Abstract
In this paper we investigate the relationship between isotopy classes of knots and links in S and the dieomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of dieomorphism types of manifolds homeomorphic to the K3 surface. In order to state our theorems we need to view the Seiberg-Witten invariant of a smooth 4-manifold as a multivariable (Laurent) polynomial. To do this, recall that the Seiberg-Witten invariant of a smooth closed oriented 4-manifold X with b2 X > 1 is an integer valued function which is de®ned on the set of spinc structures over X , (cf. [W], [KM], [Ko1], [T1]). In case H1 X ;Z has no 2-torsion (which will be the situation in this paper) there is a natural identi®cation of the spinc structures of X with the characteristic elements of H2 X ;Z. In this case we view the Seiberg-Witten invariant as
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