SLICE KNOTS IN S3

By PATRICK M. GILMER*

[Received 4th September 1980; in revised form 22nd March 1982]

Introduction

A KNOT K in S3 is said to be slice if it is the boundary of a properly andsmoothly embedded 2-disk A in D4 . Let F<= S3 be a Seifert surface for Kwith Seifert pairing 0: HX{F) x H^F) - • Z. Let L denote the doublebranched cover of S3 along K. Define e: H^F) -> H^F) by e(x)(y) =0(x, y) + 0(y, x). Let A <= H1(F)<S>Qfl denote the kernel of e<8>idQ/z. Wewill give an isomorphism of A with H'(L, Q/Z). This isomorphism isnatural up to sign. A. J. Casson and C. McA. Gordon have defined aninvariant r(K,x) of elements verf'(L,Q/Z). Since T(K,X) = T(K,~X),

we can view T as defined on A. Let A ' c A b e the subset of elements of Awith prime power order.

The following theorem combines the results of Casson and Gordon [3],[4], [5], [7] (Section 13) with the earlier work of J. Levine [11] in anontrivial way.

THEOREM (0.1). / / K is slice then there is a direct summand H of H^F)such that

1) 2 rank H = rank Hi(F).2) 0(HxH) = O.3) For all XeA'nH®Q/Z,T(K,x) = 0.

In Section 1, we will give a proof of this for homology slice knots inhomology 3-spheres. In Section 2, we will use this theorem to define ahomomorphism of the knot cobordism group to a certain Witt group F .The cobordism class of a knot K goes to zero in F if and only if Ksatisfies the conclusion t»f Theorem (0.1). We also discuss the relation ofF to previous Witt groups defined by Levine and Casson-Gordon: G_and A'.

In Section 3, we discuss r(K, x) and show how to estimate T(K, X)(actually OXT{K, X)) in terms of curves lying on a Seifert surface. Moreprecisely, if x e H^F) is primitive let Cx denote the collection of knots inS3 obtained by representing x by a simple closed curve y on F and thenviewing y in S3. Our estimates together with (0.1) give the followingcorollary.

COROLLARY (0.2). If K is slice then there is a direct summand H of

* Partially supported by N.S.F. grant MCS78-02977 and an N.S.F. Postdoctoral ResearchFellowship.

Quart. J. Math. Oxford (2), 34 (1983), 305-322

306 PATRICK M. GILMER

HX(F) such that1) 2 rank H = rank Hi(F).2) 0(HxH) = O.3) If xeH is primitive, and x ® sime A' for some 0 < s < m, then for all

J e Cx, we have

kwm)(J)l« genus (F).

Here o-(l/m) is an ordinary knot signature. Steve Kaplan has indepen-dently found a result similar to (0.2). His bound on |a(s/m)(/)| is 2(genusF ) - l . He uses the approach of [4], while we follow [3] more closely.When F has genus one, our result coincides with his. Note that in thegenus-one case, Cx is a singleton. We denote its element Jx. Also o-(j/m) ofa knot is always even. This yields:

COROLLARY (0.3). Let K be a slice knot which possesses a genus-oneSeifert surface F. Then there exists a primitive x e H^F) such that 0(x, x) =0 and cris,m)(Jx) = 0 for all 0<s<m where m is any prime power dividing

V|det(0 + 0T)|.

Corollary (0.3) lends itself to an interesting interpretation. Namely if Jx

is slice then one can use the smooth disk in D4 with boundary Jx to doambient surgery on F and obtain a slice disk for K. This is essentiallyLevine's program (which works in higher dimensions) for showing knotswith metabolic Seifert pairings are slice. On the other hand, if Jx is slice,o~>im(Jx)= 0 for all 0<s < m and m a prime power. If a genus-one Seifertsurface has a metabolic Seifert pairing, then there are exactly two (up tosign) primitive classes with square zero. Let J\ and J2 be the associatedknots. If F fails to satisfy the conclusion of (0.3), then neither / t nor J2

can be slice, and Levine's program cannot be carried out. Thus theCasson-Gordon invariants seem to be secondary obstructions to carryingout the program.

In Section 3, we also calculate T(K, X) exactly in the case of genus-oneknots. It turns out that the cobordism class of sucti a knot maps to zero inF if and only if the conclusion of (0.3) is satisfied.

Finally in section 4 we give an example of a non-slice knot which isdetected by F but not by G_ or A'.

I wish to thank R. A. Litherland for pointing out an error in the proofof Theorem (1.1) in an earlier version of this paper. At that time I knewanother correct proof for (0.1), given in [13]. However, this method couldnot be used to prove the homology slice version (1.1) below. Thus, I havemodified the original proof in a way suggested by Litherland.

Section 1

Suppose more generally we have K a knot in a homology sphere S. Bythe same transversality argument as is used in S3, we can find a Seifert

SLICE KNOTS IN S 3 307

surface FcS. Let 6 be the Seifert pairing, and define e,A, A' as in theIntroduction. Let L be the double branched cover of S along K. Thereare several ways to give the identification of A with H^L, Q/Z). Webegin with a method which relates well to the proof of Theorem (1.1). Wediscuss some other methods following the proof of (1.1).

Let X denote S slit along F, i.e., the complement of the interior of B, abicollar neighborhood of F. L can be constructed by gluing together twocopies, say Xo and Xj of X along dX0 and dXj appropriately. The decktransformation T acts on L by switching Xo and X1. Let F+ (respectivelyF_) denote the lift of F in L which corresponds to the copy of F in dX0

lieing just above (respectively below) F.As is well known, Hi(L) is an odd torsion group, see [3] Lemma 2, for

instance. Since L/T = S and H^S) = 0, a standard transfer argumentshows T cannot fix any nonzero element in H^L). Therefore x + T(x) = 0for all x in Hi(L). In other words, the action of T on H1{L) ismultiplication by — 1. It follows that the inclusion induces a surjection ofH\(X0) onto Ht(L). Thus a homomorphism ^€H'(W,Q/2) is deter-mined by its restriction x\Xa- Thus restriction gives an injective mapH1(L,Q/Z)->H1(X0,Q/Z).

We wish to determine the image. If t/» eH'(X0 , Q/Z) extends to *eH\L, Q/Z) then x\x, = -X\xoT=->l>T. Thus ip extends iff 4> agrees with-tj/T along dX0 = dX1. However T acts on dX0~F+UF_ by interchangingF+ and F_. T maps t/f|Xo+ <A |x<, to its negative. Thus i/> extends to Hi(W)iff t/>|F + i^T|F+= 0. Thus we have an exact sequence

0 -> H\L, Q/Z) -+ H\X0, Q/Z) -> H 1 ^ , Q/Z)

We have isomorphisms (with Q/Z coefficients understood) Hl(X)~H2(S,X) = H2(B,aB) = H1(B) = H1(F) given by, respectively, thecoboundary, excision, Lefshetz duality, and a homotopy equivalence.Using the identifications of Xo with X and F+ with F, we have an exactsequence

0 -* H\L, Q/Z) -+ H^F, Q/Z) -» H'(F, Q/Z).

We wish to identify the last map with e <S> Jdo/z. The universal coeffi-cient theorem gives isomorphisms: H, (F) <8> Q/Z = H,(F, Q/Z) andH\F,<Q/Z) = H\F)®Q/Z. If / is a curve on F representing x e H ^ f ) ,then x ® sld corresponds under the above isomorphisms to the elementof H2(S, X, Q/Z) = Horn (H2(S, X), Q/Z) which assigns to a 2-cycle a, thevalue (sld)(a ° J). Here ° denotes intersection number. Therefore x <8> sldmaps to the element iJ/eH1(X0, Q/Z) which assigns to a 1-cycle b, s/dtimes the linking number of b and J. Moreover (/f|F+ + t/<T|FVeH1(F+, Q/Z)is then s/d e(x). Our identification of Hl(L, Q/Z) with A depends on achoice of a component of the cover of X to be Xo. If we make a differentchoice we only change the isomorphism by a sign.

308 PATRICK M. GILMER

We say K in S is homology slice if there is some homology 4-ball Dwith 3D = S and a smooth properly embedded 2-disk A in D withboundary K.

THEOREM (1.1). If K is homology slice, then the conclusion of Theorem(0.1) holds.

Proof. Since D - A is a homology circle, a standard transversalityargument produces an embedded, oriented 3-manifold R <= D with 8R =FUA. Let HaH^F) be the inverse image of the torsion subgroup ofH\CR) under the map induced by inclusion. Since FUA is the boundaryof R, another standard argument (using Lefshetz duality) shows 2 rank Hequals rank Hj(F). Curves on F representing elements in H rationallybound surfaces in R. These surfaces can be used to calculate linkingnumbers in S. One sees 0(HxH) = O. We have just recapitulatedLevine's argument in this dimension [11].

Now let W be the double branched cover of D along the slice disk Afor K. Let i denote the inclusion of L in W, and T the coveringtransformation. By the argument in [3], Lemma 2, or using Smith theory,H^W) is an odd torsion group. Since Hi(D) = 0, T must act as multipli-cation by —1 on HxiW) (by the same argument given for H^L)).

Let Y denote the complement of the interior of a bicollar neighbor-hood of R in D. W then can be built from two copies Yo and Y1 of Y.We have an exact sequence for W analogous to that given for L above.The first map, for instance, is given by (with Q/Z coefficients understood)

« H2(R x.I,Fxr)~ H2(R, F).

Moreover, it fits into a commutative diagram (due to Litherland)

0 • H\W, Q/Z) • H2(R, F, Q/Z) > Hl(R, Q/Z)

• H\L,Q/Z) > H^F,Q/Z)-^UH\F,Q/Z)

H,(R, Q/Z)

The first vertical map is given by restriction. Now H®Q/Z is the kernelof /$ and thus the image of d. Therefore the image i* is a subgroup ofADH® Q/z.

Let {x, , . . . , Xg} be a basis for H extended to a basis{xu . . . , X,, yx, . . . , yg} for H^F). Let {xf,..., y*} be the dual basis with

SLICE KNOTS IN S 3 309

respect to ( > the nonsingular intersection pairing on H,(F). Thus{yf,..., yf} forms a basis for H x the subspace that anhilates H under (, >.Since (x, y)= 6(x, y)-0(y, x), H^H^. Since they have the same rank,H = H \ Thus {x1 ; . . . , xg, xf,..., x*} is a basis for H^F). With respectto this basis 6 is given by a matrix of the form

[ 0 C + HlCT E J

where E = E T . Let {zu ..., z2g} be the dual basis for H\F). With respectto these bases e is given by

T 0 2C+H12CT + I IE \

Let m denote |det (2C + /) | then using the above matrix representation ofe we see \A\ = m2 and \A nH®Q/Z | = m. Since A =H1(L, Q/Z), wehave |Ht(L)| = m2. By Lemma 3 of [3], the image of H,(L) in H^W) hasorder m thus |image i*\ = m. Since image i* is a subgroup of A D H ®Q/Z and they both have the same order, we can conclude they are equal.Thus if xeAnH<8>Q/Z, x extends to an element of H\W,Q/Z). ByTheorem 2 of [3], T(K, X) = 0. •

We now give some alternative descriptions of the identification of Awith Hl(L,Q/Z). Let V denote a matrix for 6 with respect to some basisfor H^F). Rolfsen, 8.D.1 (page 212) [12], shows that V+VT (a matrixfor e) is a presentation matrix for H^L). Using the above basis foriTjCF), we can write an element in A as a vector with Q/Z coefficientswhich, when multiplied by V+ VT, becomes zero. The corresponding xassigns to the ith generator for Hi(L) the ith entry in this vector. Thekernel condition insures that all the relations map to zero.

Another way this presentation arises is as follows. Let D denote thebranched cover of D along F pushed slightly into D [8], [1]. Then D maybe obtained by adding 2-handles according to a framed link in theboundary of some homology ball (D4 if D = D4). The linking matrix forthis framed link is V+ VT. The meridians for the components can beidentified with Rolfsen's generators. One may use D to understand thelinking form I on H^L). Let c denote the correlation isomorphismH J C D - ^ H ^ L , Q/Z) given by c(u) = l( , u). Define a linking form j3 onH\L, Q/Z)«= A by P(cu, cv) = -l(u, v). Then one can show

0(x <8> r, y <8> s) = rs[6(x, y) + 0(y, x)] mod 1

where f and s are rational numbers that reduce mod 1 to r and s.This means given two elements of A written as vectors with Q/Z

310 PATRICK M. GILMER

coefficients a and b, pick any two vectors d and 6 with rational coeffi-cients which reduce mod 1 to a and b, then multiply dT(V+ VT)6 outand reduce mod 1.

Section 2One may form a knot cobordism group C. Elements of C are equival-

ence classes of knots K in S3. Two knots Ko and K\ are equivalent ifthere is a smooth proper embedding / : SxxI—*S3xI with /(S'xO) =K 0 C S 3 X { 0 } , / ( S 1 X 1 ) = XI<=S 1 X{1}. The group operation is given bythe connected sum of knots. Inverses are given by taking mirror imagesand reversing string orientation. This group was first studied in 1957 byFox and Milnor [2].

One can define a related group 3? as follows. One forms a semigroup ofisomorphism classes of pairs (S, K) where S is a oriented homology3-sphere and K is an oriented knot in S. We say (S, K) is null cobordantif there is some homology cobordism W* of S to some other homology3-sphere and there is a smooth D2 <= W with dD2 = .K <= S. If both(So, Ko)#(Su Kj) and (Si, KJ are null cobordant then so is (So, Ko). Let—(S, K) be obtained from (S, K) by reversing the orientations of S and K.We say (So, Ko) ~ (Su Kx) if (So, J«C0)#-(S1, KJ is null cobordant. This isan equivalence relation. "<K is then equivalence classes of pairs withaddition given by the connected sum. There is a natural morphism

In 1968, Levine [11] defined a group G_ and a epimorphism C—*G-as part of his study of codimension-2 knot cobordism. See Kervaire'sarticle [9] for an excellent discussion. We briefly outline the definition ofG_. Elements of G_ are equivalence classes of pairs (V, 6) where V is afinitely generated free Z module and 6 is a bilinear pairing with theproperty that 0 - 0 T is unimodular. (V, 0)~(W, 0') if (V, 0)©(W, -0') ismetabolic (i.e. admits a half dimensional summand on which the formvanishes). The map from C to G_ assigns to the class of K the class of(H^F), 6). This clearly factors through 9?.

Casson and Gordon have also defined homomorphism from X to adifferent Witt group A' [5]. A' is defined to be equivalence classes oftriples (A, /3, T) where A is a finite abelian group, 0: AxA—> Q/Z isbilinear and nonsingular (in the sense that the induced map A —*Horn (A, Q/Z) is an isomorphism) and T: A —*• W(C(t), J)®Q is a mapof sets. Let A ' c A be the subset of elements of prime power order.

One defines (Ao, /30, To)©(A,, fiu T,) = ( A O © A,, ft,® Pi, T O © ^ )where j3o©/3i(*o©*i, yo© yi) = /3o(*o. yo) + 0i(*i> yi) and TO©T1(XO+xi) = T(XQ) + T(Xi). (A, p, T) is called metabolic if there exists G<^ A such

SLICE KNOTS IN S 3 311

that

1) |G|2 = |A| 2) 0 (GxG) = O 3) T ( G D A ' ) = 0

One defines -(A, 0, T) = (A, -/3, - T ) and says (Ao, /30, T0) ~ (A,, 0U Tj) if(Ao, 0o» T0) © -(A l 5 0!, Tt) is hyperbolic. One proves by a formally paral-lel proof to Kervaire's for G_ that ~ is an equivalence relation.

The homomorphism of Casson and Gordon sends the class of {K, S) tothe class of (Hl(L, Q/Z), /3, T(K, )). There is an analogous group Adefined identically except condition 3) above reads T(G) = 0.

We will next define another group F which combines G_ and A' in anontrivial way. In fact one has a commutative diagram.

In Section 4, we will give an example of an element in C whose images inA' and G_ are zero but whose image in F is nonzero.

F' is defined to be equivalence classes of triples (V, 0, T) where V is afinitely generated free Z-module, 0 is a bilinear pairing 0: VxV-+Zsuch that 6 - 0T is nonsingular, and T is a function from A to W(C(t), J)<8>Q. Here A is an abelian group associated to (V, 0) as follows: A =ker e ® idoyzC V®Q/Z, where e: V—•HomCV, Z) is given by e{x)y =0(x, y) + 0(y, x). We think of a triple (V, 0, T) as a single algebraic object.

One defines (Vo, 0O, To)©(V,, 0U n) to be (Vo© Vu BO®BU TO®TX)where 0o©0i ar>d TO©T! are defined as in the definitions of G_ and A'.(V, 6, T) is called metabolic if there exists a direct summand H of V suchthat 1) 2 rank H = rank V; 2) 8(HxH) = 0; and 3) T ( A T I H ® Q / Z ) = 0.One defines - ( V , 0 , T ) = ( V , - 0 , - T ) and (V0, flo, To)~(V,, fllf Tl) if(Vo, 0O, T 0 )©—(VL 61, Tt) is hyperbolic. We remark that one may definea similar group T in the same way except condition 3) above should read

The cancellation lemma below shows that ~ is an equivalence relation.F is then ~ classes of triples (V, 6, T). The homomorphism from X to Fsends the class of K to (H^F), 6, T(K, )). Theorem (1.1) and Proposition(3.2) allow one to see that this indeed gives a homomorphism. One finallyhas homomorphisms from F to G_, respectively A' sending (V, 0, T) to(V, 0), respectively (A,fi,y). Where A<=V®Q/Z is as above and0(x® r, y ®s) = +rs(0(x, y) + #(y, x)) mod 1. Here r,s are any elementsof V®Q which reduce to r and s.

312 PATRICK M. GILMER

CANCELLATION LEMMA. If (W, i/>, A) is metabolic and (V, 0, r)ffi(W, ip, A) is metabolic, then (V, 0, T) is metabolic.

Proof. (Again based on Kervaire's proof for G_). Let L^V(BW andK c W b e the direct summands with half the rank which make the triplesmetabolic. Let H e V be the smallest direct summand which includesp(LDV(&K) where p: V© W —* V is the projection. Kervaire showsthat 2 rank H = rank V and that 0(HxH) = O.

We only need to show that r(H®Q/ZnA') = 0. L e t B ' c f l c W ® Q / Zdenote the analogous subsets to A and A'. Thus A is defined on B, andT © A is defined on A®B. We are given that A(K®Q/ZnB') = 0 andT©A(L®Q/Zn(A©B)') = O. Let p also denote the projection p: A®B-»ALDefi_ne H as p(L ®Q/2n(V®Q/Z)©(X®Q/Z)nA ©B).

Let W <= H be the subset of elements of prime power order. If h e H',then h = v®b and / ® a = (v ® b) © (fc <8> c) where / € L, t> e V, k e K, anda, b, c eQ/Z. Moreover qrb = 0 for some prime q. Write the denominatorof c as q'd where deZ and d^Omodq. By the Chinese RemainderTheorem, there is an neZ such that dn = 1 modq'. We have dnh =v® dnb = v ® b = h, dn(l ® a) = h® (nk <8>dc) and dc has prime powerorder. So r(h) = 0. Therefore r(Hr) = 0.

It is clear that H<=H<8>Q/ZnA. it is easy to show that |H®Q/ZnA|2 = |A| (see the proof of (1.1)). On the other hand |H]2 = |A|. Thisfollows from the proof of the cancellation law for A which in turn ismodelled directly after Kervaire's proof for G_. It follows H =H ® Q / Z n A and we are done.

Section 3Recall T(K,x)eW(C(t),J)®Q, [7]. Here C(t) denotes the field of

rational functions over C in the variable t and / is the involution on C(t)that conjugates the complex numbers and sends t to f~\ W(C(t), J) is theassociated Witt group of finite dimensional hermitian inner productspaces. The signature is an isomorphism from W(R) the Witt group over Uto Z. There is a natural map W(R) -» W(C(t), J). Together these yield ahomomorphism p: Q—* W(C(f), J)®Q. There is also a homomorphismov W(C(t), J )®Q^ 'Q , [3]. It is easy to see that <rx°p is the identity.

Given a closed 3-manifold N and a map <p: H,(A/)^ Cm © C», onemay define an invariant T(N, <p)e W(C(t), J)®Q of the associated cover,in the way T(K, X) is defined in [7].

As before let L be the double branched cover of S3 along K. Let M bethe result of 0-framed surgery on L along K, the lift of K in M. There is anatural isomorphism H1(M) = H1(L)©Z. Given x- H,(L) ^ Q / Z , ^ willmap into some cyclic group Cm. Define x+- H\(.M) —> Cm © C«, by

SLICE KNOTS IN S 3 313

X+(x,n) = (x(x),tn). Then T(K, X) = T(M, x+l Define X G H ^ M , Q / Z ) byx(x, n) = xM- Since the deck transformation T, is a diffeomorphism of L,fixing K, and acting by - 1 on HX{L), T(K, x) = r{K, ~x).

Given <p G H'(N, Q/Z), define Tj(iV, <p) = dim H[(N, C). Here we use thenotation of [7] for twisted homology groups. Also define cr(N, <p) € Q as in[7]. These invariants are also discussed in [6] in slightly different guise.The relation is given in [3]. The proof of Theorem 3 in [3] shows that

) (3.1)

Let Ko and Kt be two knots and K2 = K0#Kl. For each knot K,, onehas as above L,, M,, K,, and A< = H\Li, Q/Z). We have 1^ = Lo#Lu K2 =

! and A2 = Ao® Au

PROPOSITION (3.2). 7 /x o eA o and Xi^Au then T(K0#K1,X0®XI) =

Proof. Let W be MoUMjXf together with a 1-handle joining the 2components and a 2-handle attached as indicated in Figure (la). Claim:d+W = M2- To see this slide a 2-handle to get a new description of d+W(b). Then trade the 2-handle for a 1-handle (c) yielding yet anotherdescription of d+ W. Finally remove the pair of cancelling 1 and 2 handles(d). The result is M2. We have 3W = M 2 U-M 0 U-M 1 . (Here the minussigns indicate reversed orientation). Moreover the Cm x C» covers ofMo, Mu M2 given by x~o, x t and {xo + Xi)+ extend t o a ^ x C . cover ofW.

To calculate H$(W, MoUMj, C(r)) we have a chain complex,

0 • C 2 ^

The map d2 can be calculated by reading off the intersections of theattaching circle of the 2-handle with the belt 2-sphere of the 1-handleweighted with elements of C © C . In this case d2 is multiplication by1-r and so Hi(W,MoUM,,C(t)) = 0. Thus the inclusion induces anisomorphism H*(M0UM,, C(f)) -> H't(W, C(r)) and the intersection pair-ing on H2(W,C(t)) is identically zero. One can also show Sign(W) = 0.The result follows. •

n

PROPOSITION (3.3). Let W = N*I \J h2, where the h2 denote 2-handles1 = 1

attached along curves yt in a closed 3-manifold N. Orient W so thatdW=-N\JP. LetxeH\N,QIZ) such thatx[yi] = 0 andx^O- X extendsuniquely to H^W) and thus defines x^H\P,Q/Z). Then

with equality mod 2.

314 PATRICK M. GILMER

K0#K

Fie. 1

SLICE KNOTS IN S 3 315

Proof. Here we write H'+( ) for H^( ,C). Since « ^ 0 , H^(N) = 0. LetAj = C\NxI, then H*(A) = ( S 1 ) . The Mayer Vietoris sequence forW as the above union shows H3(W) = 0. Another part gives

0-+ H'2(N) -» H^(W) -+ H{(U A) -» Hi(N) «- H{(W) - • 0

By Poincare duality H^ (N) = H[ (N), thus dim HJ (W) = dim H{ (W) + n.Let a and TJ denote the signature and nullity of the quadratic form on

Hi,(W). Then(1)

with equality mod 2. Since a matrix representing the form will alsorepresent the map H£(W)->Hi(W,NUP) we have

0 -+ C -> Hj (P) © HJ (N) -> H{ (W) -»• HJ (W, N U P)

By Lefshetz duality HJ(W, NUP) = 0. Thus we have

:)-7, (2)

Since H{(N) injects into Hi(W) and the intersection pairing vanisheson the image, -17^dim H2(N) = r}{N, x)- We may also view W a s P x Junion 2-handles. Thus T)S=T)(P, X) and

Equality mod 2 holds trivially. Using (1), (2), and (3) one has

with equality mod 2. Finally by definition

<T(P,X)-<T(N,X) = O--Sign W.

ForO<s<m, define o-^m{K) = Sign ((1-w')0 + (l-w-*)0') Tll/m(K) =nullity ((l-ws)O + ( l -w" ' )0 ' ) where w = e2m7m. Recall if m is a primepower, then r\lJm{K) = Q, [6] (3.1).

THEOREM (3.4). If x = x®s/meAc:Ht(F)®Q/Z, 0 < s < m , xeH t (F)is primitive and J e Cx, then

m

Proof. We can view F as a disk with 0i(F) twisted knotted bandsattached where the first has / tied in it and represents x. By [1], L is givenby surgery on a framed link with /3i(F) components. The linking matrix isthe matrix of 6 + 6T with respect to the basis of H^F) given by the bands.Let J* denote the knot obtained by reversing the string orientation on /.It is easily seen that J and J* have the same signatures and nullities. The

316 PATRICK M. GILMER

first component of the link is / # / * . x takes the value s/m on themeridian of this component and zero on the meridians of the othercomponents.

Let (N, x) be the 3-manifold given by 20(x, x)-framed surgery alongJ#J*. By [6] (3.6),

<r(N, X) = 2<7I/m(J)-Sign (0(x, x)) + 4(m~S)S0(x, x)m

Now L is obtained from N by O t ( F ) - 1) surgeries and M is obtainedfrom L by one more surgery. Thus M is obtained from N by 3i(F)surgeries. This gives us a W as in (3.3). M plays the role of P. MoreoverSign W = 0^(10-Sign (0(x,x)). So by (3.3),

<r(M, X)~2o-l/m(J)-4(ms)5-0(x, x) +m

This, the formula (3.1), and the triangle inequality give the result.

Remark. Given a particular choice of xe -A. there is a great deal ofchoice in applying Theorem (3.4). First there is the choice of x and thenJe Cx. The estimates of o-^riK, x)) obtained can vary widely. In particu-lar, a judicious choice of two estimates taken together can give a sharpestimate for CT1(T(K, X))- For example let K be the connected sum of ntrefoils. Take F to be the connected sum of n genus one Seifert surfacesfor the trefoil. There is an x e HX(F) coming from the Seifert surface ofthe first trefoil such that 6{x, x) = —3 and x = x®^eA. C, contains boththe unknot and the connected sum of n — 1 trefoils. This yields

and

Taken together this yields

In fact by (3.5) below a-^{K, x) — ~§• The inequalities above then read|2n — 2|*s2n. This shows that the inequality in (3.4) cannot really besharpened much.

THEOREM (3.5). J/g(F) = l, and x = x®slm, where 0 < s < m , m is aprime power and where x is primitiw, then

(K, X) = P(2a«m(Jx) + 4imm2

S)S 6(x, x ) -

SLICE KNOTS IN 317

Proof. View F as a disk with two twisted knotted bands with onerepresenting the class x with Jx tied in it. Let F' be formed by tying —Jx

in this band and putting —6(x, x) more twists in it. Let K' = dF' and ingeneral use prime to denote objects associated to K'.

Note that K' is slice. Moreover the 3-manifold R in the proof of (1.1)can be taken to be F ' x J U h 2 where h2 is a 2-handle attached along acurve representing x. So by the proof of (1.1), r(K', x') = 0. We will nowcompare T(K, X) to T(K', X')- This idea is due to Steve Kaplan.

Let N be the 3-manifold obtained by doing 6(x, x)-framed surgery toS3 along Jx. Let <p: H^N)^-^ map the meridian of Jx to e

2iril/m and<p+: HX(N) -»• Cm x Coo map the meridian to (e2irWm, 0). It is clear thatT(N, <p+) = p(a(N, <p)). By [6] (3.6),

2m

M

FIG. 2

318 PATRICK M. GILMER

Belt 2-sphere

FIG. 3

Let N* denote the result of 6(x, x) framed surgery to S3 along J*, thereverse of Jx. Then T(N*. V+) = T(N, <p+).

We build a 4-manifold W with boundary M'UNUJV*U-M such thatthe Q,, x Coo covers given by x+, X+''» a r |d <P+ extend to a cover of W. W isformed by attaching two 1-handles to (MU-NU-N*)xZ joining theseparate components and then attaching two 2-handles in a certain wayalong M # - N # - N * .

This is best described by example. Figure 2 illustrates K and x and alsoa surgery description of M obtained using [1]. In Figure 3 we showM#-N#-N* together with the belt 2-spheres of the 1-handles and theattaching circles for our 2-handles. If we ignore the belt 2-spheres, this isa surgery description of d+ W. Figure 4 shows another surgery description

SLICE KNOTS IN 319

FIG. 4

obtained by sliding 2-handles [10]. Finally we can erase the two trefoilswith framing-3 and the simply linking meridians with framing 0 as thesesurgeries cancel each other, and recognize M'.

By Novikov additivity,Sign W = a^K1) - (cr(i)(X) - 2 Sign (6(x, x))).

One then shows, as in the proof of (3.2), that the intersection pairing onH2(W,C(t)) is identically zero. Thus

T(K' , X') = T(K, X) ~ 2T(N, <p+) - p (Sign W).

This completes the proof.The results of this section hold equally well for knots in homology

3-spheres. The proofs require only minor modification.

Section 4When now give the promised example of a knot K whose image in V is

nonzero but which maps to zero in G_ and A'. Figure 5 is a picture of our

320 PATRICK M. GILMER

FIG. 5

knot K. A genus 1 Seifert surface F is clearly visible. With respect to thebasis {x, y}, the Seifert matrix for F is

ro 4iL5 0 1

Thus K has a null bordant Seifert matrix and so is zero in G_.A c H ^ F l ^ Q / Z is generated by x®J and y®£. (This is denotedA = (x <8> £, y ® £)) and is isomorphic to 2^ © Z9. There are exactly threesubgroups G such that \G\ = 9 and 0(Gx G) = 0. They are G, = (x ® > =2:9)G2 = <y®^) = Z9 and G3 = ( x ® i y ®\)^Z3®Z^. We need toevaluate T on each of these subgroups.

Let K(n, m) denote the (n, m) torus knot and J the mirror image of J.Note that JX = K(2, 7)#JC(2, 3)#K(2, 3) and Jy =JX. Using (5.1) of [6],crj(/x) = -<rj(/y) = 2 and o-j(JI) = -aj(Jy) = 0. Of course for any knotv\=<J\. By Theorem (3.5), T(x®|) = -r(y ®|) = p(2). Therefore T(G,)

SLICE KNOTS IN S 3 321

FIG. 6

and T(G2) are nonzero. Also we have

T(X ® £) = T(X <8> |) = r(y <8> i) = T(y ® §) = 0.

Now Jx+y=Jx#Jy#K(2,3)#K(2,3) so ^ ( J ^ , ) = - 4 . By (3.5),T((x + y)®i) = T((x + y)®|) = 0. Finally Jx^=Jx#Jy#K0 where Ko isshown in Figure 6. Xo is drawn with two extra crossings so that arelatively simple genus 4 Seifert surface is readily apparent. A straightfor-ward calculation shows cr$(K0) = 4. Thus T((X - y) ® i) = T((X - y) ® §) = 0.Thus T(G 3) = 0, and K maps to zero in A'.

Finally there are only two direct summands H of H^F) of rank 1 suchthat 6(HxH) = 0. They are H, =<x> and H2 = <y>. Since G; =H, ® Q / Z n A and T(GI) and T(G2) are nonzero, we conclude that Kdoes not map to zero in V.

REFERENCES

1. S. Akbulut and R. Kirby, 'Branched Covers of Surfaces in 4-manifolds', Math. Ann. 252(1980), 111-131.

2. R. H. Fox and J. W. Milnor, 'Singularities of 2-spheres in 4-space and equivalence ofknots'. Bull. Amer. Math. Soc. 63 (1957), 406.

3. A. J. Casson and C. McA. Gordon, 'Cobordism of classical knots, printed notes', Orsay,1975.

322 PATRICK M. GILMER

4. A. J. Casson and C. McA. Gordon, 'On slice knots in dimension three', Proc. Symp. inPure Math. XXX, (1978), part two, 39-53.

5. Lecture by A. J. Casson, July, 1978 in Cambridge, England.6. P. M. Gilmer, 'Configurations of surfaces in 4-manifolds', Trans. A.M.S. 264 (1981)

353-380.7. C. McA. Gordon, 'Some Aspects of Classical Knot Theory', Knot Theory, Proc. Plans-

sur-bex, Switzerland, 1977. Lecture Notes in Math. 685, Springer Verlag, (1978), 1-60.8. L. Kauffman, 'Branched coverings, open books and knot periodicity', Topology 13

(1974), 143-160.9. M. A. Kervaire, 'Knot cobordism in codimension two', Manifolds—Amsterdam 1970,

Lecture Notes in Math. 197, Springer Verlag, (1971), 83-105.10. R. Kirby, 'A calculus for framed links in S3', Inventiones Math. 45 (1978), 173-207.11. J. Levine, 'Knot cobordism groups in codimension two', Comment. Math. Helv. 44

(1969), 229-244.12. D. Rolfsen, Knots and Links, Mathematical Lecture Series 7, Publish or Perish, Inc.,

Berkeley, CA (1976).13. P. M. Gilmer, On the Ribbon Genus of Knots, Proc. of the Eleventh Annual U.S.L.

Mathematics Conference, October 1980, Research Series No. 52, Dept. of Publications,USL Lafayette, LA (1981), 63-78.

Institute for Advanced StudyPrinceton, N.J. 08540andLouisiana State UniversityBaton Rouge, LA 70803