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BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 80, Number 6, November 1974

RESEARCH ANNOUNCEMENTSThe purpose of this department is to provide early anno uncemen tof outstanding new results, with some indication of proof. Researchannouncements are limited to 100 typed lines of 65 spaces each. Alimited number of research announcements may be communicatedby each member of the Council who is also a member of a Societyeditorial committee. Manuscripts for research announcements

should be sent directly to those Council m embers whose names aremarked with asterisks on the inside back cover.

PRODUCTS OF KNOTSBY LOUIS H. KAUFFMAN

Communicated by William Browder, February 24, 19740. Introduction. Let : Cn-+C be a (complex) polynomial mapping

with an isolated singularity at the origin of C n. That is , / (0)=0 and thecomplex gradient df has an isolated zero at the origin. The link of thissingularity is defined by the formula L(f)=V(f)nS 2n~1. Here thesymbol V(f) denotes the variety of/ , and S211'1 is a sufficiently smallsphere about the origin of C n .Given another polynomial g:C m->C, formf+g with domain Cn+m =C n X Cm and consider ( + # ) c s2*1*2-1.In this note, we announce a topological construction for L(f+g)

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PRODUCTS OF KNOTS 1105manifold with boundary equal to Kx S1. If n is larger than 3, we assumetha t K is connected. Thus, by Alexander duality, ^(Ej^c^Z. One maychoose :EK-+ Sx representing the generator of ^(Ej^) so that isdifferentiable and \dEK is projection on the second factor. If n=3,then K may consist of a collection of disjointly imbedded circles. Achoice of orientations for these circles determines so that _1 (regularvalue) is an oriented spanning surface for K which induces the chosenorientat ions on each component .A knot is said to be spherical if i t is homeomorphic to a sphere.

A knot is said to be fibered if there is a choice of as above so thatcfriEx-^S 1 is a locally trivial sm oo th fibration.Now suppose that we are given knots K^Sn and L^S m and corresponding maps (f>:EK-> S1 and xp-.Ej^-S 1. If one knot is fibered, thenEK x8iE L={(x,y) e EKxEL\cf>(x)=ip(y)} is a well-defined smoothmanifold with boundary. Henceforth, when dealing with a pair of knots,we shall assum e th at a t least one k no t is fibered. W e now define a man ifoldKL and, using i ts properties, obtain the product knot KL D m+1xS1

(essentially

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1106 L. H, KAUFFMAN [NovemberGiven po lynom ials an d g as in the introduction, we have L(f)^S2n ~ land L(g)^S2m ~1. These are fibered knots. The maps to S1 are given bythe Milnor fibering (see [5]).THEOREM 3. There is a diffeomorphism L(f+g)~L(f)L( g).Furthermore, Uf+g^S2^2 '1 (naturally) and Ltf^L^g^S2^2 -1(by Lemma 2). These imbeddings are ambient isotopic.Note that , up to or ientat ions , L(f)

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1974] PRODUCTS OF KNOTS 1107has Seifert matrix V then K A has Seifert matrix V. Hence we concludeas follows.

THEOREM 6. Let C s denote the Levine knot cobordism group (see [4])of spherical knots in S2n+1 (n^.3, s=2n l). Define C s+ 4 by co(K)=KA. Then is an isomorphism.R E M A R K S . Note that A is the Brieskorn manifold 2(2, 2) . Thus co(K)is obtained by two double branched coverings as in (a). In fact , ourdescription of co coincides with tha t given by G. B redo n in [1], T o seethis , we construct 0(m) act ions on knot products .(c) Let Am=L(zl+zt+- -+z2m). Then the pair (S 2-1, A J has ana tura l 0(m) act ion so that Am is an orbit. It is then easy to see thefollowing.PROPOSITION 7. Let K