consider an oriented, based knot K in S 3 and a knot projection of K. We shall use the resolution tree for this projection as a guide for simplifying XK in a way which leads to a calculation of XK . Let us consider the "rst step, say K = K+ {K , K0 }. At the crossing of K that is in question, there is an unknotted circle U linking K algebraically 0 times, so that the result of +1 surgery on U turns K into K+ = K. (See Figure 5.) In XK we have the nullhomologous torus S 1 U . (It bounds the product of S 1 with a punctured torus in S 3 \ K .) The fact that +1 surgery on U turns K into K+ means that XK+ is the result of a (1/1)-log transform on XK along S 1 U . Let XK (0/1) denote the result of performing a (0/1)-log transform on S 1 U in XK . We now use (2.4) and (2.5) to compute SW(XK ). Two tori are central to this calculation. Letting mU be a meridional circle to U , we get the torus S 1 mU which represents the homology class of (2.4). Also, in XK (0/1), the boundary of the punctured torus described above is spanned by a disk, and we obtain a torus of self-intersection 0 representing a class such that = 1. Note that H2 (XK (0/1)) = H2 (XK ) H( , ); so (2.5) applies. Hence SW XK = SW XK + SW XK (0/1) . e e ' e ! $ % e & e e +1 K Figure 5 Recall that XK (0/1) is obtained by performing 0-framed surgery on both components of the link K U , crossing with S 1 and "ber-summing with X, using the torus T obtained from a meridian of K crossed with S 1 . Hoste's recipe, (2.6), allows us to interpret the result of 0-framed surgery on both components of K U in S 3 as s(K0 ; A2 ) where K0 is the 2component link obtained by resolving the crossing under consideration (see Figure 6), and is the linking number of the two components of K0 . ' e 0e e e & e ! e 0 $ % e e s e e K0 ! ; A2 K e e Figure 6 Knots, Links, and 4-Manifolds 21 Let X(s(K0 ; A2 )) = (s(K0 ; A2 ) S 1 )#T X where T is the product of S 1 with a meridian to either component of K0 . Because A2 sends meridians to meridians, this de"nition does not depend on the choice of component. Then XK (0/1) X(s(K0 ; A2 )); so = SW XK+ = SW XK + SW X(s(K0 ;A2 )) , (3.2) mimicking the skein move which gives the second tier of the resolution tree. Now consider the next stage of the resolution tree and the skein move L {L , L0 } where K0 = L = L+ . This move corresponds to changing a bad crossing involving both components of L. If the bad crossing under consideration is, say, right-handed, then L = L+ can be obtained from L = C1 C2 by +1-surgery on an unknotted circle Uo as in Figure 7. This means that X(s(L; A2 )) is the result of a (1/1)-log transform on the torus S 1 Uo in X(s(L ; A2 )) where is the linking number of the components C1 , C2 of L and is determined by the fact that H1 (s(L ; A2 ); Z) = H1 (s(L; A2 ); Z) = Z Z. 1 e e ' e C e e & e L !C 2 $ % +1 Figure 7 In X(s(L ; A2 )), the torus S 1 Uo is nullhomologous. This is precisely the situation of (2.7) where Z = s(L ; A2 ). We wish to apply (2.5) to this situation. Let X(s(L ; A2 ))(0/1) denote the 4-manifold obtained by performing a (0/1)-log transform on S 1 Uo in X(s(L ; A2 )). This is the manifold X#T (Z0 S 1 ) of (2.7), and the 3-manifold Z0 is the result of 0-surgery on Uo in s(L ; A2 ). Let mUo be a meridional circle to Uo s(L ; A2 ). The torus mUo S 1 in X(s(L ; A2 ))(0/1) is the T0 mentioned in the statement of (2.4). As we argued in the proof of (2.7), in Z, the loop Uo bounds a punctured torus, and this gets completed to a torus of self-intersection 0 in Z0 . Let be its homology class in X(s(L ; A2 ))(0/1) = X#T (Z0 S 1 ). Since the class satis"es the hypothesis of (2.5), we have SW X(s(L+ ;A2 SW X(s(L+ ;A2 )) = SW X(s(L ;A2 )) + SW X#T (Z0 S 1 ) . Applying (2.7), this becomes: )) = SW X(s(L ;A2 )) + (t1/2 t 1/2 )2 SW XL0 (3.3) 22 Ronald Fintushel, Ronald J. Stern where L0 is the result of resolving the crossing of L which is under consideration, and t = exp(2[T ]). In order to see that this process calculates K (t), for "xed X, we de"ne a formal Laurent series , which is an invariant of knots and 2-component links. For a knot K, de"ne K to be the quotient, K = SW XK /SW X , and for a 2-component link with linking number between its components, L = (t1/2 t 1/2 ) 1 SW X(s(L;A2 )) /SW X , where as usual t = exp(2[T ]). It follows from (3.2) and (3.3) that for knots or 2-component links, satis"es the skein relation K+ = K + (t1/2 t 1/2 ) K0 . For a split 2-component link, L, the 3-manifold s(L; A2 ) contains an essential 2-sphere (coming from the 2-sphere in S 3 which splits the link). This means that X(s(L; A2 )) contains an essential 2-sphere of self-intersection 0, and this implies that SW X(s(L;A2 )) = 0 (see [FS2]). Thus for a split link, L = 0. For the unknot U , the manifold XU is just X#T (S 2 T 2 ) = X, and so U = 1. Subject to these initial values, the resolution tree and the skein relation (3.1) determine K (t) for any knot K. It follows that K is a Laurent polynomial in a single variable t, and K (t) = K (t), completing the proof of Theorem 1.4. 4 The proof of Theorem 1.9 We "rst review the axioms which determine the Alexander polynomial of a link. The reference for this material is [T]. The fact, proved in [T], that we shall use here is that there is but one map which assigns to each n component ordered link L in S 3 an element of the "eld Q(t1 , . . . , tn ) with the following properties: 1. (L) is unchanged under ambient isotopy of the link L. 2. If L is the unknot, then (L) = 1/(t t 1 ). 3. If n 2, then (L) Z[t1 , t 1 , . . . , tn , t 1 ]. n 1 4. The one-variable function (L)(t) = (L)(t, . . . , t) is unchanged by a renumbering of the components of L. 5. (Conway Axiom). If L+ , L , and L0 are links coinciding (except possible for the numbering of the components) outside a ball, and inside this ball have the form depicted in Figure 8, then (L+ ) = (L ) + (t t 1 ) (L0 ). 6. (Doubling Axiom). If the link L is obtained by replacing the jth component Kj from the link L = {K1 , . . . , Kn } by its (2, 1) cable, then (L )(t1 , . . . , tn ) = (T + T 1 ) (L)(t1 , . . . , tj 1 , t2 , tj+1 , . . . , tn ) j Knots, Links, and 4-Manifolds 23 where T = tj i=j ti k(Kj ,Ki ) . u e e e ! e e e e u e e e e e e ! y gg g e g g g g # L+ L L0 Figure 8 The Alexander polynomial L in Theorem 1.9 is just . L (t2 , . . . , t2 ) = (L)(t1 , . . . , tn ) 1 n . where the symbol = denotes equality up to multiplication by 1 and powers of the variables. Recall the construction of E(1)L . If L = {K1 , . . . , Kn } is an (n 2)component ordered link in S 3 and ( j , mj ) denotes the standard longitudemeridian pair for the knot Kj , we let L : 1 (S 3 \ L) Z denote the homomorphism characterized by the property L (mj ) = 1 for each j = 1, . . . , n. De"ne ML to be the 3-manifold obtained by performing L ( j ) surgery on each component Kj of L. Then, in ML S 1 we use the smooth tori Tmj = mj S 1 to construct the n fold "ber sum n E(1)L = (ML S 1 )#Tmj =F j=1 E(1), the "ber sum being performed using the natural framings Tmj D2 of the neighborhoods of Tmj = mj S 1 in Mj and the neighborhoods F D2 of an elliptic "ber in each copy of E(1) and glued together so as to preserve the homology classes [pt D2 ]. It is amusing to note that this later condition is unnecessary in this special situation. For, since E(1) \ F has a big diffeomorphism group, we can "ber sum in the E(1) with any gluing map and end up with diffeomorphic 4-manifolds. Now let be that function which associates to every ordered link L of n-components, the polynomial (L)(t1 , . . . , tn ) = SW E(1)L (t2 , . . . , t2 ), 1 n with tj = exp(2[Tmj ]). We show that satis"es the above stated axioms. However, there is a small obstruction to doing this in a straightforward 24 Ronald Fintushel, Ronald J. Stern manner: SW X is only de"ned when b+ > 1 and when L has but one component E(1)L has b+ = 1. Although our Theorem 1.9 is stated only for n 2, the axioms insist that we consider 1-component links. We overcome this problem as follows. Given an ordered n-component link L we always "ber sum in the K3-surface (rather than E(1)) to ML S 1 along Tm1 . In the case that n 2, the resulting manifold, which we temporarily denote by E(2, 1)L , has SW E(2,1)L = (t1 1/2 t1 1/2 ) SW E(1)L We shall complete the proof of Theorem 1.9 by showing that (L)(t1 , . . . , tn ) = SW E(2,1)L (t2 , . . . , t2 ) n 1 t1 t 1 1 satis"es all the axioms. Axiom 1 is clear. For Axiom 2 note that if L is the unknot, then E(2, 1)L is the K3surface, so that (L)(t) = SW E(2,1)L (t2 )/(t t 1 ) = 1/(t t 1 ). To verify Axiom 3, consider an (n 2)-component link L with components K1 , . . . , Kn . We need to see that the only possible basic classes of E(2, 1)L are the classes Tmi (which are identi"ed in E(2, 1)L with the "ber classes Fi ). A Mayer-Vietoris sequence argument shows that H2 (E(2, 1)L ) im( ) G = where n 1 H2 (E(2) \ F ) 1 H2 (E(1) \ F ) H2 ((S 3 \ L) S 1 ) H2 (E(2, 1)L ) G n and G is isomorphic to the kernel of 1 H1 (T 3 ) H1 ((S 3 \ L) S 1 ). Now H2 (E(2) \ F ) 2E8 2H 3(0) and H2 (E(1) \ F ) E8 3(0) = = where H denotes a hyperbolic pair. Each copy of E8 is represented by eight 2-spheres in the usual con"guration, say W , with W = (2, 3, 5), the Poincar homology sphere. Since W embeds in E(2), whose only basic e class is 0, and since (2, 3, 5) has positive scalar curvature, a rudimentary gluing formula implies that each basic class of E(2, 1)L is orthogonal to the image of the E8 summands. Each of the two hyperbolic pairs H is the homology of a nucleus in E(2) \ F and is generated by a torus of self-intersection 0 and a sphere of self-intersection 2 which intersect at one point. For any basic class of E(2, 1)L , the adjunction formula Knots, Links, and 4-Manifolds 25 implies 0 2 + | |; so is orthogonal to . Also, + is represented by another torus of self-intersection 0; so k is in fact orthogonal to H. Furthermore, each of the summands (0) in H2 (E(1)\F ) and H2 (E(2)\F ) is represented by a torus in the boundary of the tubular neighborhood of F , and each of these tori is glued to a torus in (S 3 \ L) S 1 in E(2, 1)L . It follows that the only possible basic classes lying in the image of in fact lie in the image of H2 ((S 3 \ L) S 1 ). This image is spanned by the classes of the tori Tmi , i = 1, . . . , n and the tori Vj , j = 1, . . . , n 1 where Vi is the boundary of a tubular neighborhood of Ki in S 3 . The nonzero elements of G determine classes in E(2, 1)L with nonzero Mayer-Vietoris boundary. These are generated by classes i , i = 1, . . . , n and j , j = 1, . . . , n 1. A representative of i is formed as follows. Let Si denote intersection of a Seifert surface for the knot Ki with the link exterior. The intersection of Si with Vj (j = i) consists of k(Ki , Kj ) copies of mj . Each of these is glued to a circle on the "ber F of the corresponding E(1) or E(2), and this circle bounds in the elliptic surface. The same is true for the longitude of the knot Ki . The result represents i . Note that i Fj = ij and i Vj = 0 for each j = 1, . . . , n. The generators j are constructed by starting with an arc Aj in S 3 \ L from a point on Vn to a point on Vj . The boundary of Aj S 1 consists of two circles, and each is identi"ed with a circle in H2 (E(1) \ F ) or H2 (E(2) \ F ). In the elliptic surfaces, these circles bound vanishing cycles, disks of self-intersection 1. Thus i is represented by a 2-sphere. We have j Fi = 0 for all i, and j Vi = ij . Suppose we have a basic class n n 1 n n 1 = i=1 ai Fi + j=1 bj V j + k=1 ck k + =1 d Since Fi is a torus of self-intersection 0, the adjunction inequality implies that Fi = 0, i.e. that ci = 0. Similarly, Vj is a torus of self-intersection 0; so 0 = Vj = dj . Also +V is represented by a torus of self-intersection 0; thus 0 = ( + V ) = = b . Axiom 4 is clear. Axiom 5 is veri"ed in the spirit of Theorem 1.5. However, we must "rst construct an auxiliary manifold E(1)L as follows. In E(2, 1)L , L an n-component link, let F1 , . . . , Fn be tori with the torus F1 the elliptic "ber in the K3-surface and, for j 1, Fj the elliptic "ber in the (j 1)st copy of E(1). (Note that Fi = Tmi in E(2, 1)L .) Now perform (n 1) internal "ber sums, identifying F1 with F2 , a parallel copy of F2 with F3 , and so on. The homology classes represented by the Fj in E(1)L are all equal, and we denote this homology class by [F ]. Let t = exp(2[F ]). It follows from the gluing formula Theorem 2.3 that SW E(1)L (t) = SW E(2,1)L (t, . . . , t) (t1/2 t 1/2 )(2n 2) , 26 Ronald Fintushel, Ronald J. Stern or by de"ning SW E(1)L (t) = SW E(2,1)L (t, . . . , t), SW E(1)L (t) = SW E(1)L (t) (t1/2 t 1/2 )(2n 2) . Suppose now that L+ , L , and L0 are links which coincide (except possibly for the numbering of the components) outside a ball, and inside this ball have the form depicted in Figure 8. Furthermore, assume L has n components. Then, as in the proof of Theorem 1.5, there is a nullho mologous torus T in E(1)L so that E(1)L+ is the result of a (1/1)-log transform on T. By the log transform formula (Theorems 2.4 and 2.5), SW E(1)L = SW E(1)L + SW E(1)L + (0/1) . There are two cases. For the "rst case, the two strands of L+ are from distinct components Ki1 , Ki2 of L+ . The manifold E(1)L (0/1) is obtained as follows: Perform surgeries on the link components {Ki } of L with surgery coef"cient L ( i ) on Ki and perform 0-surgery on the unknotted component U which links Ki1 and Ki2 as shown in Figure 9. ( i1 ) ( i2 ) L0 ( 0 ) mi1 mi2 ' $ & Ki1 Ki2 m0 m0 K0 0 % Figure 9 Let M 3 be the resulting 3-manifold. Next form E(2, 1)L (0/1) = (M 3 S 1 )#Tm1 =F1 E(2)#Tm2 =F2 E(1)# #Tmn =Fn E(1). (4.1) Finally, E(1)L (0/1) is obtained from E(2, 1)L (0/1) by performing n 1 internal "ber sums, as described above. If we slide the handle corresponding to Ki2 over the handle corresponding to Ki1 then we obtain a new Kirby calculus description of M 3 : it is obtained from surgery on the link L0 with surgery coef"cients again given by the L0 ( i ), as in Figure 9. (Note that if the new component is called K0 then for j = i1 , i2 , the linking number kL0 (K0 , Kj ) = kL (Ki1 , Kj ) + kL (Ki2 , Kj ); so the total linking number for the longitude 0 is L0 ( 0 ) = L ( i1 ) + L ( i2 ) 2 kL (Ki1 , Ki2 ) which is exactly the framing which is given to K0 by the handle slide.) Knots, Links, and 4-Manifolds 27 Now the link L0 has n 1 components. We see that the difference be tween the constructions for E(1)L (0/1) and E(1)L0 is that an extra copy L and then an extra internal of E(1) needs to be "ber-summed into E(1) 0 "ber sum needs to be performed on the result, in order to get E(1)L (0/1). Theorem 2.1 and Theorem 2.3 then imply that SW E(1)L (0/1) (t) = (t1/2 "ber sum with E(1), and (t1/2 t 1/2 )2 comes from the extra internal "ber sum. We have: SW E(1)L (t) + SW E(1)L (t) = 1/2 + (t t 1/2 )(2n 2) SW E(1)L (t) + SW E(1)L (0/1) (t) = (t1/2 t 1/2 )(2n 2) = SW E(1)L (t) + t 1/2 )3 SW E(1)L (t), where one factor (t1/2 t 1/2 ) comes from the extra 0 (t1/2 t 1/2 )3 SW E(1)L (t) 0 (t1/2 t 1/2 )(2n 2) (t1/2 t 1/2 ) SW E(1)L (t) 0 = SW E(1)L (t) + (t1/2 t 1/2 )(2n 4) 0 = SW E(1)L (t) + (t1/2 t 1/2 ) SW E(1)L (t), as desired. For the second case, the two strands of L+ are from the same component (say the j th ) of L+ ; so L0 has (n + 1) components. Then E(1)L (0/1) is obtained by "rst performing surgeries on the components {Ki } of L with surgery coef"cient L ( i ) on Ki and then performing 0-surgery on an unknotted circle which links Kj twice geometrically and 0-times algebraically as in Figure 10. ( j ) e e ' e e & e e ! mj $ K s e e e 0 % e ! m ; An e e Kj Figure 10 K This gives a 3-manifold, M 3 . Then we form E(2, 1)L (0/1) as given by (4.1), and E(1)L (0/1) is obtained from this by performing (n 1) internal "ber sums. Denote the components of L0 by K1 , . . . , Kj 1 , K , K , Kj+1 , . . . , Kn . 28 Ronald Fintushel, Ronald J. Stern Let Lj denote the 2-component link Lj = {K , K }. It follows from Hoste's theorem (2.6) that M 3 may be obtained from the sewn-up link exterior s(Lj ; An ) by further surgering the Ki , i = j with framing L ( i ) (where n = L ( j ) + 2 k(K , K )). Again see Figure 10. Because L0 ( ) + L0 ( ) = i=j k(Ki , Kj ) + 2 k(K , K ) = n, and also L ( i ) = L0 ( i ), i = j, the discussion preceding Proposition 2.7 relates the Seiberg-Witten invariant of E(2, 1)L (0/1) with the Seiberg-Witten invariant of E(2, 1)L0 . We need to keep in mind, however, that because L0 has n+1 components, there is an extra "ber sum with E(1) in the construction for E(2, 1)L0 . Thus, SW E(2,1)L (0/1) = (tj 1/2 tj 1/2 ) SW E(2,1)L0 |t =t =tj Furthermore, in E(1)L (0/1) there is one more internal "ber sum than in L ; so E(1) 0 SW E(1)L Thus SW E(1)L (t) = + (0/1) (t) = (t1/2 1 SW E(1)L (t). 0 t 1/2 ) SW E(1)L (t) + = SW E(1)L (t) + SW E(1)L (t1/2 t 1/2 )(2n 2) (0/1) (t) (t1/2 t 1/2 )(2n 2) = SW E(1)L (t) + = SW E(1)L (t) + 1 SW E(1)L (t) (t1/2 t 1/2 ) 0 (t1/2 t 1/2 )(2n 2) (t1/2 t 1/2 ) SW E(1)L (t) 0 (t1/2 1/2 t 1/2 )2n 0 = SW E(1)L (t) + (t t 1/2 ) SW E(1)L (t), completing the proof of Axiom 5. Finally, to verify Axiom 6 we "rst note that E(2, 1)L is obtained from E(2, 1)L by performing an order 2 log transform on the torus Tj = Kj S 1 j is the core of the surgered Kj in ML . The homology class of the where K resulting meridian mj is twice that of mj so that [Tmj ] = 2[Tmj ]. The log transform formulas of [FS1], then state that SW E(2,1)L (t1 , . . . , tj 1 , tj , tj+1 , . . . , tn ) = (tj 1/2 1/2 + tj ) SW E(2,1)L (t1 , . . . , tj 1 , t2 , tj+1 , . . . , tn ). j Knots, Links, and 4-Manifolds 29 The result now follows since [Tj ] = [Tj ] + i=j k(Kj , Ki )[Ti ]. 5 Examples with b+ = 1 In this section we shall discuss examples which have b+ = 1. For such manifolds, the Seiberg-Witten invariant depends on a choice of metric and selfdual 2-form as follows. Let X be a simply connected oriented 4-manifold 2 with b+ = 1 with a given orientation of H+ (X; R) and a given metric X + g. Since bX = 1, there is a unique g-self-dual harmonic 2-form g 2 2 H+ (X; R) with g = 1 and corresponding to the positive orientation. Fix a characteristic cohomology class k H 2 (X; Z). Given a pair (A, ), where A is a connection in the complex line bundle corresponding to k and a section of the bundle W + of self-dual spinors for the associated spin c structure, the perturbed Seiberg-Witten equations are: DA = 0 + FA (5.1) + = q( ) + i + where FA is the self-dual part of the curvature of A , DA is the twisted Dirac operator, + , is a self-dual 2-form on X, and q is a quadratic function. Write SWX,g, + (k) for the corresponding invariant evaluated on the class i k (= 2 [FA ]). As the pair (g, + ) varies, SWX,g, + (k) can change only at those pairs (g, + ) for which there are solutions of (5) with = 0. These solutions occur for pairs (g, + ) satisfying (2 k + + ) g = 0. This last equation de"nes a codimension 1 subspace (`wall') in H 2 (X; R). The point g lives in the double cone CX = { H 2 (X; R)| > 0}, and, if (2 k + + ) g = 0 for a generic + , SWX,g, + (k) is well-de"ned, and its value depends only on the sign of (2 k + + ) g . A useful lemma, which follows from the Cauchy-Schwarz inequality (see [LL]) is: Lemma 5.1 Suppose and are nonzero elements of H 2 (X; R) which lie in the closure of the same component of CX . Then 0 with equality if and only if = for some > 0. It follows from this lemma that that SWX,g, + (k) depends only on the component of CX which contains the g-self-dual projection of 2 k + + . Fur+ thermore, if the gi -self-dual projections of 2 k + i lie in different compo+ + nents of CX and are oriented so that (2 k+ 1 ) g1 > 0 > (2 k+ 0 ) g0 , then SWX,g1 , + (k) SWX,g0 , + (k) = ( 1) 2 (k)+1 1 0 1 (5.2) 30 Ronald Fintushel, Ronald J. Stern where (k) = 1 (k 2 (3sign + 2e)(X)) is the formal dimension of the 4 moduli spaces. Thus, as the pair (g, + ) is varied, there are exactly two values of SWX,g, + (k). Furthermore, in case b 9, for any "xed a H2 (X; Z) with (a) = a2 + b 9 0, the self-dual projections of 2 a all lie in the same component of CX ; so, if a = k is characteristic, then for small enough perturbations, the Seiberg-Witten invariants agree, independent of metric [KM,S2]. Suppose that X contains a smooth essential torus T of self-intersection + 0. By Lemma 5.1, the class [T ] orients CX by declaring CX to be the component of CX which contains classes with [T ] > 0. Denote the other com ponent by CX and the corresponding Seiberg-Witten invariants by SW ; X + i.e. SWX (k) = SWX,g, + (k) where the g-self-dual projection of 2 k + + has positive intersection with [T ], and we de"ne SW (k) similarly. The X [T ] -restricted Seiberg-Witten invariants are de"ned to be SW = X,T k [T ]=0 SW (k) tk X where the variables satisfy ta+b = ta tb as above, and ta is identi"ed with exp(a). When k2 0 and k [T ] = 0, Lemma 5.1 implies that k = [T ]. In particular, if b 9 and (k) 0, then k = [T ] if k [T ] = 0. X Lemma 5.2 Let X be a simply connected smooth 4-manifold with b+ = 1, X and suppose that X contains a smooth homologically nontrivial torus T of self-intersection 0. Let t = exp(2[T ]). Then (t1/2 t 1/2 ) SW + = (t1/2 t 1/2 ) SW . X,T X,T Proof The coef"cient of exp(k) in (t1/2 t 1/2 ) SW + is X,T c+ (k) = SW+ (k [T ]) SW+ (k + [T ]). X X Since k [T ] = 0, we have (k [T ])2 = (k + [T ])2 ; so the wall-crossing formula (5.2) implies that c+ (k) = SW (k [T ]) SW (k + [T ]), the X X coef"cient of exp(k) in (t1/2 t 1/2 ) SW . X,T For example, consider the case of the rational elliptic surface E(1) and its "ber class [F ]. For a Kahler metric g on E(1), the Kahler form is selfdual, and since F is a complex curve, [F ] > 0. So the self-dual projection of [F ] is a positive multiple of , and we see that the small-perturbation + component of CE(1) is CE(1) for n[F ], n > 0, and similarly is CE(1) for n[F ], n < 0. Since E(1) carries a metric of positive scalar curvature, this means SW+ (n[F ]) = 0 for n > 0, and SW (n[F ]) = 0 for n < 0. E(1) E(1) The wall-crossing formula (5.2) implies that (up to an overall sign) SW ((2n + 1)[F ]) SW+ ((2n + 1)[F ]) = 1. E(1) E(1) Knots, Links, and 4-Manifolds 31 Hence (up to that same sign) SW ((2n + 1)[F ]) = E(1) In other words, SW E(1),F = where t = exp(2[F ]). Similarly, 1, 0, n 0 n < 0. t(2n+1)/2 n=0 SW + E(1),F = n=0 t (2n+1)/2 . Notice that both and (t1/2 t 1/2 ) SW E(1),F equal 1, as Lemma 5.2 demands. Let X be a simply connected smooth 4-manifold with b+ = 1, and X suppose that X contains a c-embedded oriented torus T . Suppose further that 1 (X \ T ) = 1. Then for any knot K in S 3 we can form XK , which is homeomorphic to X. Since [T ] = [Tm ] orients CXK , we have invariants SW and SW K . Our discussion of 3 applies in this case as well, once X X we have the analogues of the log transform and gluing formulas of 2. These formulas are also due to Morgan, Mrowka, Szabo, and Taubes cf. [S2,MMS]. Theorem ([MMST]) Consider the simply connected 4-manifolds X1 and X2 with b+ 1 = 1 and b+ 2 1. Suppose that X1 and X2 contain cX X embedded tori T1 and T2 . Then (t1/2 t 1/2 ) SW + E(1),F SW X1 #T1 =T2 X2 = SW 1 ,T1 SW 2 ,T2 (t1/2 t 1/2 )2 , if b+ 2 = 1 X X X SW 1 ,T1 SW X2 (t1/2 t 1/2 )2 , if b+ 2 > 1 X X where t = exp(2[T ]) and [T ] = [T1 ] = [T2 ] H2 (X1 #T1 =T2 X2 ). It follows from Lemma 5.2 that these formulas actually make sense. The restriction to SW 1 ,T1 is accounted for by the fact that b+ 1 #T =T X2 3, X X 1 2 and so any basic class in H2 (X1 #T1 =T2 X2 ) must be orthogonal to [T ]. In order to state the internal "ber sum formula, let X be a simply connected 4-manifold with b+ = 1. Suppose that X contains a pair of cX embedded tori T1 , T2 , and let XT1 ,T2 denote the internal "ber sum. Denote by SW 1 ,T2 the ([T1 ], [T2 ]) -restricted Seiberg-Witten invariants of X, X,T SW 1 ,T2 = X,T k [Ti ]=0 i=1,2 SW (k) tk . X 32 Ronald Fintushel, Ronald J. Stern Theorem ([MMST]) Let X be a simply connected 4-manifold with b+ = X 1. Suppose that X contains a pair of c-embedded tori T1 , T2 representing homology classes [T1 ] and [T2 ]. SW XT1 ,T2 = (SW 1 ,T2 )| X,T 1 = 2 (t1/2 t 1/2 )2 where i = exp([Ti ]) and t = exp(2[T ]). Of course we also need the log transform formula: Theorem ([MMS], [T6], [MT]) Let Y be a smooth 4-manifold with b+ = 1, and suppose that Y contains a nullhomologous torus T . Let be the homology class of T0 in Y (0/1). For each characteristic homology class H2 (Y ; Z) and p = 0, SW (p/q) ( ) = pSW ( ) + q Y Y SWY (0/1) ( + 2i ). i= Now the arguments of 3 go through verbatim to give: Theorem 5.3 Let X be a simply connected smooth 4-manifold with b+ = X 1, and suppose that X contains a c-embedded oriented torus T . Suppose further that 1 (X \ T ) = 1. For t = exp(2[T ]) the [T ] -restricted SeibergWitten invariants of XK are SW K ,T = SW K (t). X X,T As in the b+ 3 case, if X is a symplectic 4-manifold with b+ = 1 containing a symplectic embedded torus T of self-intersection 0 and K is a "bered knot, then XK is a symplectic 4-manifold. Conversely, if X is symplectic, choose a metric g for X so that the symplectic form is self-dual, and let X denote the canonical class. In [T1,T2], Taubes shows that for r << 0, we have SWX,g,r ( X ) = 1 and also that if SWX,g,r (k) = 0 then X k , with equality only when k = X . Now let b+ = 1, X and let T be an embedded torus of self-intersection 0, such that [T ] > 0. For example, this holds if T is symplectically embedded. Then for any k H2 (X; R), we have (2 k + + r ) [T ] < 0 for r << 0. This means that SWX,g,r (k) = SW (k). Hence: X Proposition 5.4 (Taubes) Let X be a symplectic 4-manifold with b+ = 1 containing an embedded torus T of self-intersection 0 such that [T ] > 0. Then SW ( X ) = 1 X and if SW (k) = 0, then X X k with equality only when k = X . Knots, Links, and 4-Manifolds 33 Lemma 5.5 Let X be a simply connected smooth 4-manifold with b+ = X 1, and suppose that X contains a c-embedded oriented torus T and that 1 (X \ T ) = 1. Suppose also that SW X#T =F E(1) = 0. Then there is a Laurent polynomial SX such that SW X,T = SX n=0 t (2n+1)/2 or SX n=0 t (2n+1)/2 . Proof The gluing formula implies that 1/2 t 1/2 )2 SW X#T =F E(1) = SW SW X,T E(1),F (t = SW (t1/2 t 1/2 ) X,T by Lemma 5.2 and the above calculation of SW E(1),F . Because the man+ = 3, its Seiberg-Witten invariant is a Laurent ifold X#T =F E(1) has b polynomial, SX = 0; so the lemma follows. Lemma 5.6 Suppose that b+ = 1 and H 2 (X), then for any pair X (g, + ) with + g = 0 and |r| >> 0, we have | SWX,g,r + ( ) SWX,g,r + ( ) | = 1. Proof Since FA = FA and q( ) = q( ), if (A, ) is a solution to the Seiberg-Witten equations for corresponding to (g, r + ), then (A, ) is a + ). Thus solution to the equations for corresponding to (g, r SWX,g, r + ( ) = ( 1)(sign+e)(X)/4 SWX,g,r + ( ). For |r| >> 0, the signs of ( 2 r + ) g and ( 2 + r + ) g are opposite. Thus the wall-crossing formula implies that | SWX,g, r + ( ) SWX,g,r + ( ) | = 1 and the lemma follows. In the symplectic case, the above lemma is essentially that of [MS, Prop.2.2]. Corollary 5.7 Let X be a symplectic 4-manifold with b+ = 1 containing a symplectic c-embedded torus T . Suppose also that SW X#T =F E(1) = 0. If K is not monic, then XK does not admit a symplectic structure. 34 d Ronald Fintushel, Ronald J. Stern Proof Write K (t) = a0 + j=1 aj (tj + t j ) with ad = 0. Suppose that XK admits a symplectic structure with symplectic form and canonical class . If [T ] = 0, it follows from Lemma 5.1 that [T ] = for some = 0. Such a symplectic form is clearly nongeneric and we may perturb it so that [T ] = 0. We may also assume that T is oriented so that [T ] > 0. The adjunction inequality of Li and Liu [LL, Thm.E] then implies that [T ] 0. The hypothesis that SW X#T =F E(1) = 0 and Lemma 5.5 imply that there is a Laurent polynomial SX = 0 such that, for t = exp(2[T ]), one of SW = SX X,T SW = SX X,T t(2n+1)/2 n=0 (5.3) (5.4) t (2n+1)/2 n=0 holds. Let be any class such that the coef"cient of exp( ) in SX is nonzero. There are "nitely many such classes; so we may list the integers m1 < < mr such that the coef"cient SX ( +mi [T ]) of exp( +mi [T ]) in SX is nonzero. Theorem 5.3 implies that if the case (5.3) holds, then SW K ( + (m1 2d + 1)[T ]) = ad SX ( + m1 [T ]) = 0 X and if the case (5.4) holds then SW K ( + (mr + 2d 1)[T ]) = ad SX ( + mr [T ]) = 0. X (5.6) (5.5) If < 0 then, by a result of Liu and Ohta and Ono, XK is the blowup of a rational or ruled surface [MS, Cor.1.4]. Every such surface has a metric of positive scalar curvature; so it follows from the wall-crossing formula that for each , SW K ( ) = 1 or 0. Equations (5.5) and (5.6) X imply in this case that ad = 1, i.e. that K is monic. Thus we may assume that > 0. This means that and [T ] both lie in the same component of the cone CXK ; so Lemma 5.1 implies that [T ] 0. Since we already have the opposite inequality, we must have [T ] = 0. By Lemma 5.6, | SW K ( + 2m[T ]) SW K ( 2m[T ]) | = 1 X X for each m. However [T ] > 0; so for m large, we have ( 2m[T ]) < . Thus Proposition 5.4 implies that SW K ( 2m[T ]) = 0 X for m >> 0. This means that SW K ( + 2m[T ]) = 1 for m >> 0. But X SW K ,T = SW K (t) and ( + 2m[T ]) [T ] = 0; so the case (5.3) X X,T must hold. Knots, Links, and 4-Manifolds 35 Again by Proposition 5.4, we have SW K ( ) = 0; so we may write X = + 2n[T ] where SW ( ) = 0 and |n| d, and again we have X m1 < < mr and (5.5). From (5.3): SW ( ) = X mi odd<0 SX ( + mi [T ]). But ( +(m1 2d+1)[T ]) = +(m1 +1 2(d n))[T ] because m1 < 0 (since SW ( ) = 0) and d n 0. Thus Proposition 5.4 X implies that m1 = 1 and n = d; so = + 2d[T ]. This means that 1 = SW K ( ) = ad SW ( ); so ad = 1, i.e. K is monic. X X As in the b+ > 1 case, if X contains a surface g of genus g disjoint from T with 0 = [ g ] H2 (X; Z) and with [ g ]2 < 2 2g if g > 0 or [ g ]2 < 0 if g = 0, then XK with the opposite orientation does not admit a symplectic structure. These results along with the blowup formula of [FS2] imply: Corollary 5.8 For any knot K in S 3 whose Alexander polynomial K (t) is not monic, the manifolds E(1)K admit no symplectic structure with either orientation, and this remains true even after an arbitrary number of blowups. The "rst examples of this sort were obtained by Szabo [S2]; in fact they are the manifolds E(1)Kk where Kk is the k-twist knot. For these examples (with T = F ) SW K E(1) SW + K E(1) ,T k = (kt (2k + 1) + kt 1 ) t(2n+1)/2 n=0 n=0 k ,T = (kt (2k + 1) + kt 1 ) t (2n+1)/2 Szabo computes the `small perturbation' invariant SW 0 K . 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