mm; 20 o a LIBRARY Michigan State University This is to certify that the dissertation entitled Computations of Floer Homology and Gauge Theoretic Invariants of Montesinos Twins presented by Adam C. Knapp has been accepted towards fulfillment of the requirements for the Doctoral degree in Mathematics mam Major Professor’s Signature 03/17/03 Date MSU is an affirmative-action, equal-opportunity employer ‘-_.—~.—.—.—-—.—-—-—u—o—a—.—-—-—-—o-g-——A-—- PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 KIProi/AocaiPresIClRCIDateDueindd COMPUTATION S OF FLOER HOMOLOGY AND GAUGE THEORETIC INVARIANTS FOR MONTESINOS TWINS By Adam C. Knapp A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2008 ABSTRACT COMPUTATIONS OF FLOER HOMOLOGY AND GAUGE THEORETIC INVARIANTS FOR MONTESIN OS TWINS By Adam C. Knapp I compute the Lagrangian Floer cohomology groups of certain tori in closed simply connected symplectic 4-manifolds arising from Fintushel - Stern knot/ link surgery. These manifolds are usually not symplectically aspherical. As a result of the computation we observe examples where H F (Lo) ”—3 H F (L1) and Lo and L1 are smoothly and Lagrangian isotopic but L0, L1 are not symplectically isotopic. and In [Gil82], C. Giller proposed an invariant of ribbon 2-knots in S4 based on a “Conway cal- culus” of crossing changes in a projection to R3. In certain cases, this invariant computes the Alexander polynomial. Giller’s invariant is, however, a symmetric polynomial - which the Alexander polynomial of a 2—knot need not be. After modifying a 2-knot into 3 Mon- tesinos twin in a natural way, we show that Giller’s invariant is actually the Seiberg—Witten invariant of the exterior of the twin, glued to the complement of a fiber in E (2) To Melissa iii ACKNOWLEDGMENTS I would like to thank my advisor, Ronald F intushel. TABLE OF CONTENTS LIST OF FIGURES 1 Computations of Lagrangian Floer Homology 1.1 Introduction .................................... 1.1.1 Symplectic Manifolds .......................... 1 .1.2 Lagrangian submanifolds ........................ 1.1.3 Lagrangian Floer Homology ....................... 1.2 Construction ................................... 1.3 Calculation of F loer Cohomology ........................ 1.4 Example ...................................... 2 Gauge-theoretic Invariants of Twins 2.1 Introduction .................................... 2.1.1 Twins ................................... 2.1.2 Definition of the invariant for twins ................... 2.1.3 Construction of Knotted twins ..................... 2.1.4 Ribbon Knots and Twins ............ g ............ 2.1.5 Projections ................................ 2.1.6 Virtual knot presentation ........................ 2.1.7 Surgery diagrams ............................. 2.1.8 Giller’s Polynomial ............................ 2.2 The 4-dimensional Macarefia .......................... 2.2.1 3-dimensional Hoste Move ........................ 2.2.2 4-dimensional Hoste Move ........................ 2.2.3 4-Dimensional Crossing Change ..................... 2.3 Calculation of the Invariant for Certain Ribbon Twins ............ 2.3.1 Relation to Giller’s polynomial ..................... 2.3.2 The Class of Ribbon Twins ....................... BIBLIOGRAPHY iv (DOIHI—‘H 16 18 25 32 32 33 34 35 39 41 45 49 51 53 53 55 59 63 66 66 72 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 LIST OF FIGURES '7,- intersecting pairwise transversely in single points .............. Basis for monodromy on trefoil ......................... An isotope of '71 bounding meridians to K1 and K2 .............. Isotopy of 71 ................................... Isotopy of 72 ................................... Standard Twins .................................. Fox’s Roll Spin .................................. Trivial addition / deletion, Band slide, and Band pass ............. Local models for a family of double points and a triple point ........ Projection of a ribbon singularity and corresponding double points ...... A sphere and a torus with crossing information ................. A 2—knot with AA = 1 — 2t and AG = r2 — 1 + t2 .............. Neighborhood of the intersection of spheres in a twin ............. Crossings in virtual knots versus crossings in projections ........... Fixing a “bad” ribbon crossing by isotopy ................... Reidemeister Moves for Classical knots ......... > ............ “Reidemeister” Moves for Virtual knots .................... “Reidemeister” Move F for Twins ........................ Giller’s example and twin version ........................ Projection and Virtual Knot surgery diagrams for a twin and torus. Resolution of a knot crossing .......................... 3-dimensional Hoste move ............................ Round handle becomes a 1 and 2 handle .................... Band sum ..................................... Annulus B used for smoothing ......................... Ribbon intersection, smoothed ......................... Hoste Move on two tori ............................. Annulus B used for crossing change ...................... Two new sets of double points .......................... Isotopy in 3-dimensional picture ........................ Isotopy of surgered + crossing to -—- crossing .................. Isotopy of surgered — crossing to + crossing .................. Hoste move nearby 7' ............................... 26 27 28 30 34 38 41 42 43 43 44 45 46 47 47 48 49 50 51 52 53 54 55 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 Giller Twin TwG with highlighted crossing ................... I(Twc) = I(A) + (t — t—1)I(B) ......................... Isotopy of A and B from Figure 2.31 with highlighted crossings ....... I(Twc) = 1(0) + (t — t"l)I(D) + (t -— t—1)I(E) + (t — t“1)2I(F) ...... A twin in which both 2—knots are unknots ................... A negative crossing from Figure 2.34 . . . .l .................. I(Twu) = I(H) — (t — t‘l)I(J) ......................... I(J) = I(K) — (t — t’1)I(L) ........................... vi 68 68 68 69 70 70 70 71 CHAPTER 1 Computations of Lagrangian Floer Homology 1. 1 Introduction 1.1.1 Symplectic Manifolds An even dimensional manifold 1W?” together with a two form to is said to be symplectic if w is a nondegenerate and closed. The condition of non-degeneracy is equivalentlto w“ being a volume form on M. In particular, M must be oriented. The prototype of a symplectic n manifold is R2" given coordinates (3,331,) and symplectic form we 2 Z dxi /\ dyi. For example, all orientable 2-manifolds are symplectic with symp=lelectic forms equal to their volume forms. In fact, Moser’s stability theorem gives us that, up to diffeomorphism, symplectic forms to on closed 2-manifolds Z are determined entirely by f w. )3 Theorem 1.1.1 (Moser). Let M be a closed manifold with cut a smooth family of coho- mologous symplectic forms. Then there is a family of diffeomorphisms (pt : M —> [If so that (150 = id and 43:0”) 2 too. From a manifold topologist’s point of view, symplectic manifolds are interesting due to the fact that the symplectic structure has no local invariants. Hence, when we obtain invariants of symplectic manifolds, we get information about the global topology of the manifold. The statement of “no local invariants” is more precisely Darboux’s theorem: Theorem 1.1.2 (Darboux). Every symplectic form w on M is locally diffeomorphic to the standard form 0.20 on 1R2". A chart of such local diffeomorphisms is called a Darboux chart. Not all manifolds admit a symplectic form. Consider a closed symplectic manifold M. As w" is a volume form, it orients M and gives a non-zero class in the top dimensional cohomology group H 2"(M ;R) 53’ R. Thus w must represent a non-zero class in H 2(M ;IR) in order for [can] 2 [w]" to be non-zero. In particular, even dimensional spheres other than 5'2 cannot be symplectic. On a closed 4-dimensional manifold X, once we pick an orientation [X] 6 H4 (X ;Z) C H4(X;IR) we get a symmetric inner product QX on H2(X;IR) by a - ,6 = (a A fi)([X]). Let b+ and b- be the dimensions of the maximal positive and negative definite subspaces for this inner product, respectively. If X is to admit a symplectic form with this orientation, we must have b... _>_ 1. Now, @2 (the complex projective space (CP2 with the opposite orientation), has Qfiz _—_ (—1). Hence W2 does not admit a symplectic form with its given orientation. On the other hand, CP" with its complex orientation, admits a symplectic form To, the Fubini-Study form. In the local coordinates [2:0 : 21 : : 23-1.: 1 : 2,41 : : 2"] we can write In particular, (CF:2 is symplectic. An almost complex structure on N12" is a bundle map J : TIM —) TM with J2 = —I. If It! is symplectic, we say that w tames J if w(X, J X ) > 0 for any nonzero tangent vector X. A tame almost complex structure is said to be compatible with w if w(J X , J Y) = w(X, Y). If J is compatible with w, the symmetric 2-tensor g(X,Y) = w(X , J Y) is a Riemannian metric on M. The space of compatible almost complex structures is contractable and so one can make sense of invariants that depend on an almost complex structure such as the Chern classes g(X) of a symplectic manifold (X ,w). A Kahler manifold is a symplectic manifold (X ,w) which admits an integrable almost complex structure. That is, X has a set of complex charts with holomorphic transition functions so that J corresponds to multiplication by 2' = \/-——1. Complex projective spaces are Kahler as are all complex submanifolds of a Kahler manifold. In particular, smooth projective subvarities of CP" are Kahler. A technique which can be applied to a Kahler or symplectic manifold (X ,w) is the blow up at a point. This procedure replaces a point p E X with the pro jectivized tangent space to p. The procedure results in another smooth manifold and can also be done in the algebraic setting. In the 4—dimensional case, this is smoothly equivalent to connect summing X with c792. Suppose that we take a generic pencil of cubics on CPL). Then there are 9 points common to each cubic. If we blow up at each of these 9 points, we get the manifold E (1) = CP2#9U52 which admits a holomorphic map E(1) —r (CF1 where the fibers are the cubics in (CP2. Genericity of the pencil ensures that a generic fiber of this map will be a smooth elliptic curve. Now, take two of these generic fibers, F1, F2. The n-fold branch cover of E (1) along F1, F2 is the manifold E (n) The manifolds E (n) also admit holomorphic maps to (CF1 where the generic fiber is an elliptic curve and are examples of elliptic surfaces. There are other prototypes of symplectic manifolds: Suppose that we have a manifold 23" with cotangent bundle T‘E. Take a coordinate chart 2:,- : U —+ IR on U C 2. Over each point a: in U, the differentials (dxi),E form a basis of TD: and give coordinate charts (Ii, yi) : T*U —+ R2. We can then form the canonical 1-form n acan = E yidflii i=1 which is independent of the coordinates chosen. In this case the form wean = —dacan = 71 2 dz.- /\ (131,- is a symplectic form on T‘Z. These manifolds occur in Hamiltonian mechanics fiftiere the base is interpreted to be a configuration space and the fibers as momentum coordinates. Suppose that we are given two compact symplectic manifolds (X 1,w1), (X2,w2) of the same dimension each with a codimension 2 symplectic embedding of (Q, wQ). Assume that 62,- C X,- has trivial normal bundle for each 2'. Then for each embedding of Q, we have the symplectic neighborhood theorem: Theorem 1.1.3 (Symplectic Neighborhood). Suppose that there is an isomorphism (I) : V(Q1) —+ V(Q2) of symplectic normal bundles which covers a symplectomorphism d) : Q1 -+ Q2. Then a extends to a symplectomorphism w : N (Q1) —+ N (Q2) of tubular neighborhoods so that dill = (I) along Q1. As Q is codimension 2 in each Xi, the isomorphism type of the symplectic normal bundle is determined by its first Chern class. We will be interested in the case where the dimension of X,- is 4, where the Chern number is the same as the self intersection number of the surface Q. For each i, there are symplectic embeddings f,- : Q x 82m) —» X.- so that f,*(w,-) = wQ +d:r/\dy. Suppose 0 < 61 < 62 and let A(el, 62) = 32(62)\B2(€1). There is an area and orientation preserving map ¢ which exchanges the two components of 8A(61, 62). Then we form the fiber connected sum: X1#X2=1(X1\Q)U(X1\Q) Q 45 There are more general versions of this important construction in [Gom95] and in [MW94]. Our principal interest will be in gluing symplectic 4—manifolds along square zero symplectic tori such as we find in elliptic surfaces. Having constructed a number of symplectic manifolds with the above procedures, we arrive at the question: How flexible is a symplectic structure a) on a fixed manifold X? Or rather, what does the space Symp(X,w) of diffeomorphisms of X which preserve 0.) look like? Because it is non-degenerate, the symplectic form gives a bijection V —+ L(V)w between vector fields and 1-forms on X. Suppose that V; is a time—dependant vector field on a sym- plectic manifold (X,w). Cartan’s formula and dw 2' 0 gives £v,w :- L(X)dw + d (L(Vt)w) : d(L(Vt)w). We say that Vt is a symplectic vector field if d(t(Vt)w) = 0. i.e. if the corre- sponding 1-forms are closed. The flows of such time-dependant vector fields give elements in the identity component of Symp(X,w). A time dependant Hamiltonian vector field is a vector field VH, such that L(VH, )w = dHt for functions Ht : X ——i IR. i.e. the corresponding 1-forms are exact. The flows of such vector fields are a subset of the symplectic diffeomorphisms and are. called Hamiltonian symplectomorphism or Ham(X,w). If b1(X) = 0, then Ham(X,w) is the same as the identity component of Symp(X,w) as all closed forms are also exact. One of the natural, early conjectures concerning symplectic or Hamiltonian diffeomor- phisms was that of Arnold, who conjectured that the number of fixed points of a Hamiltonian diffeomorphism on X was always greater or equal to the rank of the singular homology of X. If we restrict to the Hamiltonian diffeomorphisms generated by generic time-independant vector fields this fact is comparatively easy. Hamiltonian vector fields VH correspond to smooth functions H : X —’ IR, modulo constants. Generic smooth functions are Morse and the Morse inequalities give that the number of critical points of H is at least as large as the rank of the homology of X. Then since L(VH )w = dH, VH must be zero exactly when H has a critical point. The general case is, of course, not this easy and was proved for symplectic manifolds with 7r2 = 0 by A. F loer in [F1087]. This result has been extended several times by a number of authors. 1 . 1 . 2 Lagrangian submanifolds A Lagrangian submanifold L of a symplectic manifold (ll/I2",w) is a n-dimensional subman- ifold of If! such that w|L E 0. Some examples include RP" C CP" and the graphs of closed l-forms within cotangent bundles with their canonical symplectic form. The closed condition which defines Lagrangians suggests that they should be fairly rigid objects, and this indeed seems to be the case. The question we seek to answer is: How rigid are they? First, similar to symplectic submanifolds, Lagrangian submanifolds have “nice” neigh- borhoods: Theorem 1.1.4. Let (It! ,w) be a symplectic manifold and L C .M a compact Lagrangian submanifold. Then there exists a neighborhood N (Lo) C T‘L of the zero section, a neighborhood N (L) C M of L, and a diffeomorphism 43 : N (Lo) —+ N (L) such that (25"(w) = ~dacan (where a is the canonical 1-form) and ¢|L0 E id. That is, Lagrangian submanifolds have neighborhoods which are symplectomorphic to the zero sections of their cotangent bundles. Therefore, when we look to dimension 4, a Lagrangian L is 2—dimensional and has self-intersection equal to —X(L). This has immediate consequences, for example the only closed orientable Lagrangian submanifolds of R4 are tori. We will assume that we are in 4-dimensional symplectic manifolds unless otherwise noted. We also require that Lagrangian submanifolds be orientable so that they represent homology classes. For a pair L0,L1 of Lagrangian submanifolds in a symplectic 4-manifold, there are several types of isotopy that we may consider: 0 Smooth isotopy. o Lagrangian isotopy. Lo and L1 are Lagrangian isotopic in X if there is an smooth isotopy of L0 to L1 through Lagrangians. ' o Symplectic / Hamiltonian isotopy. These are Lagrangian isotopies of L0 to L1 which ex— tend to Symplectic / Hamiltonian isotopies of the ambient manifold X. These concepts coincide when b1(X) = 0. Let us consider the case of Lagrangian tori in R4. Consider R4 E C2 and the unit circle S1 C (C. Then it is easy to verify that S1 x S1 C C2 is a Lagrangian torus. This can be reinterpreted as a “spun knotted torus”. i.e. Take a knot K lying away from the boundary in the half space 1R3+ and form 5'1 xK C (S 1 XR3+)/ ~'where (01, 3:1) ~ (02, 33;») iii 2:1 = .132 e are“. Now, (51 xn3+)/ «a R4 and 7r1(lR4\Sl xK) = 1r1(R3+\K) = «1(S3\K). Therefore, distinct knots K determine distinct isotopy types of “spun knotted tori” in R4. In [Lut95], K. Luttinger showed that, out of these isotopy classes, only the spun unknot can have a Lagrangian representative. His theorem is based on a kind of surgery which can be performed on Lagrangian tori which we describe here. We know that a Lagrangian torus L will have a trivial normal bundle since x(T2) = 0. Pick a trivialization of the normal bundle, L x D2, so that L x {pt} is Lagrangian for all pt 6 D2. Such a trivialization is canonical. In fact, since these tori are nullhomologous, there is another trivialization of the normal bundle coming from a push-off of the Lagrangian into the Seifert 3-manifold that it bounds. For Lagrangian tori in R4, Luttinger shows that the framing coming from Lagrangian push- offs and the nullhomologous framings coincide. Suppose that L is a Lagrangian representative of a the isotopy class of the spin of a knot K. Give L x D2 coordinates 2:,y E IIRQ/Z2 on L and polar coordinates (r, 0) on D2. We can then write w = d(r(cos(21r9)d:r + sin(21r6)dy)). The map Fm,n(:r,y,r,l9) = (a: + m9, 3/ + 710, r, 0) preserves the restriction of w to. 6(L x D2). Therefore, we can excise L x D2 and glue it back in using Fm," for any m, n. Whenever the knot K is not the unknot, nontrivial surgeries will result in manifolds with 1r1 nontrivial. However, there is a theorem due to Gromov with a refinement due to McDuff, which says Theorem 1.1.5 (Gromov, McDuff). If (X ,w) is a symplectic 4-manifold which is symplec- tomorphic to R“ at infinity, then (X ,w) is symplectomorphic to R4 blown up finitely many times. Since Luttinger’s surgery only affects a compact region, this theorem holds. Therefore, K could not have been nontrivial so there are no Lagrangian representatives of the isotopy classes of spins of nontrivial knots. Further restricting the isotopy classes of Lagrangians, Eliashberg and Polterovich showed in [EP96] that a Lagrangian R2 in R4 which is asymptotic to a Lagrangian plane at infinity is Hamiltonian isotopic to the planar embedding. This rules out the idea that you could have potentially chosen a Darboux chart centered at a point in the Lagrangian and performed a local knotting operation. i.e. a connect sum of a 2-knot in 5'". Another result from Eliashberg and Polterovich ([EP93]) shows that if L is a Lagrangian sphere or torus embedded in T’L, if L is homotopic to the zero section then it is smoothly isotopic to the zero section. The previous three results each speak of topological unknottedness of Lagrangians. Things become more subtle when the symplectic invariants come into play. For exam- ple, in [EP97] Eliashberg and Polterovich gave examples of Lagrangian tori in R4 which are Lagrangian isotopic but not Hamiltonian isotopic. These examples require the introduction of the Maslov class. Suppose that we have a loop 7 on a Lagrangian L C X with a trivial- ization of TX over '7. If X is simply connected (or if ['y] = 1 E 7r1(X)), such a trialization may be given by fixing a disc D which '7 bounds and trivializing the pullback of TX over D. Then the h‘laslov class ,u.('y) is a winding number of the Lagrangian planes TL around '7 relative to this trivialization. For a general manifold, this may only be well defined up to adding of spheres S to D which changes the Maslov class by c1 (X) - 5'. However, R4 is contractable so this issue is immaterial here. It is elementary to see that [1 gives a homo- morphism H1(L) -—-> Z and so can be thought of as an element of H1(L; Z). It is also easy to see that p is a Lagrangian isotopy invariant. Now, thinking of R4 as T‘RZ, we have the canonical 1-form a. Let f : T2 —+ R4 give the embedding of L. We call f *a E H1(T2) the symplectic area class of L. The torus L is monotone if f *a = An for /\ E IR+. Consider the following Lagrangian tori in 1R4 A2. C2: 2a 2b La,b : {(21932) E C2 I '21' = ———,|z2| -: _} 7T 7T The La“; are monotone while Lmb with a # b are not. Clearly any Lay, and Lgd are La- grangian isotopic and so have the same Maslov classes. They must have different symplectic area classes then — which are invariants of Hamiltonian isotopy. Thus these Lagrangian tori are smoothly isotopic through Lagrangians, but are not isotopic by Hamiltonian diffeo- morphisms. We can also look at Lagrangian tori in closed symplectic 4-manifolds and manifolds with nontrivial topology. The construction which we use for our theorem is originally due to Vidussi in [Vid06]. It will be described more throughly later. Roughly, however the construction starts with the Fintushel-Stern knot surgery manifolds. That is, a fibered knot K in S3 is chosen and zero surgery on that knot is performed to get 5'3 (K) Then with m a meridian, E(2)K = E(Q) # 51X 38(K) F=Slxm is a symplectic manifold. Vidussi pointed out that loops on the fiber of K cross S 1 corre- spond to Lagrangian tori. He showed that there are an infinite number of these Lagrangian tori which are smoothly nonisotopic inside E (2) K by computing relative Seiberg—Witten invariants and showing that the degrees go to infinity. In the related paper [F 804], Fintushel and Stern defined an integer valued smooth invari- ant (extracted from the relative Seiberg-Witten invariants) of the same sort of Lagrangian tori, distinguishing the smooth isotopy class of an infinite family. In fact, the tori in ques— tion are nullhomologous and hence have a Lagrangian push-off and Seifert pushoff framing. Unlike in R4, where Luttinger proved that the two coincide, Fintushel-Stern’s integer in- variant detects the difference between the two and is shown to take an infinite number of values. The previous two results can be interpreted as stating that Lagrangians may be smoothly knotted - without giving any symplectic results. To get such results, a new tool is needed. This is Lagrangian Floer Homology, introduced by Floer in [F1088]. The original purpose of Lagrangian F loer homology was to answer a variant of Arnold’s conjecture. The variant of the conjecture in question is this: Suppose that L C X is a Lagrangian submanifold of the symplectic manifold (X ,w). Then given a Hamiltonian diffeomorphism (b on X so that L and ¢)(L) are transverse, IL 0 q5(L)| is at least as large as the rank of the singular homology of L. 1.1.3 Lagrangian Floer Homology Floer’s solution to Arnold’s (Lagrangian) conjecture is an invariant of Lagrangian subman- ifolds, now called Lagrangian Floer Homology, which is well defined up to Hamiltonian isotopy. We will now sketch a basic “User’s Guide” to the construction. Suppose that (X, a), J) is a symplectic manifold together with an almost complex struc- ture and that L C X is a compact Lagrangian submanifold of X. Let ()5 be a Hamiltonian diffeomorphism of X so that L and ¢(L) intersect transversely. Then L and cb(L) will intersect in a finite number of points. Define the chain group CF(L,¢(L)) = 69 Z23: xeLn¢(L) With this definition in mind, we can see that any homology built out of this chain group will have at least as many generators as there are points of L H ¢(L). Thus, once a set of invariant homology groups are built, its rank will give a lower bound on IL 0 cb(L)|. Consider the moduli space M(:r,y) of maps 21 of the infinite strip S = {z E C | 0 s 39(2) 3 1} to X such that 0 lim {is =.rand lim {is =1, 9(;)—ooo ( ) ‘3(z)—t—oo ( ) J 0 '&(0 + iy) C L and 11(1 + iy) C (ML), and o 71 is J-holomorphic. i.e. with Jo the standard complex structure on (C, J odfl = dfioJo The first two conditions define what we refer to as a “topological Floer disc” connecting a: and y. Together with the third we define a “Floer disc” connecting a: and 3;. There is a free R-action on M(:1:, 3/) given by precomposing with a reparameterization of the infinite strip 7‘ : a: + iy —+ a: + i (y + r). Also, we can consider maps u from the unit ball around zero D2 C (C by a conformal map S ——> D2 \ {:lzi} which sends ((‘9D2 fl {%(2) g 0}) \ {ii} to {0 + iyly e IR} and (or)2 n {§R(z) 2 0}) \ {ii} to {1 + iyly e R}. Homotopy classes of topological Floer discs form a space 1r2 (X, L) that has an action of 7r2(X) on it. These split M(:r,y) into sets of connected components divided by their homotopy class. For a moment, assume that 7T2(X) = 0. Then for a generic choice of J, M(a:, y) is a smooth orientable manifold of dimension g(x, y) where g(x, y) a Maslov class defined as follows: Take a topological disc in the (unique) homotopy class connecting a: and y. Trivialize the (pullback of the) tangent bundle of X over this disc. Then p(:r,y) is i the winding number of T'L and T'¢(L) with respect to this trivialization. A more precise definition can be found in [R893]. When 7T2(X , L) # 0, we must specify the homotopy class in order to define p(:z:,y). If D1,D2 are two such homotopy classes with [6D1] = [8D2] E ,7r1(L), then D1 — D2 is a element of 7rg(X). It is elementary to compute that up, (:r, y) — p02 (as, y) = c1(X)[01 — D2]. Define 6.7: = Z # (Mum/1133):; y E L 0 ¢(L) M1331) = 1 where # (M(:r, y) / IR) is the point count modulo 2 of the 0-dimensional space. Compactness arguments show that this sum is finite. Gromov showed in [Gro85] that when bounds on symplectic energy (integral of the symplectic form over the surface) are satisfied, the spaces M (3:,y) can be compactified by adding in configurations of nodal curves to the boundary. There are three types of degeneration, which in a high dimensional moduli space may each happen together and / or more than once: 0 Breaking into trajectories. i.e. A Floer disc from a: to z may degenerate into a Floer 10 disc from a: to y and one from y to z. 0 Bubble sphere off interior. i.e. A Floer disc D1 from a: to 3/ may degenerate into another Floer disc D2 from :r to y plus a J-holomorphic sphere S. In this case, Hal-16,31): #02 + 61(X)[5]- 0 Bubble disc off boundary. i.e. A F loer disc D1 from x to y may degenerate into another Floer disc D2 from :r to y plus a J-holomorphic disc D3 with‘boundary entirely on L or (ML). As J-holomorphic curves are necessarily symplectic, any non-constant J-holomorphic spheres S or J-holomorphic discs D with boundary on L have / w, / w > 0. There- S D fore, when 7T2(X, L) = 0 or / w|,,2( X, L) :— 0 there are no non-constant spheres or discs with boundaries entirely on one Lagrangian.‘ For a moment assume that / col,“ X, L) E 0. Now, for 6 to be a differential, we must have 02 E 0. With the assumption that / wl,,2(' X, L) E 0, the boundary of a compactified 1-dimensional moduli space M (x,z) /R of F loer discs consists only of products of zero dimensional moduli spaces M (x,y) /IR x .M(y, z)/IR for y E L D cb(L). Then we compute, 0%: = Z Z # (Mam/R) # z zeLn¢(L) yeLmML) 1421.2) = 1 Many) = 1 = Z # (6(M(:v. 2WD) 2 z E L O (ML) m. z) = 2 which is zero as M(:L', z)/1R is a 1-dimensional space and so has an even number of boundary components. Thus 62 = 0. Floer showed that his homology is a Hamiltonian isotopy invariant and independent of almost complex structure by continuation. Suppose that we have two generic almost complex structures J0, J1. Then the chain group CF (L,¢(L)) has two differentials 80,81 each defined as above but counting Jo- and Jl-holomorphic Floer discs respectively. As the space of almost complex structures is contractable, there is a path J, connecting J0 and J1. 11 Now define roux) = 2 Manny, y E L 0 ¢(L) Many) = 0 a map from (CF(L,¢(L)),81) to (CF(L,¢(L)),60). Here, M(:r,y) is a zero dimensional moduli space of Floer discs where the almost complex structures vary. That is, maps 21 of the infinite strip S = {z E C | 0 3 31(2) S 1} to X such that 0 lim A =azcd I'm “3:, g(z)__*o'o'11(z) an NHL-001“) y 0 11(0 + iy) C L and {1(1 + iy) C ¢(L), and 0 {1 has — JO 0 dfi = dfi o Jstd over points z with 8(2) 3 0, — J1 o dfl = (111 o Jstd over points z with 8(2) 2 1, and —— J, o dfi = d2} 0 Jstd over points z with 8(2) = t 6 [0,1], where Jstd is the standard almost complex structure on C. Note that there is no R-action on M (:17, y) defined this way so when M (:13, y) is O-dimensional, it need not be empty. To prove that (1)01 is a chain map, and thus induces a map on homology, we must show that 80 o01 — (D01 0 (‘31 = 0. This is accomplished by describing the boundary of the compactification MOB, z) of M(:r,z) when p(:r,z) = 1. In this case M(:r, z) is compact, one dimensional and so modulo two the count of points in its boundary is zero. In this setup, Gromov compactness means that elements of NICE, 2:) will degenerate into two types of curves in the limit: 0 Jo-holomorphic Floer discs in M(:1:, y) with Mr, y) = 1 and elements of M(y, z) with MIN) = 0 0 Elements of M(:1:. y) with 11(3, y) = 0 and Jl-holomorphic Floer discs in M(y, z) with 12 These configurations are exactly those counted by (‘30 04101 and 01 081 respectively. There- fore, as lam] = 0 mod 2, 80 o01 — 4’01 o 81 = 0 mod 2. So (1)01 is a chain map. Now we describe why (1)01 is well defined (on homology) independent of the path Jt. Suppose that we had two paths Jt, J,’ interpolating between J0, J1. We then get two map 01, R be such that r(—t) = r(t) —t and when t >> 0, r(t) = 0. Then letting H z r o h, we get a Hamiltonian flow a. = 0,1(Hul|(u,v). Then as r'(0) = g, the 27r map can be extended over the zero section as the antipodal map. We then get the symplectic diffeomorphism , : 021rr’(|]u|])(uav) if u ¢ 0 T(U, 'U) { (0, —"U) If u z 0 which is compactly supported and called a Dehn twist. As 7' is compactly supported, it ex- tends to a symplectomorphism TL of X which is the identity outside a tubular neighborhood of L. The action of TL on H2(X) is (TL),(;1:) : a: + (1' ~ L)L 15 and since L2 = —2, (TL)... has trivial square. In fact, as a smooth map TE is isotopic to the identity. Now let X be a linear plumbing of three T‘SQS, L1,L2,L3 so that L1 0 L2 = ptl, L2 n L3 = ptg, with pt1,pt2 antipodal, and L1 0 L3 -= ll). Then c1(X) and wx are both nullhomologous. Consider the sequence of Lagrangian 528 L" = 7123(L1). As 1'2 is smoothly isotopic to the identity, each L? is smoothly isotopic. In the symplectic world, consider the intersections L? (1 L3. By hypothesis, L1 0 L3 = 0 so we must have H F (L1, L3) E’ 0. As X is a plumbing, L1 and L2 intersect so that, in the tubular neighborhood N (L2) in which the support of T22 lies, L1 0 N (L2) is a normal disc. By the description previously, we can then see that the image TEAL} 0 N (L2)) projects to a double cover N (L2) —) L2. As pt1,pt2 are antipodal, they are each other’s conjugate locus with the round metric. Thus T22 (L1 0 N (L2)) intersects the fiber over ptg in an S 1 so L] 0 L3 E’ SI. Moreover, this is a clean intersection - TL] 0 TL3 = T(L] 0 L3). By similar argument, L? 0 L3 is a clean intersection that consists of n copies of S 1. Pozniak’s result then gives us the E2 page of a spectral sequence converging to H F (L'l‘, L3): E2 2’ H... (S1; Z2)®". Seidel then proves that Theorem 1.1.6 (Seidel). HF(L1,L3) 9.” 0 and HF(L'I‘,L3) 7L 0 by showing that, due to large (2 2) differences in Maslov class of discs connecting the components of L]1 0 L3, at least one term of the spectral sequence does not vanish. Thus L and L? are Hamiltonianly isotopic. In fact, all L'l' and L? are Hamiltonianly non-isotopic for r > s, as we can pull back via TE: and see that L1 and L?” are Hamiltonianly non-isotopic. 1 .2 Construction Vidussi’s symplectic version of the Fintushel-Stern link surgery can be described as follows: Consider M L, a 3 manifold obtained from zero surgery on a nontrivial fibered m component link L in S3 with a fibration 7r : Ill L —> S 1. Choose metrics on AI L and S1 appropriately so that the fibration map 7r is harmonic. (Without loss of generality assume that the metric on S1 gives it volume 1.) Then S 1 x [W L has a symplectic form 1..) = d0 /\ d7l' + *3d7r. Here 9 is a coordinate on S1 and *;;dr indicates the pullback of *d7r in Al L via the projection 16 S1 x .M L —» 11/11,. The form to is closed since 1r is harmonic and nondegenerate since L is fibered. If m,- are meridians to the components K,- of L, then S1 x m is a symplectic torus of square zero. Let X,- be symplectic 4-manifolds each with a symplectic torus F.- of square zero and tubular neighborhood N (E) Suppose that 1r1(X,~ \ N (E)) = 0, then the symplectic fiber sum X1, = 51 X ML # X,- i=1,...,m Fg=Slei is simply connected and symplectic. Symplectic and Lagrangian submanifolds of each X,- and of S 1 x M L which do not intersect the F,- = S 1 x m,- remain so under this process. (We also note that on each link component the choice of meridian m,- does not matter since isotopies of m,- induce deformation equivalences of symplectic structures on X L-) Let '7 be a loop on a fiber of 1r. Then with the specified symplectic form, L, = S l x 7 is a Lagrangian torus in S1 x M K- When 7 and the m,- are disjoint, L7 is also naturally a Lagrangian torus in X L- It is this class of Lagrangian tori which we will be considering here. There is one more observation of note. In the 3-manifold [W L there is a natural con- struction of a vector field 11, namely the vector field uniquely determined by L(p)(*d1r) E 0 and «*(dvolsrflp) E 1. By construction, the time t flow of 11 preserves the fibers of it, moving them in the forward monodromy direction. Thus the time 1 flow of 11 on M L gives the monodromy map when restricted to a fiber of 7r. If we extend a to a vector field on S1 x M L which we also call M, then we note that L(u)w is a closed, but not exact 1-form. Thus we get a 1-parameter family of symplectomorphisms gbt on S1 x [if L which are not Hamiltonian. Consider the action of a, on our Lagrangian torus L7. When t 93 Z, qbt(L7) 0 L7 = 0 as 7 is moved to a disjoint fiber. However, when t e Z it is possible that ¢t(L7) and L7 intersect. Further, when the monodromy is of finite order, we can find a good choice of meridian m so that the symplectic isotopies of L7 to its iterates under the monodromy stay away from the S 1 x m,. These symplectic isotopies survive as Lagrangian isotopies in X L- 17 1.3 Calculation of Floer Cohomology we use the variant of Lagrangian Floer cohomology over the universal Novikov ring A in [F000]. In this theory, the construction of the Floer cohomology groups is defined when certain obstruction classes vanish. These classes. count pseudoholomorphic discs with boundary on the Lagrangians. The following lemmas show that we are in the situation where these classes vanish and serve to compute the homology. Lemma 1.3.1. Suppose (S1 x 111 L,w) is as above and that the link L is nontrivially fibered in the sense that the genus of the fiber is at least 1. Then S 1 x [W L contains no pseudoholo- morphic spheres. Further, suppose that 7,, i = 0,1, are loops on a fiber of 7r which meet transversely in exactly one point and let L,- = L“. Then all pseudoholomorphic discs in S 1 x M L with boundary on L0 or on L1 are constant and there are no nonconstant F loer discs for L0, L1. Note that we cannot extend this lemma to say that there are no pseudoholomorphic representatives of 7r2(S1 x AIL, Lo U L1). We see such a counterexample in Section 1.4. Proof of Lemma 1.3.1. As was assumed, 111 L is a fibration over S1 with fiber 29 of genus g 2 1 and projection 71'. As [70] - [71] = :l:1 in the homology of the fiber no nonzero multiples of the two may be homologous. Thus they represent distinct infinite order elements of 7r1(29, 70 n 71) for which no powers 1‘, j 7é 0 give [70? = [71]j. By considering the universal abelian cover 29 x IR, we see that «1(29) injects into 7r1(M L) by the inclusion of a fiber. Then the subgroups generated by 70,71 intersect trivially in 7r1(ML,70 n 71). Since the genus of the fiber 9 Z 1, 7T2( 1W 1,) = 0 and thus 7r2(S1 x NIL) = 0. Now consider the exact sequence 0 = «2(51 x M) —>vr.(81 x ML,L.) —+vr.7r1(51 x ML) It follows that 1r2(S1 x ML,L,-) = ker(i) = 0 by our assumptions on 7,. Therefore, the homomorphism / w on 7r2(Sl x M L, Li) given by choosing a representative and integrating the pullback of 112 over the disc is the zero homomorphism. Let Q(L0,L1) denote the space of paths 6 : ([0,1],0,1) —+ (S1 x AIL,L0.L1) and 00(L0,L1) be that subset whose members are homotopic to a point. As L0 0 L1 is con- 18 nected, 90(Lo, L1) is also connected. Let i0, il be the inclusions of L0 and L1 into S1 x 11/1 L respectively. There is an evaluation map p : 90(L0,L1) ——> L0 x L1, p(6) = (6(0),6(1)). This is a Serre fibration whose fiber is homotopy equivalent to 90(51 x NIL,:1:). Thus 7rk(p"1(60, 61)) E 7rk+1(Sl x ML) and there is the exact sequence . 4.11 7r2(51 X ML) _>7F1(90(L0,L1))LW1(L0) X “1(L13°‘_1>,r1(51 X ML) 0 —————-—>7r1(Qo(Lo, L1)) ————>z2 x z? -——>-Z x «I(ML) For the last map, the sequence is exact in the sense that im(p..) = ker(i0..,.i1“*1)= {(a, b) E 1r1(L0) x 7r1(L1) | i0...(a) - (i1...(b))—l = e}. Then 7r1(Qo(Lo, L1)) E ker (i0. - i121). Since the 7,- are nontrivial and nontorsion each 10..., i1... is individually injective on in. So the kernel of i0... - if: depends only on the intersection of the images of i0... and il. in Z x 7r1(ML). Then as we know [the] subgroups generated by 70, 71 in 7r1(ML) intersect trivially, we see that 1r1(Qo(Lo, L1)) E Z. Consider D2 as the unit disc in C and let 8+ = 8D2 0 {z E C I 312 Z 0} and 3.. ._—_ 002 fl {2 E C ] 312 g 0}. Consider A as the annulus {2 E C [ 1 _<_ [2] S 2} with boundary components 6W1 and alzl=2' We may represent elements of 7r1(Qo(Lo,L1)) as maps of the annulus (Afllzlflfilzlfl) into (S1 x .ML,L0,L1). As with 7r.2(Sll x ML,L,-) there is a homomorphism on 7r1(§20(L0, L1)) which we shall call / to which is given by integrating the pullback of (1) over (in this case) the annulus. We now show / w on 7r1(Qo(Lo, L1)) is the zero homomorphism. To see this, consider a certain generator for 7r1(f20(L0, L1)) E Z represented by a map 11 : (Afilzlzgfllzlfl) —» (S1 x ML,L0,L1) for which 11(A) C Lo 0 L1 and a]z]=118]z[=2 both map to :t the generator of 7r1(L0 0 L1) = Z. Clearly fu‘w : 0. Then fa) E 0 on 7r1(Q(L0,L1)). A If we have topological Floer disc, that is, a map 11:(D2,6_,(9+)——>(S1 x ML,L0,L1), then the images of ii lie in LoflLl. As LoflLl E S 1 is connected, we can connect the images of :l:i by an are 7 : (—-72:, g], {—:2r—}, {g}) —> (L0 0 L1,u(—i),11(i)). There are, of course, 19 two ways of doing this but that will be immaterial. Then define 11 : (Aralz|=21a|2|=l) —> (51 X ML,L0,L1)by 1'2 (7)_{uoq’> if§Rz<0 7 7(6) if 892202 =re7‘9,—12’-g OS cola Where ()5 is a diffeomorphism from the interior of D = AF] {2 E C [ 322 _<_ 0} to the interior of D2 whose extension to the boundary takes D n 8|z|=2 t0 6-, D ne|,,=1 to 6+, the part of D lying on the positive imaginary axis to i and the part of D lying on the negative imaginary axis to —z'. Then [171.) = / u'w. Now, 17. defines an element of 7r1(Q(Lo,L1)) so / u’w = A D2 02 {fa} = 0. Thus as all nonconstant pseudoholomorphic curves have positive symplectic A area, there are no nonconstant Floer discs. [:1 Lemma 1.3.2. Suppose that the Lagrangian tori L,- = S1 x 7,- meet in an S1 in (S1 x ML,w) as in Lemma 1.3.1. Let m,- be meridians to each component K,- of Leach away from the 7, and (Xg,wx,) be a collection of symplectic 4-manifolds each containing an embedded symplectic torus E- of square zero. Then, given any bound on energy E, there exists an almost complex structure on the fiber sum manifold X L (which on each side of the fiber sum is sufficiently close to one for which the E and S l x m,- are pseudoholomorphic) for which all (perturbed) pseudoholomorphic discs in X L with boundary on L0 or on L1 and all Floer discs for L0, L1 have energy greater than E. Note that if Jt is a loop of almost complex structures, starting at a J as described, which is contained within a small neighborhood of J, then Jt-pseudoholomorphic strips (Floer discs) can be considered as solutions to the perturbed pseudoholomorphic curve equations. Proof of Lemma 1.3.2. In [IP04], Ionel and Parker construct a 6 dimensional symplectic manifold Z with a map to D2 so that over /\ E D2 \ {0} the fiber is the symplectic sum XL and over 0 the fiber is the singular manifold S1 x AIL U Xi. Each fiber XL,,\ is i=1,...,m H.231 X111,- 20 canonically symplectomorphic to S1 x M L \ N (S 1 x mi) 0 X ,- \ N (E) away from the fiber sum region. i=1 Further, the almost-complex structure J z on Z is chosen so that in the singular fiber X14), S 1 x m,- is a pseudoholomorphic torus and so that the restriction of J z to each fiber is an almost complex structure. (The singular fiber is pseudoholomorphic in the sense that each inclusion of X and S1 x M L is pseudoholomorphic.) Suppose that 7 is a smooth embedded path in D2 which passes through zero and that {An},,°°:0 —1 0 with An E 7\ {0}. Since the L,- are disjoint from the fiber sum region, there are Lagrangian submanifolds L,- in Z for which in every fiber X L, A" above An, L,- 0X L, A" = Li," which is mapped to L,- under the canonical identification. Suppose that we have a family of pseudoholomorphic discs 2,, in X L, A" that have bound- ary in L,- for each n and have energy bounded by E. Then by Gromov compactness for pseudoholomorphic curves with Lagrangian boundary conditions, this sequence has a sub— sequence which converges to a pseudoholomorphic curve 11 with image in the singular fiber X14) E S1 x 114;, U i=1?” X,- and boundary on Lip E L. The domain C of 11 is then a collection of spheregjrfdxd’iscs. We next state Lemma 3.4 of [IP03] within our context. Note that here, V is a perturba- tion of the 5 J operator. (We suppress the perturbation elsewhere.) Lemma 1 (Ionel,Parker). Suppose that C is a smooth connected curve and f : C ——1 S1 x [if L is a (J, u)-holomorphic map that intersects S1 x m at a point p = f(2o) E S1 x m. Then either 1. f(C) C S1 x m,- for some i or 2. there is an integer d > 0 and a nonzero a0 6 C so that in local holomorphic coordinates centered at p f(z. 5) = (Pi + 002]), aozd + 0(|z|d+1)) where 0([2]k) denotes a function which vanishes to order k at 2 = 0. We note that no irreducible component of ii is mapped entirely into any S1 x m,- as there are no nonconstant holomorphic maps of spheres into tori and all discs have boundary on 21 L,- away from each S1 x mi. Part (2) of the lemma shows us that on each component of the domain, ii meets the F,- as algebraic curves do. Thus we know that 131 intersects each S 1 x m.- at a finite number of points and that the image of each spherical or disc component of the domain of 12 will lie entirely in one of the X,- or. in S1 x M L- Then our map 11 splits into two pseudoholomorphic maps 111 : (C1, 8C1) —-1 (S 1 x M L, Li) and 112 : C2 —> H Xi. Here C = C1UC2 with 01 a collection of spheres and discs and C2 1' a (possibly empty) collection of spheres so that C1 fl C2 C 11—1 (O S1 x m.) are nodes of C. i=1 Thus we have obtained 111, a pseudoholomorphic curve in S1 x M L with boundary on L. By our Lemma 1.3.1 this map must be constant. Thus the image of 111 is in L,- disjoint from the fiber sum region. Then since the images of 112 and 111 are connected, 112 must have empty domain. Therefore, 11 is constant. This implies that for /\ sufficiently small, all of the discs 2,. must have been contained in S1 X It! L disjoint from the fiber sum region. Therefore by Lemma 1.3.1, they were constant in the nonsingular fiber sum. Let x\' be one such value. Then picking J equal to the fiber—wise almost complex structure J ,V on X L, y, we have the desired result. The proof follows identically if we consider pseudoholomorphic Floer discs for L0, L1. El Theorem 1.3.3. Let Lagrangian tori L,- = S1 x 7,- in (S1 x ML,w) meet in a S1 as in Lemma 1.3.1. Then [{F.91x1\-IL(LOaLO)CE HFSI xML(L1.~L1) 8’ HilTQ) (53> A and HFSlxML(L01L1) g Hilsl) ® A- Corollary 1.3.4. The Lagrangian tori L,- = S1 x 7,- are not Hamiltonian isotopic in (S 1 x Albee). 22 Proof of Theorem 1.3.3. We begin by computing H F (L0,L0). Since we have found in Lemma 1.3.1 that 1r-2(S1 x AIL, L0) = 0, we have that HF(LO, L0) 2 H*(T2) e A as in Floer’s original work [F1088]. Similarly, H F (L1, L1) = H * (T2) ® A. Now we consider H F(Lo, L1). As L0, L1 intersect cleanly, Proposition 3.4.6 of [Poz’94] implies that in some neighborhood N (Lo 0 L1) of Lo 0 L1 the Floer complex and Morse complex for some Morse function f : Lo 0 L1 —> 1R coincide. This allows us to consider a slight modification of the action spectral sequence of [FOOO]. The universal Novikov ring A can be written as the ring of formal sums Z aiTA‘eni 1 with 1. AiERandnz-EZ 2. for each X' E R, #{ilAi S )3} < 00 Here the TA parameter will be used to keep track of the action of a pseudoholomorphic disc. The formula deg T’\e" = 2n determines a grading on A and we denote by A” the homogeneous degree It part. An IR+ filtration on A is given by FAA —_— {Z aiTA‘en‘ | A, 2 A} We can then get a Z filtration by picking some X“ E 12+ and setting qu = F q’vA. The homogeneous elements of level q are then grq (.7: A) = qu /fq+1A. Then as in Theorem 6.13 of [FOOO] we have a spectral sequence Ef’q with E5“? 2 EB H"(L0 n 1.1; o) e grq(A(p_k-)) k Thus E2 E H *(S 1) (8 A. We now want to see that all higher order differentials vanish. Recall that in Lemma 1.3.1 we showed that 0J[Tr2(Slx ML,LoU L1) E 0. Since the Hamiltonian perturbation may be chosen to be very small so that the local curves are of area less than X“, Lemma 1.3.1 shows that we have found all of the discs to be counted. Therefore, there are no higher order differentials and HF(L0,L1) = H*(Sl) @A 23 E] In [Che98], Chekanov showed that when the Hofer energy of a Hamiltonian symplecto- morphism is less than the minimum symplectic area of pseudoholomorphic sphere or disc with boundary on a Lagrangian, a modified Lagrangian F loer homology can be computed which has rank equal to the rank of the singular homology. The Hofer norm of a Hamiltonian function H : [0,1] x P -—> IR on a symplectic manifold P is defined as l [IHH 2/ (max H(s, r) — min H(s,r)) ds. 0 rEP TEP This extends to a definition of Hofer energy of a Hamiltonian symplectomorphism ¢ by E(gb) = inf{l|H|[ [gb is the time 1 flow of H} Theorem 1.3.5. For all E > 0, let XL be the fiber sum of S1 x NIL and the X3, L.- the image of S1 x 7,- in the fiber sum with 7 as .in 1.3.1, J as found in Lemma 1.3.2, and 4) a Hamiltonian symplectomorphism which has Hofer energy less than B. Then, supposing that for each L,, the Maslov class ,1 : 7rg(XL,L,-) —> Z takes only even values, in the fiber sum manifold X L, HEXL(L0,L0; J, a) a HFXL(L1,L1; .13) a H*(T2) e AE where A13 is the truncated Novikov ring A/ F E A. Note that this version of Lagrangian F loer homology is not invariant under change of 45 to a Hamiltonian with larger Hofer energy. Corollary 1.3.6. Under the conditions of Theorem 1.3.5, the Lagrangian tori L,- = S1 x 7,- are not Hamiltonianly isotopic in X L- In section 1.4, We give examples which are Lagrangian isotopic. As these examples are in simply connected X L, we may strengthen the corollary to exclude symplectic isotopies. Proof of Corollary 1.3.6. From Theorem 1.3.5 we see that for any energy bound E on gb, H F (L,, Li); J, 4)) has 4 generators so #Li H ¢(L,-) 2 4 for any Hamiltonian symplectomor- phism of energy at most E. Also, there is a Hamiltonian isotopy which is given by smoothly 24 extending a perfect Morse function on S1 2 L0 0 L1 to be constant outside a neighborhood of L0 0 L1. Letting one of the L,- flow by this Hamiltonian, we get a pair of Lagrangians with two intersection points. Therefore, Lo and L1 cannot be Hamiltonian isotopic. Cl Proof of Theorem 1.3. 5. This is proved using the same methods as in the previous theorem. Lemma 1.3.2 ensures that the obstructions to defining a small perturbation Lagrangian Floer cohomology on the L,- C X L vanish and we make make the following local (in J and H) computation: Using the action spectral sequence for H F(Lo, Lo) we see that E2 = H ‘(T2) 8) A 3. By Lemma 1.3.2, we see that all the higher order differentials vanish for a small perturbation as all flow lines either have already been counted via d1 or are of energy higher than E and thus die in A E. (This can be seen by Lemma 6 of [Che98]) In this case, the calculation of H F (L0, L0) goes through as in the case of Theorem 1.3.3 and we find that HF(LO, Lo; J, ob) E H*(T2) <8) AE. Similarly, we get HF(Ll, L1; qu5) E H*(T2) (81 AE. Now we show that the computed groups are invariant under the Hofer energy hypothesis. Invariance is guaranteed as long as in 1-parameter families, there are no discs with boundary on L0, L1 of index —1 which bubble off.Our restriction on the Maslov class ensures that there are no index -—1 discs which appear in the boundary of 1-parameter families. Thus there is a continuation isomorphism and we have a well defined invariant of (energy bounded) Hamiltonian isotopy in X L. 1.4 Example If we let L be the union of the right-hand trefoil knot K1 and one of its meridians K2, the loops 71,72,73 in Figure 1.1 are all freely smoothly isotopic in M L and meet transversely pairwise.1 We select the fibration 7r : A! L -1 S1 so that the Seifert surface containing 7,- shown in Figure 1.1 is the fiber. The smooth isotopies of 71 and 72 to a common curve are shown in Figures 1.4 and 1.5. In each of these figures going from (1) to (2) involves sliding over the 0-surgery on the meridian, going from (2) to (3) is an isotopy. To relate 1Though they are freely isotopic, the proof of Lemma 1.3.1 shows that they are not equal in m i.e. fixing a basepoint. 25 Figure 1.1. 7;, i = 1, 2, 3 intersecting pairwise transversely (within the fiber) in single points the end results we have the additional move of “twisting up the corkscrew” which takes the curves each (3) to the other. Note that this smooth isotopy is different than the Lagrangian isotopy which we will mention later. Since the 7,- are smoothly isotopic, the Lagrangian tori L, = S1 x 7, are smoothly isotopic. As Lagrangians do not have canonical orientations we neglect the orientations of the loops here. Addressing the comment made after Lemma 1.3.1, we note that we can choose the almost complex structure J so that the Seifert surface 2 (a T2) is pseudoholomorphic. Then choosing any pair 71,71- (1' E j), E \ (7,~ U7!) is a disc. This is however, not a Floer disc as it does not satisfy the correct boundary conditions. For the left and right handed trefoils, the monodromy is of order 6 and in the basis A, B shown in Figure 1.2 is given by the matrix 1 1 —1 0 On the (positive/negative) Hopf link the monodromy is a (positive/ negative) Dehn twist about a curve parallel to the components. The connect sum of fibered links is fibered with monodromy which splits around the connect sum region. From this computation of monodromy, we see that 72 is i the 2nd and 5th image of 26 Figure 1.2. Basis for monodromy on trefoil 71 under the monodromy map and that 73 is :l: the lst and 4th image of 71. Finally, 71 is sent to —71 under the third iteration of the monodromy. Thus, as the monodromy gives a symplectic isotopy (c.f. Section 1.2) in S1 x M L the tori L1, L2,L3 are all symplectically isotopic. In X L they are all Lagrangian isotopic as the fiber sum is taken away from the isotopy. Despite the existence of these symplectic isotopies in S1 x M L, Theorem 1.3.3 shows that the L, = S1 x 7,- are not Hamiltonian isotopic there. Now we consider how Theorem 1.3.5 applies. That all the hypotheses of the theorem are satisfied, except that on the Maslov class, is clear. With the following lemma we see that the remaining condition is satisfied. Then Theorem 1.3.5 shows that the L,- are not symplectically isotopic in X L- Lemma 1.4.1. For the Lagrangian tori Lg, we may choose X,- = E (1), and the particular identification of F, and S1 x m.- so that “L. : n2(XL, L,) —> Z is even. Proof. As L is a two component link with odd linking number, X L is a homotopy E (2) and thus is spin. See [F398]. In fact E (2)L is E (2)1340“. We shall write E (2) L := X L- For such a 4-manifold, the first Chern class is an even multiple of the fiber. Note that ILL,- factors through 7r2(E(2)L,L,-) —1 H2(E(2)L, Li). As H1(E(2)L) = 0, the Meyer-Vietoris sequence gives that the group H2(E(2)L,Li) is generated by elements of H2(E (2)1.) / (Li) and relative classes with boundary spanning H1(L,-). The Maslov index of 27 a class B, fiL,(fl), B E H2(E(2)K, Li), will change by an even amount whenever an element of H2(E(2)L)/ (L,) is added as c1(E(2)L) is divisible by 2. Thus if we can find a pair of relative discs whose boundaries generate H1(L,-) and whose Maslov indices are even, we have shown that 11 is even. For i = 1, 2, we will choose the identification of E and S1 x 771,- so that 1. pt x m,- is identified with a vanishing cycle on F,- and 2. S1 x pt is identified with the sum of two vanishing cycles on F,- whose boundaries meet once, transversely, in E. Because 711(E(1) \ F.) = 1, we may select an elliptic fibration on E(1) with nodal fibers having vanishing cycles a and b where a, b generate 7r1(F,-). With the decomposition E- = a x b, identify a with pt x m and a + b with S 1 x pt. This gives us the desired identification of F,- and S1 x mi. +0\ R —\ / 9 \— Figure 1.3. An isotope of 71 bounding meridians to K1 and K2 Each of the 7,- bounds a four times punctured disc Dom- in A! L where three punctures are meridians of K1 and one is a meridian to K2. See Figure 1.3. By our choice of fiber sum gluing, each meridian of K1 bounds a vanishing disc on its E(1) side of the fiber sum. Similarly, the meridian to K2 bounds a vanishing disc on its E (1) side of the fiber sum. Take three copies of the vanishing disc D1, D2, D3 from the E (1) fiber summed to 28 S 1 x ml and one vanishing disc D4 of from the E(1) fiber summed to S1 x m2 and form D7,. = 00.7,- UD1UD2UD3UD4. Each 73’s Lagrangian framing relative to that induced by trivializing over D7, is —2 and is given by a pushoff in the Seifert surface. See [FS04]. The framing coming from this disc is then —1 — 1 — 1 —1—(—2) = —2 and gives us p7,.(D7i) = —2. By our choice of gluing, the S1 x pt C L,- is bounded by a pair of vanishing discs. This pair of vanishing discs intersects at one point on F and so can be smoothed to a disc D31 xp, with relative framing —2. For this loop the framing defect from the torus is 0 (given by pushoff in the monodromy direction) and so pL,(D31xpt) = —2. Thus we have found a basis on which a is even, hence 11 is even. [:1 This example generalizes to similar links with K1 = T2,2n+l where we find many La- grangian but not symplectically isotopic tori. 29 6x623? / / < w .DC DC 2 49 B > CHAPTER 2 Gauge-theoretic Invariants of Twins 2. 1 Introduction A 2-knot K is an embedded $2 in S“. We say that K is unknotted when it bounds a D3. In contrast to the classical case of embedded S 1s in S3, 2-knots display many pathologies; including that their exteriors need not be K (m, Us and that the homeomorphism type of the knot exterior does not determine the knot’s isotopy class. [For instance, see [AC59]) and [Gor76]. From a gauge theoretical perspective, 2-knots are uncomfortable to deal with as their exteriors are homology S1 x D38 and even after the surgery which replaces N(K) = S2 x D2 with D3 x SI, we obtain homology S1 x S35. With b2 = 0 for these manifolds, current tools for smooth 4-manifolds offer little help. On the other hand, the natural generalizations of gadgets like the Alexander ideal/polynomial can be computed for 2-knots. In [Gil82], C. Giller proposed a defini- tion and method of computation for an invariant we will call Ag. The proposed invariant is derived from the projection of 2-knots to R3 and for a certain class of 2-knots is known to compute the Alexander polynomial. This is a result of the proposed invariant obeying a re- lation similar to Conway’s for computing the Alexander polynomial for classical knots. The relation, which we discuss more thoroughly in section 2.1.8, always results in a symmetric 32 polynomial. However, the Alexander polynomial of a 2—knot need not be symmetrizable. For example, the example given in Figure 2.7 has Alexander polynomial AA = 1 — 2t while Giller’s polynomial is AG = t"2 — 1 + t2. Further complicating matters is that Giller’s polynomial is not actually known to be an invariant. If instead of 2-knots, we look at “twins” in S4, we have objects to which standard gauge—theoretic methods can be applied. It is in this context we see that Giller’s polynomial computes, in the relevant cases, the Seiberg-Witten invariant of the exterior of the twin. 2.1.1 Twins In [Mon83] and [Mon84], José Maria Montesinos introduced the concept of a twin in S4. Such an object consists of two embedded S23 which meet exactly twice, transversely. As the second homology of S4 is trivial, these intersection points have opposite orientations. By “standard twins”, I will mean that both S23 bound embedded B33 and that the exterior of the pair is T2 x D2. In general, the exterior of such a twin is only a homology T2 x D2 with boundary T3. Standard twins may be described as follows: Take a 3-ball B3 in R3. Form the space (S1 x B3)/ ~ with (60, 1:0) ~ (61,321) iff 11:0 = 2:1 E 833. This space is S“. Then consider the image of {1} x BB3 and the S 1 x 2-axis under ~. Call these 51, S2 respectively. S1 bounds {1} x B3 and if we take S 1 times the half plane {y = 0, :1: Z 0} then after quotienting by ~ we get a B3 which S2 bounds. Also we can see that the exterior of these twins is S1 times the exterior of the 2-axis in B3 — i.e. S1 x (S1 X D2) = T2 x D2. See Figure 2.1. In fact, all twins in S4 have as their exterior a homology T2 x D2 with T3 boundary. As 7r2(SO(2)) is trivial, an orientation on an S2 C S4 determines a trivialization N (S2) E S2 x D2. Thus, for a 32 C S“, all framings are equivalent to the Seifert framing. Fix orientations on the 2-spheres, K1, K2, in a twin 'll‘w and consider 6N('ll‘w) E T3. Take a simple closed curve 7 on the twin passing through the intersection points of K1, K2 and lift it to 8N('ll‘w) using the framings. All such 7 are isotopic on the twin and as each K,- has a single framing, this lift is canonical. This decomposes 6N('ll'w) = 7 x T2 The boundaries of normal discs Df, D3 to K1, K2 complete the decomposition to give 8N('ll‘w) = 7x6D¥ XBDg. With this decomposition in mind, we define a standard surgery a twin Tw in S4: form 33 Figure 2.1. Spin to get standard twins S4('ll‘w) = (S4 \Tw) (g(T2 x D2) where d) : T3 —> T3 identifies (‘9D2 and 7. Up to isotopy, any two such d) differ by a diffeomorphism T2 x pt ——> T2 x pt. Since any such map extends over T2 x D2, the resulting manifold only depends on the embedding of the twin. Also, H,.(S4('l['w);Z) E H...(T2 x SQ;Z) so that the core T2 of the surgery is identified with T2 x pt C T2 X 32 as a homology class. 2.1.2 Definition of the invariant for twins Let F be a fiber in an elliptic fibration of the K3 surface, E (2). Then N (F) comes with a trivialization N (F) = D2 x T2 from the fibration map which induces 6N (F) = ('9D2 x F. Taking a twin 'll'w C S4, we also have a decomposition 6N('ll‘w) = 7 x T2. Fix an identification of this T2 and F and using an orientation reversing diffeomorphism between (9D2 and 7 we obtain an identification (b of 6(E(2) \ N(F)) and (9(S4 \ Tw). Form E(2)rw == (E(2) \ N(F)) 5,434 \ TWl Although the resultant manifold may depend on the precise choice of d), we will omit the distinction. As 7r1(E(2)\N(F)) = 0 and the image of in (T3) normally generates 7r1(S4\'l[‘w), E (2)1“; is simply connected. This procedure may also be thought of as the generalized fiber sum of E (2) and S4 (TW) along F and the core T2 of the surgered twin. As such, we will sometimes write E (2)1rw as E(2) # 34(rw). F=T2 The invariant of the twin 'll‘w which we will consider will be the Seiberg-W’itten invariant of E(2)Tw- i.e. 34 Definition 2.1.1. I ('ll‘w) :2 SW(E(2)—3w) thought of as an element of the group ring ZlH2(E(2);Z)l- Precisely, SW(E(2)-3w) lives in Z[H2(E(2)TW;Z)] and we identify H2(E(2)-N) with H2(E(2)) by a homomorphism which extends the identity map on H2(E(2)Tw\(S4\'ll‘w)) —+ H2(E(2) \ F). While E(2)Tw may depend on the choice of the gluing map (b, the Seiberg-Witten invariant does not. This is due to the gluing formulas in [MMS97]. The invariant is well defined up to a Sign which depends on a homology orientation of Home» a det H2(E(2)) 69 11111212» As standard twins 'll‘wstd have exterior equal to T2 x D2, the gluing map (1) extends over ‘ the interior so E(2)1rws,d E E(2) and I(Twstd) = SW(E(2)) = 1. (2.1) Now suppose that we have a twin 'll'w and a disjoint torus T in S4. T has a canonical framing given by pushing off into its. Seifert manifold. Thus we can form the self fiber sum S4('lfw) # , where T Tw is the core of the standard surgery on 'lfw with its default framing. Thig‘VIZZnifold is well defined with a choice of an identification of T'rw and T. Again, any choice of identification will have the same Seiberg-Witten polynomial. Then form E(2)'ll'w,T = E(2) F 36;, (S4('Il‘w) # T with vaw a pushoff of TN. Now, E(2)-Ir,“~ = N N: is a homology E(2) # with F, F’ elliptic fibers and SW(E(2) # ) = (t — t-l)2 with F=F’ F=F’ t = exp([F]). This is a result of the gluing theorems of [MMSQ7]. We define Definition 2.1.2. I(TW,T) :2 (t - t‘1)—ISW(E(2)TW,T) thought of as a element of the group ring Z [H2(E(2); Z)]. 2.1.3 Construction of Knotted twins We will, of course, want to consider twins other than the standard ones. There are several techniques we will consider. Construction 2.1.3 (Connect sum). Take K0, a knotted S2, and a twin 'll‘w = KIUK2 in S4. Then, selectii‘ig one of the spheres K1 in the twin, we form the connected sum 35 (S4 # 5", K0 # K1) at some point away from the 2 double points of the twins. (This con- struction is not independent of the choice of K1, K2 in 'll'w in general.) If we take Tw to be the standard twins, then this construction is independent of the choice of 32s and provides a handy method for studying 2-knots via twins. This inde- pendence is due to the existence of a orientation preserving diffeomorphism p of S4 which interchanges K1,K2 in standard twins. The diffeomorphism p is constructed as follows: View S4 as N ('lI'wstd)LJT2 x D2. Define p on N ('ll‘wstd) to be the obvious map which interchanges K1 and K2. Then p induces the map Ol-‘O OCH 0 0 1 on BN('ll'wstd) = T 2 x (9D2 = T3 under the basis for H 1 (T 3) given by BDfivanfi, and (9ng where 8D; is the boundary of the normal bundle to S. This map then extends over T2 x D2 giving p on S3. Construction 2.1.4 (Artin Spin). In the construction in the introduction, if we replace the 2-axis with a knotted arc, K, meetin BB3 at the north and south oles, the procedure 3 P gives us a twin 'll‘wK whose complement is S1 x (S3 \ K) with K = K U{the international date line of 6B3} and consists of a unknotted sphere (the image of (983) and an Artin spun knot (the image of K). This spinning construction is originally due to Artin in [Art26]. When the standard twin surgery is performed on twins formed by Artin spinning K, the result is the manifold S1 x 58(K). (Where 58(K) is the result of zero surgery on K C S3.) This case is identical to the knot surgery considered by Fintushel and Stern in [F898] and so we have Theorem 2.1.5 (Fintushel,Stern). I (TWK) = AK(t2) where AK(t2) is the symmetrized Alexander polynomial of K and t = exp([F]). Construction 2.1.6 (Twist Spin). As before, take a knotted arc K1 in 83 with boundary on the north and south poles. For each 0 E S 1 = lR/27rZ we take the image of K rotated by 116 36 radians. Then in S l x B3 we have an annulus formed by the rotated K Is. This descends to a knotted S2, K2, in S4 = S1 x B3/ ~ which together with the image of {6} x 8B3, forms a twin. We call K2 the n-twist spin of K1 and write K2 = T"K1 and 'll'an K, for the associated twin. This comes from Zeeman, who in [Zee65], showed that the n-twist spin of a knot was not isotopic to any 2—knot obtained by Artin spinning when n > 1. The n = 1 case is interesting in that the 1-twist spin, K2 is unknotted independently of choice of K1. Construction 2.1.7 (Roll Spin). Similar to the twist spin, this construction involves a de- formation of a knotted arc K1 in B3 fixing the north and south poles which returns the arc to the starting point. Take a international date line of 6B3 union K1 and push it into B3 \ K1 so that it is null homologous. Call this K1. Then consider the l-parameter family of diffeomorphisms given by pushing a base point a: n times along K1. This gives us a diffeomorphism of the quadruple (B3,6B3, K1,1:) which is the identity on all but the first component. Proceed as before and quotient S 1 x B3 by ~ to get K2 to be the image of the K1 in each {6} x B3. We call K2 the n—roll spin of K1 and write K2 2 p"K1 and 'll‘wanl for the associated twin. This idea is due to Fox who, in [Fox66], showed that, for K1 2 41, the knotted 2-sphere K2 coming from the deformed arc is not isotopic to any n-twist spun knot. In this case, the 1-roll spin had a corresponding visualization of the motion of K1 in B3 which explains why “roll” was chosen to describe this. I duplicate the rolling move in Figure 2.2. Note that both twist and roll spinning can be described in terms of certain diffeomor- phisms of B3 which keep BB3 and K1 fixed identically. With this in mind, we now consider their mutual generalization, Deform spinning: Construction 2.1.8 (Deform Spin). Let g be a self diffeomorphism of B3 keeping BB3,K and a base point :1: fixed identically. Then the mapping torus of g is S l x B3 E ([0, 1] x B3)/ ((0,:r)r:u(1, g(r))) with an embedded annulus K2 which is the quotient of I x K1. Then after quotienting by ~ as before, we have a knotted 2—sphere K2, the image of K2, which together with the image of {0} x 683, forms a twin. We write K2 = 9K1 and 'll'ng, for the twin pair. The isotopy class of K2 and of 11‘ngl is determined solely by the isotopy 37 Figure 2.2. Fox’s Roll Spin 38 class of g. This construction was introduced by Litherland in [Lit79]. Diffeomorphisms such as g are called “deformations” and form a group D(K1) of deformations modulo isotopy. D(K1) is isomorphic to the group Auta(7rl(S3 \ K1)) of automorphisms of 7r1(S3 \ K1) preserving some fixed peripheral subgroup. In this setup we find that 7' corresponds to conjugation by the meridian of K1 and p corresponds to conjugation by the longitude of K1. Then Lemma 2.1.9 (Litherland). If K1 0 is not a torus knot, D(K1) E ZT EB Zp. o is a torus knot, D(K1) E Z, and qup = id. 2.1.4 Ribbon Knots and Twins We say that a 2-knot K is ribbon if it is formed by the following construction: Let D = HD3 (bases), B = 111)2 x I (bands) eech be embedded in R4 with (6D) 0 B = 111)2 x :tl. If a band intersects a base elsewhere, (D2 x (—1,1)) 0 D3 = D2 x t, t E (—1,1) and (D2 x (—1,1)) 0 (6D3) 2: 0. The second type of intersection is called a ribbon intersection of K or ribbon singularity of D U B. Then K 2 (6D \ LID2 x i1)U(U(6D2) x I) W—J in BB is a ribbon knot with ribbon presentation given by DU B. we can define ribbon surfaces of arbitrary genus in the same manner. Suppose that we have a twin 'll‘w = K1 U K2 for which the K,- are ribbon. Then for each K ,- we have a set of bases D,- and bands B,. We will say that 'll‘w is ribbon if 0 Bl r1 B2 = 0 0 D1 0 B2 2 HD2 x t, t E (——1, 1) with (BDI) 0 D2 x(—1,1)= 0 for each band in B2 (ribbon intersection) 0 D2 0 B] = HD2 Xt,t€ (—1,1)Wlth (602)fl D2 X (—1,1)= 0 IOI~ each band in Bl (ribbon intersection) 39 0 D1 0 D2 = 2D2. More specifically, D1 and D2 meet only in two balls D1, D2’ from each and at these intersections (corresponding to the intersection points K1 0K2), we have the following local model: D’1={(21,z2)e 122111222 31,822 = o} c «:2, D; = {(21,22) E C2 [3121 S 1,821: 0} C C2, So that taking the boundary of each gives us the cone of the positive Hopf link. The '1',D§’ case is the same but with orientations reversed on D3, giving us a negative Hopf link cone boundary. In this case we say that 'lI‘w is a ribbon twin. It is known that of the deform spun knots, only the Artin and 1-twist spun knots are ribbon. Artin spun twins are also ribbon. It is not known to the author if 1-twist spun twins are ribbon (and suspected not to be'the case). All other deform spun twins are not ribbon. There are several operations which will be useful to perform on ribbon presentations. Addition of a trivial base/band pair, sliding the disc to which a band attaches (band slide), and moving a ribbon intersection along a base/band sequence (band pass) are shown in Figure 2.3 and together with isotopy generate stable equivalence of ribbon presentations. Clearly stable equivalence of ribbon presentations generates isotopies of the corresponding ribbon knot but the converse also holds —— isotopic ribbon knots have stably equivalent ribbon presentations. For a proof of this, see [Mar92]. It will occasionally be easier to deal with simplified ribbon presentations. Let F = F (D, B) be the graph which has vertices corresponding to bases and edges given by bands, connected in the natural way. It is clear that b1(I‘) (thought of as a cell complex) is the genus of the ribbon surface specified by the ribbon presentation. Restrict ourselves to the case where the ribbon surface is a sphere or torus. Suppose that F has a vertex :1: of valence 3 or greater. Then one of the outgoing edges of :1: has a path, never returning to 3:, which ends at a vertex y which has only one incoming edge. Perform the band slide corresponding to this path to get a new ribbon presentation I" with the same set of bases. In F', the 40 Figure 2.3. A) Trivial addition/deletion B) Band slide C) Band pass valence of :1: has decreased by 1 and the valence of y is now 2. Continue this procedure until we arrive at a graph F (and corresponding ribbon presentation) for which each vertex has valence at most 2. Then, as cell complexes, F is‘either an interval or an S 1 as the ribbon surface is a $2 or a torus. We will call such ribbon presentations linear. Consider the connect sum of a ribbon 2—kn0t K0 with standard twins 'll‘w = K1 UK2. Standard twins have a simple ribbon presentation given by two bases and a band each. (Each base is for one of the twin intersection points.) Stabilize the band in K1 by switching it for two bands and a base. Then the connect sum K0 # K1 is formed by taking K0 and K1 each in a D3 and adding a band from an endpoint base D’ of K0 to the “middle” base of K1. Then sliding the band from the “middle” base of K0 to the 9 along Ko’s linear ribbon presentation to the other endpoint base, we get a linear ribbon presentation of K0 # K1 which has the twin intersection bases as its endpoints. 2.1 .5 Projections In the study of classical knots in IR3, their generic projections to R2 together with crossing information completely determine their isotopy type and have proved extremely useful. Projections are also quite useful for twins and surfaces in R4 = S4 \ pt. Giller proves in [Gil82] that projections of surfaces in IR4 to R3 with only double and triple points exist and are generic. In these generic projections, the double points either 41 exist in families which are either simple closed curves or embedded open intervals whose closed endpoints are triple points. See Figure 2.4. In the same paper, he gives methods of decorating these projections with over/ middle/ under crossing information and a way of determining if an arbitrary set of crossing information gives a lift of such an immersion of a surface in HR3 to an embedding in R4. Figure 2.4. Local models for a family of double points and a triple point We will only consider those knots and twins which admit a projection which contain no triple points. Not all twins or surfaces have such a projection and those that do are said to be simply knotted. First examples of simply knotted 2-knots include Artin spun knots and ribbon 2-kn0ts. For Artin spun knots, we can get a projection with no triple points by doing the same spinning construction (one dimension down) to the projection (to R2) of the original, classical knot. This creates an S 1s worth of double points for each crossing. in the classical knot’s projection. We will call a twin 'lI‘w = K1 U K2 simply knotted if both K ,- are simply knotted and pairwise have no triple points. Ribbon knots have embedded projections away from the ribbon singularities — the intersections of the interiors of bands with interiors of the bases. (This is in contrast with ribbon l-knots, for the which projections of bands may have crossings. The analogous situation here is an under/over crossing of the whole band — which does not result in a crossing in the projection.) Nearby the ribbon singularities, we have projections which appear as in Figure 2.5. It was proved in [Yaj64] that all simply knotted 528 are ribbon. When we have a S 1 family of double points, we have local neighborhoods around each which appear as in the first picture in Figure 2.4. This gives the neighborhood of the family the structure of an bundle over S 1, possibly with nontrivial monodromy. As the surfaces in IR4 are orientable and the two preimages of the double points are separated, the monodromy must be trivial. This means that, local to the S 1 family of double points, the projection is that of a classical knot crossing times S 1. Then, for a simply knotted projection of a (oriented) surface in R4, it is sufficient to label one of the surfaces as being over crossing at each family of double points. We will use 42 ............... .. O O ------ Q Figure 2.5. A projection of the neighborhood of a ribbon singularity (shaded disc) and the corresponding two S 1s of double points. “+” to denote this. Figure 2.6. A sphere and a torus with crossing information. For a twin, a few additional pieces of information are needed. We need to keep track of the two intersection points of the spheres. In S4, the neighborhood of each is diffeomorphic to the cone on a positive or negative Hopf link. Then the (undecorated) projection of such a neighborhood appears as does a neighborhood of double points. We decorate the projection with a solid dot to indicate the intersection point of the $23 and + signs to indicate over/under crossings on the double point arcs which emanate. We switch from over to under at the intersection of the spheres in twins. See Figure 2.8 43 Figure 2.7. A 2-kn0t with Alexander polynomial 1 — 2t and Giller polynomial AG = 1‘2 — 1 +12 44 Figure 2.8. The projection of the neighborhood of the intersection of spheres in a twin. To the left, the vertical plane crosses above and to the right, the horizontal does. 2.1.6 Virtual knot presentation In [SatOO], Satoh showed how to represent ribbon surfaces of genus 0 and 1 in IR4 by means of virtual knots/links. For our purposes, a virtual knot (or link) is a diagram in R2 of embedded, oriented arcs which end either at “crossings” as in the (top) first two pictures in Figure 2.9 or at endpoints as in the (top) third picture. Each such diagram corresponds to a collection of immersed surfaces in R3 by replacing each of the crossings and endpoints in Figure 2.9 with the corresponding surfaces in IR3 under them. These are then connected via tubes parallel to the embedded arcs. Thus, any virtual link corresponds to the projection, with crossing information, of a collection of ribbon 2-spheres and tori in R4. Conversely, linear ribbon presentations of knots correspond to virtual knots. Take a projection of K C IR4 —+ R3 having only double points at ribbon singularities. For each band in the linear ribbon presentation, consider the image of its core in IR3 extended to the center of the bases to which the band attaches. This gives an immersed (at ribbon singularities) arc K in 1R3. Taking a generic projection IR3 —> IR“2 we get an arc K immersed in IR2 with two kinds of singularities: 0 double points of the projection K C R3 —> IR2 3 K and o projections of immersion points K ¢—+ IR}. 45 f’\ \J EDGE) (ZZZ) O O Figure 2.9. Correspondence of crossings in virtual knots to crossings in projections of surfaces to IR3 Each of the first kind of double point corresponds to a virtual crossing. For the second kind of crossing we must first consider a diversion about orientations. The endpoints of K each correspond to a base with only one band attached —— here, K locally consists of the discs D1, D2. With a fixed orientation on K, we orient the boundary of the D,- with the outward normal. We then say that the endpoint of K is out/in as the boundary orientation on the D, is counterclockwise or clockwise, respectively (when D.- is orientation-preserving identified with the unit complex disc.) This orients K. Then, with K oriented, we can check that the second type of immersion point corre- sponds to the ribbon intersection in Figure 2.9. If it does not (i.e. the two crossings have the opposite under/over information) then perform the isotopy in Figure 2.10. Once this has been done, we may use our correspondence from Figure 2.9 to label each of the immersion points of K as virtual knot crossings. In addition to the “classical” Reidemeister moves in Figure 2.11, associated to a virtual knot, we have the series of “virtual” Reidemeister moves in Figure 2.12 giving allowable isotopies. Notice that move D is one of the forbidden moves of the virtual knots of Kauffman. The type of virtual knot we consider here is sometimes referred to a being weakly virtual but 46 Figure 2.10. Fixing a “bar ” ribbon crossing by an isotopy which creates two virtual crossings in the spirit of brevity we will omit “weakly” in this paper. These concepts of virtual knots are inequivalent as there are virtual knots (in the sense of Kauffman) which are knotted (nonisotopic to a standard configuration) which, when move D is allowed, are unknotted. An example of this is given in [Sat00]. \H l "Xi K" A1 A2 Figure 2.11. Reidemeister Moves for Classical knots x] We will add two more items to these diagrams. For a ribbon twin 'lfw = K1 U K2, there are two bases D2, D2’ in the ribbon representation of each K,- which correspond to the twin intersection points K1 (1 K2. Perform band slides until the ribbon presentations of the K,- . . . I II . . . are linear With endpomts D,, D,- . Then we have corresponding Virtual knot representations 47 Figure 2.12. “Reidemeister” Moves for Virtual knots 48 of the K,- which have identical endpoints. We will use ti), 9 to mark each of these as they correspond to the cone on the positive and negative Hopf bands at the intersection points. With this in mind, we get the moves in Figure 2.13. | i 0 0 F1 3., F2 [ Figure 2.13. “Reidemeister” Moves F 1,F 2 for Twins, versions for e are identical Now let us consider the connect sum of a ribbon 2-knot K0 with standard twins Tw = K1 U K2 as described in Section 2.1.4. Notice that K2 will not have any ribbon singularities with Ko # K1. In fact, we can drag the bases of K1 and K2 corresponding to one of their twin intersections along together when we perform the band slide described previously. This creates only virtual crossings between the new diagrams for K0 # K1 and K2. By the “Reidemeister” moves B 1 — B3 for virtual knots (Figure 2.12) or the standard Reidemeister moves A1 — A3, we can separate the are for K2 entirely from that for K0 #K 1, except for their common twin points. This means, in practice, we can obtain the virtual knot presentation of the connect sum of a ribbon 2-knot and ribbon twins by: 0 using move E of Figure 2.12 to obtain a diagram where the endpoints are on the boundary of a D2. 0 connecting these points via an unknotted arc in the complement of the D2 and chang- ing the source point to a 69 and the sink point to a 9. For example, see Figure 2.14 2. 1.7 Surgery diagrams As discussed earlier, a twin in S‘f‘l has a canonical surgery associated to it. Since our decorated projections determine isotopy type, no additional information is needed to carry 49 f\ A T \J - A [G \J v-— 11—— Figure 2.14. Giller’s example (left, projection version in Figure 2.7) and twin version (right) out surgery. For a T‘2 in S4, however, we will need additional information. As any T2 C S4 is nullhomologous, it bounds a Seifert manifold1 which, via its inward normal, gives a Seifert framing for the T2. This gives us a decomposition, 6N(T2) = T2 x S 1 . As surgery replacing N (T2) with T2 x D2 is determined by theimage of 3D2, we see that we can entirely describe surgery by specifying a curve on T2 and an integer giving the winding about a meridian (boundary of normal disc) to T2. See Figure 2.15. When the T2 is ribbon with a linear ribbon presentation and corresponding virtual knot diagram, we can decompose T2 in the following manner: Let C be the core of the ribbon presentation, projected to R3. Let a be an essential loop on T2 which, when projected to IR3 is null-homologous in IR3 \ C. Any such loop represents the same homology class on T2. Let B be ('9D2 x {t} in a band in the ribbon presentation. Orient a to coincide with the orientation of the virtual knot diagram. Then orient B so that a - B = +1 with respect to the orientation on T2. So T2 E a x B. Then, in the virtual knot diagram, labeling the knot corresponding to T2 with (7, B / 01) where 7 is the winding number of the attached 8D2 with respect to the Seifert framing and fi/oz is the slope of 8D2 projected to T2. We will write an S together with *s on the appropriate components when we wish to lA Seifert manifold for a surface 2 in S4 is a 3 manifold M with boundary diffeomorphic )3, smoothly embedded in S4 so that 3M = X]. As in [R0176], Seifert manifolds exist because of the following: Consider the map 2 x 0D2 —» 802 which is given by a trivialization of the normal bundle of E. Obstruction to extending this map over all of S4 \ N (2) vanish, giving us a map S4 \ N (2) -> S 1. We can homotope this map to a smooth map which remains equal to the projection )3 x 6D2 ——1 BD2 on a tubular neighborhood of the boundary. Then as S4 \ N (2) is compact there are a finite number of critical points. Let A! be the preimage of one of the regular points. 50 Figure 2.15. Projection and Virtual Knot surgery diagrams for a twin and torus. denote this surgery. 2.1.8 Giller’s Polynomial In [Gil82], C. Giller defines a polynomial Ac(t) of simply knotted S23 in S4. This supposed invariant obeys a “Conway calculus” relation similar to that of the Alexander polynomial for classical knots. That is, consider a embedded circle of double points in a projection of a (collection of) oriented sphere or torus in IR“. As mentioned before, we can trivialize the neighborhood of the double points so that we have the neighborhood of a classical knot crossing times S1. All surfaces in question are oriented and so orient the double points of their projection — this orients both strands in the classical picture. We can then replace this neighborhood with S1 times any of the 3 options in Figure 2.16, obtaining The invariant is then defined by the relation: Add.) — Add-) = (W2 — t—l/2)A0(L0) (22) together with Ag(unknotted sphere) = 1 (2.3) 51 Figure 2.16. Resolution of a knot crossing and Ac(surfaces separated by an S3) = 0. (2.4) Giller also describes AG in a manner similar to that of the Alexander polynomial. That is, letting M be a Seifert manifold for K, he forms the infinite cyclic cover X of S4 \K and presents H1(X) as a Q[t,t'l] module. Then AG is defined by T = Q[t,t’1]/(Ac) where T is the Q[t, t’l] torsion part of H1 (X) Whenever K is ribbon, we can choose AI to be a punctured nSl x S2 given by the ribbon presentation. It is easy to verify by standard arguments that isotopies and band- stabilizations of the ribbon presentation yield the same Ac. Therefore, AG is well-defined for ribbon knots. For Artin spun knots, Giller’s polynomial is the Alexander polynomial. In the case that we apply these computations to the projection of the knotted sphere in an Artin spun twin, Giller’s polynomial is the Seiberg-Witten polynomial. (as shown in [FS98]). Interestingly, 2-knots and twins need not have a symmetric Alexander polynomial. Giller’s polynomial and the Seiberg-Witten polynomial, however, are symmetric. For ex- ample, see Figure 2.7, which is the spun right hand trefoil with crossing changes. The natural questions to ask are then: Is Giller’s polynomial an invariant of 2-knots? If so, is it equal to the Seiberg—Witten polynomial for the corresponding twin? For twins, what is the relationship between the Alexander polynomial and the Seiberg-Witten invariant? Our invariant provides suggestive evidence that the second question, at least, should be answered in the affirmative. 52 2.2 The 4-dimensional Macarefia 2.2.1 3-dimensional Hoste Move The main theorem of Fintushel and Stern in [F898] gives a way of computing the Seiberg- Witten Invariants of classical-knot surgered 4—manifolds in terms of the symmetrized Alexander polynomial of the knot. The proof relies on a technique J. Hoste developed in [H0384] which is a method for obtaining Kirby calculus diagrams for so called “sewn-up r-link exteriors” in S3. (Like F intushel and Stern, we will only consider the case where r-links are actually knots and links.) We discuss a simplified but sufficient version of the original move below so to demonstrate the ideas involved. 1 K1 K2 Figure 2.17. 3-dimensi0nal Hoste moVe A sewn up knot exterior is formed by taking either two oriented knots in one copy of S3 or in two separate copies, excising a normal neighborhood of each knot, and gluing the resulting boundary T23 by a diffeomorphism. For our purposes, we will let the diffeomorphism be the one which identifies oriented meridians and longitudes for the Seifert framings of each knot. This procedure does two things, it removes two copies of S 1 x D2 with a chosen framing and orientation, and it replaces them with an S1 x S 1 x I. Together, these are the boundary of S1 x D2 x I. Thus we may think of forming a sewn up link exterior as the result (on the boundary) of adding a round 4-dimensional 1-handle to B4 so that the feet of the round 1-handle are the two knots, each with the proper framing. Now, consider a projection of a link L in S3 with oriented components K1, K2 and a 53 small region in the projection where K1 and K2 run parallel but in opposite directions. We can then connect K1 and K2 via an arc. See Figure 2.17. Attach the round handle as above to form the sewn up link exterior for K1 and K2. Note that we can choose the attaching map of the round handle so that in the corresponding Morse-Bott function, the points p1, p2 on K1, K2 where the arc touches each knot are both connected to the same point p on the critical S 1 by gradient flow lines. Take a perfect Morse function on the critical S1 of the round handle so that the index zero critical point is p. This decomposes the round 1-handle into a 1-handle and 2-handle corresponding to the 0- and 1—handles of the Morse function on S 1. "III-"II| Figure 2.18. Round handle becomes a 1 and 2 handle Attaching the 1-handle to S3 results in self connect summing S3 at the points p1,p2. By standard tricks, this is the same as zero surgery on the unknot around the arc in the second drawing in Figure 2.17. The attaching circle of the 2—handle is the “band sum” K of K1 and K2 as shown in Figure 2.18 and the second picture in Figure 2.17. Attaching the 2-handle to the result (an 54 S1 x S2) of the previous surgery is then merely zero surgery on K. 2.2.2 4—dimensional Hoste Move In [F898], the Hoste move shows up in 4—dimensions with an S1 equivariance as we cross the 3—manifold with S 1. When that is done, the surgeries on knots show up as surgeries on square zero tori which, by using [MM897], are amenable to computations of the Seiberg- Witten invariant. The 4-dimensional version of the Hoste move we will discuss here does not assume this S1 equivariance, although local S1 equivariances will occur. Proposition 2.2.1. Consider two embedded, oriented square zero tori T1,T2 in a 4-manifold X. Suppose that T1, T2 are connected by an annulus A = S1 x I, embedded in X, so that EA consists of an essential curve on each torus. Let each T,- be framed so that A n 6N(T,) is in the subspace of H1(6N(T,)) = H1(T3) generated by the pushoffs of loops on T,- with respect to the framing. Let :2 be a diffeomorphism T1 —+ T2 which identifies the components of 6A in T1 and T2. Then the self fiber sum, X # , is also the result of surgery on two tori: the “band sum” of the tori along A and toru:¢gii/en by the loop in Figure 2.19 in the neighborhood of 0 x I C A for each 6 E S 1. Figure 2.19. Band sum Proof of Proposition 2.2.1. We can reinterpret the fiber sum as the result (on the boundary) of adding a 5-dimensional toric 1-handle (a. T2 x D2 x I) to X x I so that the attaching region, T2 x D2 x i1 is identified with the normal bundles to T1 and T2 with their chosen framing. This results in deleting the two T2 X D23 and replacing them with a T2 X 8D2 X I. 55 This identification is determined by choice of framings for T1,T2 and a diffeomorphism (15 between them. Choose a factorization T1 E4, T2 E S}, x 8}, so that the S; factor is 0A in both Tis. The critical T 2 for the 5-dimensional toric handle is identified with the Tis by the gradient flow. Pick a perfect Morse function on S; and perturb the Morse-Bott function on the 5-dimensional handle by an extension of it. This gives us a reinterpretation of the 5- dimensional toric 1-handle as two round (S 1) handles — a round 2-handle and a round 1—handle — corresponding to the critical points of the Morse function on 32,. Consider the round 1-handle first. Such a handle is a S 1 x D3 x I so that it attaches along S1 x D3 x :l:1. By construction, the two S1 x D35 are neighborhoods of the components of 6A, with framing given by the inward normal along A, a vector field along 8A parallel to T1, and a third vector field defined by orthogonality to these and the tangent space to 6A. Consider a neighborhood of A which is S 1 equivariant, matching the S1 equivariance of A = I x S]. When small, such a neighborhood is diffeomorphic to S1 times the “H” in Figure 2.19. (The vertical lines are in T,; the horizontal, slices of A.) Attaching the round l-handle is the same as (equivariantly) self-connect summing at the places where the vertical lines intersect the horizontal core of the band. In each 3-manifold slice, this is equivalent to performing zero surgery on the loop linking the band in Figure 2.19. Then, in turn, this gives us a square zero torus L and a surgery to perform on it within the neighborhood of A. Now, a 5—dimensional round 2-handle is a S l x D2 x D2 attached along S1 x D2 x 6D2. Outside of the neighborhood of A, the attaching torus is equal to the Ti. (two annuli) Inside the neighborhood of A, the attaching torus T is given by S1 times the boundary of the band in Figure 2.19. (two more annuli) The framing of this torus is given by the framings of the T,- outside the neighborhood of A and by the inward normal to the band on (each slice of) the inside. This can be seen by band summing pushoffs of the Ti. (By hypothesis, A has zero winding with respect to our framing so this can be done by band summing in the S l trivialization. ) T inherits a factorization S; x S; from the T; by noting that the S}, factors of the T,- survive and that the S [13 factors are themselves band summed. Attaching the round 2—handle then performs a surgery on this torus which sends CD2 to the 8,] factor. CI 56 4-Dimensional Hoste Move for a Twin and Torus Let us now examine how Proposition 2.2.1 can be described in terms of surgery information on projections. First we will look at the case when the tori come from surgeries on a twin and a torus. Let X be the result of standard surgery on a ribbon twin 'll‘w and (0,0) surgery on a ribbon torus T in S4, both specified by a virtual knot diagram as in sections 2.1.6 and 2.1.7. Note that the cores TTMTT of each surgery inherit preferred framings from their surgery description. Suppose that 'll‘w and T have a classical knot crossing in the virtual knot diagram and hence a ribbon intersection. Then, in a projection to R3, there is a neighborhood as in Figure 2.20. Consider the annuli shown in the figure. Each of these annuli are isotopic. This can be seen by the fact that on the left of each picture, the horizontal surface overcrosses the vertical surface so the annulus must lie completely under the horizontal surface to the left of the ribbon singularity. Thus we can isotope the annulus freely on the left of the ribbon singularity. Similarly, on the right the annulus lies completely above the horizontal surface and so we can isotope it freely on that side. Now let B be the particular representation of the annulus corresponding to the particular orientations of the virtual knot crossing depicted below it. Now, B connects an equator 73“, to one of K,- in 'll‘w = K1 U K2 to a essential curve 7T on the torus. Since we have done (0, 0) surgery on the torus (in virtual knot notation), the surgery curve (in the projection notation) meets 7T once. The method of constructing B ensures that BB consists of essential curves on the cores of the twin and torus surgeries when B is extended by the projection to the surgered manifold. This means that we can apply proposition 2.2.1 once we have chosen the diffeomorphism <23 between TymTT. We have already required that (I) identify the components of 68. (p is then determined when we require that it identify the following: o the projection to TT of ptl x BDQ, where ptl E T, T x D2 C S4 the normal bundle, and o the projection to Tin-W of pt? x BDQ, where pt2 E Tw \ (S2 intersection points) and Figure 2.20. Annulus B used for smoothing lies in the 32 which does not contain the preimage of the double points. 8D2 is the boundary of the fiber of the normal bundle to this S2. With this data fixed, proposition 2.2.1 can be applied. The result is that X \ (N(Trw) UN(TT)) = 34 \ (N(TW) U N(T)) sewn up by (b is diffeomorphic to s4 \ (N(TwfiT)U7-)UT2 x D2UT2 x D2 where o 'll‘w #ST denotes the “band sum” of 'll‘w and T along B, o the first T2 X D2 is glued to 0N(TW%T) by the standard surgery on twins, 0 T is the torus given by the loop in Figure 2.19, and o the second T2 x D2 is glued to 6N(r) so that (9D2 goes to the nullhomologous pushoff of the loop in Figure 2.19. Finally, we can isotopy this region in Tw#T to be as shown in one of the pictures in B the top row of Figure 2.21. The appropriate smoothing depends on the orientation of the horizontal surface and corresponds to the selection of band previously. In Figure 2.21, the correspondence of the orientation of the horizontal surface to the virtual knot diagrams is 58 +- .__/K _L\\ Figure 2.21. Two smoothings of the intersection on Figure 2.20 with corresponding virtual knot diagrams show by the original diagrams to the lower left of each smoothing and the smoothed virtual knot diagrams to the lower right. 4-Dimensiona1 Hoste Move for two Tori We will require the Hoste Move between two tori in only one case. Suppose that both tori lie in a neighborhood diffeomorphic to S1 x D3 so that in each 0 x D3 the tori are as shown in Figure 2.22. The we only need to describe one aspect of the 3—dimensional local picture —7— the gluing map for the sewn up exterior for the right hand side of Figure 2.22. This map is given by identifying meridians to each loop and the pushoffs along the obvious once punctured discs to each. 2.2.3 4—Dimensional Crossing Change Consider a classical crossing in a virtual knot diagram for a twin and/or a torus and the corresponding annulus from Figure 223. Notice that the correspondence is reversed from that of the 4D Hoste move. As before. both bands shown are isotopic. Push the horizontal surface along the annulus in Figure 2.23 to get the configuration in Figure 2.24. 59 O O L(_I LC_I r i re? Figure 2.22. S1 times the left is S1 times the right with the two starred tori sewn up Figure 2.23. Annulus B used for crossing change 60 Figure 2.24. Diagram obtained from the first picture in Figure 2.23 by pushing horizontal tube along band, over crossing the vertical tube in the two new sets of double points. The new pair of crossings is locally modeled on 5'1 times the diagram in the lower right. The diagram corresponding to the second picture in Figure 2.23 is similar. Focus our attention to the lower of the two new loops of crossings in Figure 2.24 (or the corresponding picture for the other band.) Call this crossing C. Local to C, we have the model of S1 times a 3-dimensional oriented knot crossing. Note that if we form an annulus by taking a path from the lower to the upper double point in each 3-manifold picture, we get an annulus A which is isotopic to the annulus B from before. Perform one of the surgeries on a torus indicated by the 3-dimensional pictures in Fig- ure 2.25 localized at C in Figure 2.24 (or the corresponding picture for the other band.) The appropriate surgery is the top for the first picture of Figure 2.23 and the lower for the second. This changes the crossing C’ from an overcrossing of the horizontal surface to an under crossing, resulting in Figures 2.26 and 2.27. Finally, we perform the isotopies indi- cated in these figures and see that our result has changed the crossing type from a classical + to — or from a classical — to + in the virtual knot diagram. It is important to note that the torus which we have surgered is isotopic to the surgered torus 7' of section 2.2.2. We identify the two via the isotopy of the band B. 61 Figure 2.25. Isotopy in 3-dimensional picture - — l = v - - Figure 2.26. Isotopy of first picture in Figure 2.23 after surgery and the corresponding virtual knot crossing of result 62 /—\ v + O O " ' 0 Figure 2.27. Isotopy of second picture in Figure 2.23 after surgery and the corresponding virtual knot crossing of result 2.3 Calculation of the Invariant for Certain Ribbon Twins Consider a twin TW 2 K1 U K2 given by a virtual knot presentation and the manifold X ('ll‘w) : E (2)-N. Suppose that the virtual knot presentation for K1 contains a classical crossing; so K1 has a ribbon intersection with itself. Let Tw+, Tw- and Two be the results of replacing the crossing in the virtual knot diagram with the three options in Figure 2.28. Note that Two will actually be a twin and a torus. /\ \/ /+ -\/ 0. Therefore, the Seiberg—Witten invariant of this surgery vanishes. It follows that SW(E(2)-EMT) = 0 and that I(TW,T) = 0 for 'll'w,T split. (2.5) Now let us consider the Conway-style relation for Giller’spolynomial we initially dis- cussed in Section 2.1.8. This relation involves crossing changes and resolution at individual loops of double points. Now, in our crossing change surgery, we had a similar action of changing the lower crossing from Figure 2.24. In the diagrams for our 4D Hoste move, we took a different local projection to illustrate the appropriate surgery. However, smoothing the lower crossing from Figure 2.24 also yields a smoothing in the virtual knot diagram. In other words, selecting a particular set of double points to apply the relation in Equa- tion (2.2) to, Ac('ll'w) computes I (Tw). Therefore, Aa(TW) = I ('ll‘w) for ribbon knots. We cannot make a stronger statement of equality however, as the relation from Equation (2.2) allows us to move into configurations of surfaces which are inaccessible to the invariant I. 2.3.2 The Class of Ribbon Twins we now make some remarks on computations. 66 Now suppose that 'll‘w = K1 U K2 is ribbon with a virtual knot presentation which only has virtual crossings. Then we can use the virtual Reidemeister moves B and F from Figures 2.12 and 2.13 to completely unknot the diagram for Tw. Therefore, 'Il‘w = Twstd and so I(Tw) = I(Twstd) = 1. Suppose that 'll‘w = K1 UK2 is a twin possibly with accompanying torus T with the configuration ribbon. Suppose that we reverse the orientation of one of the K.- or of T. This reverses the orientation of the torus T-rw (or TT) and induces a chance in homology orientation from the change of sign in pairing with T1,”. Then, if Tw = TGU K2, I (Tw) = ——I('lI‘w). Similarly, I(Tw,T) = —I(']I‘w,T). Currently, the author is unaware if crossings can be chosen so that the tree of terminates in standard twins and unlinked twin/ torus pairs. The previous work of Fintushel and Stern guarantees that the process terminates when K1 in Tw = K1 UK2 is knotted with only classical crossings and K2 is unknotted with no ribbon intersections with K1. The presence of virtual crossings in the diagrams complicates the general case. Additionally, the author has yet to find a general method of dealing with ribbon intersections between the Kg. However, there seem to be a fairly large number of new examples which we may compute using the current tools. The first we will compute is the twin version of the example from Giller’s paper. Call this twin 'll'wG. This was encountered previously in Figure 2.14. Follow the computation through Figures 2.30, 2.31, 2.32, and 2.33. In Figure 2.33, we arrive at configurations C, D, E, and F. Here C and F are isotopic to standard twins, so I (C) = I (F) = 1. The other configurations D, E differ by the orientation of the torus and so I(D) = —I(E). Therefore, I(Twc) = I(A) + (t — t‘1)I(B) = I(C)+(t—t‘1)I(D)+(t—t‘1)I(E)+(t—t‘1)21(F) = 1+0+(t—t‘1)2 = r2 — 1 + t2 Now let us look at the twin in Figure 2.34, a twin in which both 2-knots are unknots but which pairwise have ribbon intersections. Call this twin 'll'wU. Our current tools do not allow us to deal directly with pairwise ribbon intersections. 67 f\ \J j» Figure 2.30. Giller Twin 'll‘wG with highlighted crossing Figure 2.31. I(TwG) : I(A) + (t — t‘1)I(B) /\ \J I» m I Af'\ ‘\/ l-u Figure 2.32. Isotopy of A and B from Figure 2.31 with highlighted crossings 68 0 {H9 A/\ ‘1 cg ELL \/ Figure 2.33. I(Twc) = I(C) + (t — t‘1)I(D) + (t - t—1)I(E) + (t — t‘1)21(F) Follow the computation through Figures 2.35, 2.36, and 2.37. We arrive at configurations H, K, and L. Here H and L are isotopic to standard twins, so I (H ) = I (L) = 1. The other configurations K contains a separated torus so I (K) = 0. Therefore, I(Twu) = I(H)-(t—t‘l)I(J) = I(H)—(t——t‘1)I(K)+(t—t1‘1)2I(L) = 1+0+(r—r—1)2 = 2:“2—1+t2 Finally, we remark on uniqueness and related topics. In what is our Artin spun case, Fintushel and Stern have conjectured that their knot surgery construction yields nondif— feomorphic manifolds for “essentially different” knots. (Here, “essentially different” means that two knots are not isotopic nor are they mirror images of each other.) The Alexander polynomial does not completely distinguish knots however, so the Seiberg-Witten invariants in their current form do not shed any light on their conjecture. Similarly, it seems doubtful that the manifolds E (2)11‘wa and E (2)-1pm, are diffeomorphic, but with the Seiberg—Witten invariants being equal, we have no obvious way in which to distinguish them. In particular, it seems possible that E(2)Tw,, is diffeomorphic to E(2)11-WK where 'll‘w K is the Artin spin 69 of the left handed trefoil. Also, results of C. Taubes in [Tau94] show that a manifold X with b+ > 2 admits a symplectic form, the leading term in SW(X) will have coefficient equal to one. The converse to this statement is known to be false by work of Fintushel and Stern in [F897]. In the classical (or Artin spun) case, it is possible to construct a symplectic form on E (2)1-w when the classical knot K from which 'll‘w is constructed is a fibered knot. While it may be possible to rephrase this construction in terms of twins, it is unclear what topological conditions are required on 'll‘w to achieve the same result. (A sufficient condition is that S4('Ifw) fibers over T2 or S2.) 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