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Proceedings of 8th Gökova Geometry-Topology Conference, pp,1 – 12 Variations on Fintushel-Stern Knot Surgery on 2 0 4-manifolds 0 2 n Selman Akbulut 1 a J Abstract. We discuss some consequences Fintushel-Stern ‘knot surgery’ operation 7 on4-manifoldscomingfromitshandlebodydescription. Wegivesomegeneralizations 1 ofthisoperationandgiveacounterexample totheirconjecture. ] T G . 1. Introduction h at Let X be a smooth 4-manifold and K ⊂ S3 be a knot, In [FS] among other things m Fintushel and Stern had shown that the operation K → X of replacing a tubular K neighborhood of imbedded torus in X by (S3−K)×S1 could results change of smooth [ structure of X. In [A] an algorithm of describing handlebody of XK in terms of the 1 handlebody of X was described. In this article we will give some corollaries of this v construction, and present a counterexample to conjecture of Fintushel and Stern which 6 5 was overlookedin [A]. First we need to recall the precise description of XK: Recall that 1 the first picture of Figure 1 is T2×D2, and the second one is the cusp C (i.e. B4 with 1 a 2-handle attached along the trefoil knot with the zero framing). Clearly the cusp C 0 contains a copy of T2×D2. 2 0 / h 0 0 t a m -1 : v -1 i X r a Figure 1 1991 Mathematics Subject Classification. 57R65,58A05,58D27. 1SupportedinpartbyNSFfundDMS9971440. 1 AKBULUT In[FS]animbeddedtorusT2 ⊂X iscalledac-imbeddedtorusifithasacuspneighbor- hood in X, i.e. T2 ֒→C ֒→X as in Figure 1. Now let N ≈K×D2 be the trivialization of the open tubular neighborhood of the knot K in S3 given by the 0-framing. Let ϕ : ∂(T2×D2) → ∂(K ×D2)×S1 be any diffeomorphism with ϕ(p×∂D2) = K ×p, where p∈T2 is a point, then define: X =(X −T2×D2)⌣ (S3−N)×S1 K ϕ LetSpin (X)bethesetofSpin structuresonX,e.g. ifH (X)hasno2-torsionthen. c c 1 Spin (X)={a∈H2(X;Z)|a=w (TX)mod2} c 2 Recall that Seiberg-Witten invariant SW of X is a symmetric function X SW :Spin (X)→Z X c It is known that the function SW is nonzero on the complement of a finite set B = X {±α ,±α ,..,±a } which is called the set of basic homology classes classes. By setting 1 2 n α =0 and t =exp(α ), the function SW is usually written as a single polynomial 0 j j X n SW = SW (α )t X X j j Xj=0 Now if T is a c-imbedded torus in X , and [T] be the homology class in H (X ;Z) 2 K induced from T2 ⊂ X, and t = exp(2[T]), and ∆ (t) the Alexander polynomial of the K knot K (as a symmetric Laurent polynomial), then Fintushel and Stern [FS] theorem says: Theorem 1.1. SW =SW .∆ (t) XK X K Recallthatin[A]the algorithmofdrawingthehandlebodyofX fromX isdescribed K asfollows: Firstweidentify the core circles ofthe 1-handlesofthe handlebody ofS3−K core circles K Figure 2 Then when we see an imbedded cusp C in the handlebody of X as in the first picture ofFigure3,wechangeittothe secondpictureC˜ ofFigure3. This meansthatwechange oneofthe1-handlesofT2×D2insideofC tothe“slice1-handle”obtainedfromK#(−K) 2 AKBULUT (i.e. remove the obvious slice disk which K#(−K) bounds from B4), and connect the core circles of the knots K and −K by 2-handles as shown in Figure 3. More precisely, there is a diffeomorphismbetween the boundaries of manifolds C and C˜ of Figure 3, and theoperationX →X correspondstocuttingoutT2×D2 fromX andgluingthesecond K manifold of Figure 3 by this diffeomorphism(in the figureK is drawnas the trefoil not). -1 -1 0 -K 0 -1 0 -1 K Figure 3 Since the attaching circles of the other 2-handles of X could tangle to the boundary of T2×D2, it is important to indicate where the various linking circles of the boundary are thrown to by the diffeomorphism of Figure 3. This is indicated in Figure 6. Recall that since 3- and 4- handles of four manifolds are attached in the canonical way, to describe a 4-manifold it suffices to describe its 1- and 2- handle structure. So, in order to visualize (S3−K)×S1, which is obtained by by identifying the two ends of (S3−K)×I,itsufficestovisualize(B3−K )×I withitsendsidentified,whereK ⊂B3 0 0 isaproperlyimbeddedarcwiththeknotK tiedonit(therestisa3-handle). Thesecond picture of Figure 4 gives the handlebody picture of (B3−K )×I. 0 Identifyingtheendsof(B3−K )×I (upto3-handles)correspondstoattachinganew 0 1-handle,and2-handles,wherethe new2-handlesareattachedalongthe1-handlesofthe twoboundarycomponentsof(B3−K )×I asindicatedinFigure5(morespecificallythe2- 0 handlesareattachedalongtheloopsconnectingthecorecirclesoftheknotcomplements). ToseethediffeomorphismofFigure3(i.e. toseethattheboundaryofthefirstpicture inFigure 5is standard),wesimply removethe dotonthe “slice”1-handle(i.e. turn itto a 2-handle) and slide it over the two 2-handles (as indicated by the arrows) in the first picture of Figure 5. This gives the second picture of Figure 5. After sliding 2-handles over each other of second picture of Figure 5, and cancelling the resulting S2×D2 with the 3-handle we obtain T2×D2. Also, to see the inverse boundary diffeomorphism from 3 AKBULUT K 0 Figure 4 0 0 0 0 Figure 5 T2 ×D2 to the first picture in Figure 5, we remove the dot from the 1-handle of the second picture of Figure 5 and perform the handle slides indicated by the arrows. Now putting these together in Figure 6 we see where the boundary diffeomorphism takes various linking circles of ∂(T2 ×D2). In particular the linking circle c of the 2- handle is thrown to the loop which corresponds the zero push-off of K in K#(−K). Figure 7 is the same as the second picture of Figure 6 except that the slice disk complement, which K#(−K) bounds, is drawn more concretely. Also note that, though our discussion is for general K, for the sake of concreteness, we have drawn our figures by taking K to be the trefoil knot. 2. Applications In[FS]FintushelandSternconjecturedthatifX isthe Kummer surfaceK3,thenthe associationK →X gives an injective map from the isotopy classes of knots K in S3 to K 4 AKBULUT b b a c c 0 0 0 a Figure 6 0 0 b 0 a Figure 7 the set of diffeomorphism classes of smooth structures on X. The following provides a counterexample to this conjecture: Theorem 2.1. XK =X−K Proof. There is an obvious self-diffeomorphism of the second picture in Figure 3 (i.e. (S3−K)×S1) exchangingroles of K and −K ; i.e. the diffeomorphisminduced by 180o 5 AKBULUT rotation of R3 around the y-axis. It is easily check that this diffeomorphism extends to the interior of (S3−K)×S1, implying the desired result. The following says that all smooth manifolds X obtained from X by using from dif- K ferent knots K become standardafter single stabilization. This resultwas independently observed by Auckly [Au]. Theorem 2.2. X #(S2×S2)=X#(S2×S2) K Proof. X #(S2×S2) isobtainedby surgeringanyhomotopicallytrivialloop(with the K correct framing). We choose to surger X along the trivially linking circle of its slice K 1-handle (the knot K#(−K) with a dot). This corresponds to turning the slice 1-handle to a 0-framed 2-handle (i.e. replace the dot with 0 framing), hence we are free to isotop the attaching circle of this 2-handle to the standard position as indicated in Figure 5. In particular, this makes the boundary diffeomorphism between the two handlebodies of Figure 5 extend to a 4-manifold diffeomorphism. So, Surgered X is diffeomorphic to K the surgered X which is X#(S2×S2). Notethatthoughweindicatedthe argumentforthe trefoilknotK inourpictures,the sameapplies for ageneralK (i.e. in Figure5 the knotK#(−K)unknots in the presence of the 2-handles) NoticethatX canbeviewedasX =X =(X−T2×D2)⌣ (S3×S1−U),where K K f ϕ U is an open tubular neighborhood of an imbedded torus f : S1×S1 → S3×S1, with Image(f)=K×S1. The map f is induced fromthe obviousimbedding K×I →S3×I by identifying the ends. More generally to any concordance s from K to itself, we can associate an imbedding of a torus f :S1×S1 ֒→S3×S1, hence getting map s Diffeomorphism classes of C(K)−→ (cid:26) smooth structures onX (cid:27) definedbys→Xs,whereC(K)={s:S1×I ֒→B3×I |s|S1×0 =s|S1×1 =K }. Itisan interestingquestionthathowthediffeomorphismclassofX dependsontheconcordance s class s of K? The following says that the above map is not injective. Theorem 2.3. If K is the trefoil knot, there is s∈C(K#(−K)) such that X =X s Proof. Let s be the concordance of K#(−K) to itself, given by connected summing the twoobviousslicediscswhichtwocopiesofK#(−K)boundinB4 asinthesecondpicture of Figure 8. Now if we use the product concordance s from K#(−K) to itself, i.e. the 0 first picture of Figure 8, our algorithm says that changing the cusp neighborhood by (S3−K#(−K))×S1 isgivenbythehandlebodyofFigure9,whichisthesameasFigure 10 (where the slice 1-handle is drawn as a usual handlebody). Whereas if we use the concordances, describedabove,we getFigure11. Byanisotopywe see that Figure11is diffeomorphic to Figure 12 which is diffeomorphic to Figure 13, and Figure 13 is isotopic toFigure14. ByhandleslidesindicatedinFigures14and15weobtainthesecondpicture of Figure 15. By cancelling a 1-and 2- handle pairs we get the first picture of Figure 16. 6 AKBULUT K # -K K # -K Figure 8 Then by a 2-handle slide, and cancelling an unknotted 0-framed 2-handle by a 3-handle, we obtain the last picture of 16 which is the cusp C. So we proved C = C, but since s C ⊂X and every self diffeomorphism of ∂C extends to C we conclude X =X s Remark 2.1. This theorem says that taking different elements s ∈ C(K) can result changing the smooth structure of X . For example, if take any c-imbedded torus in a s smooth manifold X with SW 6= 0, and K is the trefoil knot, and if s ∈ C(K#(−K)) X 0 is the product concordance,then by Theorem 1.1 SWXs0 =SWXK#(−K) =SWX.∆K#(−K) 6=SWX hence X 6=X. But on the other hand Theorem 2.3 says that there is s∈C(K#(−K)) s0 with X = X, so X 6= X . In particular, this shows that the concordances s and s s s0 s 0 are different. This gives a hope the that hard to distinguish knot concordances might be distinguished by the Seiberg-Witten invariants of the associated manifolds X . s Remark 2.2. Lets∈C(K),andf :S1×S1 ֒→S3×S1bethecorrespondingimbedding s One can ask whether Theorem 1.1 generalizes to SW = SW .∆ (t)?, where ∆ (t) is Xs X s s the Alexander polynomial associated to this imbedding. 2.1. A twisted version of XK Another version of the operation X → X that was previously introduced in [CS], K which, in a sense, is the square root of this operation: Let K is an invertible knot, i.e. an orientation preserving involution τ : R3 → R3 (e.g. 1800 rotation) restricts to K as an involution with two fixed points, and let N be the open tubular neighborhood of K. Then we can form the following S3−N bundle over S1: (S3−N)×˜S1 =(S3−N)×[0,1]/(x,0)∼(τ(x),1) 7 AKBULUT Define a D2-bundle over the Klein bottle K2 by C∗ =S1×D2×[0,1]/(x,0)∼(τ(x),1). Then(S3−N)×˜S1 andC∗ havethesameboundaries,andsoifX isasmooth4-manifold with C∗ ⊂X, we can construct X∗ =(X −C∗)⌣ (S3−N)×˜S1 K ϕ where ϕ : ∂C∗ → ∂(K ×S1)×˜S1 is a diffeomorphism with ϕ(p×∂D2) = K ×p. The operation X → X∗, is a certain generalization of the Fintushel and Stern operation K X → X done using a ‘Klein bottle’ instead of a torus. This operation was previously K discussed in [CS]. By using the previous arguments one can see see that the handlebody picture of the operation X →X∗ is given by Figure 17. The first picture of Figure 17 is K C∗ andthe secondis(S3−N)×˜S1. The restofX∗ isobtainedbysimplybydrawingthe K images of the additional handles under the diffeomorphism ϕ : ∂C∗ → ∂(S3−N)×˜S1. Forconvenience,inFigure17theimagesofthelinkingcirclesa,b,cunderϕareindicated. Now, call an imbedded Klein bottle K2 ⊂X c-imbedded Klein bottle, if K2 ⊂C∗ ⊂U ⊂X where U is either one of the manifolds of Figure 18, and π (U) → π (X) injects (notice 1 1 π (U) = Z ). Then it is easy to see that the obvious 2-fold cover X˜ → X contains 1 2 a cusp C (so it contains a c-imbedded T2), and the operation X → X∗ lifts to the K usual Fintushel-Stern knot surgery operation X˜ → X˜ (done using this T2). Hence if K SW 6=0and∆ (t)6=0,theoperationX →X∗ changesthesmoothstructureofX,i.e. X˜ K K X 6= X∗. For example, X can be a manifold with boundary, which is 2-fold covered by K a Stein manifold X˜ (so X˜ compactifies into a closed symplectic manifold which Theorem 1.1 applies). It is easy to check that the first manifold of Figure 18 is such an example. -1 0 0 0 -1 Figure 9 8 AKBULUT 0 -1 0 0 0 0 -1 Figure 10 -1 0 0 0 0 0 --11 Figure 11 References [A] S.Akbulut,A Fake Cusp and a Fishtail.TJM,vol23,no.1(1999), 19-31 [Au] D. Auckly, Families of four dimensional manifolds that become mutually diffeomorphic after one stabilization.GT/0009248 [CS] S.CappellandJ.Shaneson, On4-dimensional s-cobordisms.JDG,vol22(1985), 97-115 [FS] R.FintushelandR.Stern,Knots, links, and 4-manifolds.Invent.Math.134,no.2(1998), 363–400. MSU,Deptof Mathemtics,E.Lansing, MI,48824,USA E-mail address: [email protected] 9 AKBULUT -1 0 0 0 0 0 -1 Figure 12 -1 0 0 0 0 0 0 -1 Figure 13 10