Knots, links, and 4-manifolds. (English) Zbl 0914.57015

This paper gives a construction which allows the authors to construct from a smooth 4-manifold \(X\) satisfying certain hypotheses and a knot \(K\) in the 3-sphere a new 4-manifold \(X_K\) which is homeomorphic to \(X\) but which can be distinguished smoothly from \(X\) by the Seiberg-Witten invariant whenever the Alexander polynomial of the knot is non-trivial. \(X_K\) is constructed from \(X\) by gluing in the knot complement crossed with a circle to the complement of a trivial normal bundle of torus \(T\) in \(X\). The required technical conditions on \(X\) are that it is simply connected with \(b_2^+ > 1\), that \(T\) lives in a cusp neighborhood in \(X\) (such as the neighborhood of a cusp fiber in an elliptic surface), and that the complement of \(T\) in \(X\) is simply connected. The formal relationship of the Seiberg-Witten invariants of \(X\) and \(X_K\) is \(\mathcal{SW}_{X_K} = \mathcal{SW}_{X} \cdot \Delta_K(t)\). Here \(\mathcal{SW}_K\) is regarded as a finite Laurent polynomial with variables being exponentials of the basic classes (its symmetric form using the fact that \(-\beta\) is a basic class whenever \(\beta\) is with corresponding value), and \(\Delta_K(t)\) is the symmetric Alexander polynomial with \(\Delta_K(t^{-1}) = \Delta_K(t)\). Moreover, when the knot is fibered and \(X\) has a symplectic structure, the construction can be performed symplectically so that \(X_K\) has a symplectic structure. The authors use C. H. Taubes’s results [Math. Res. Lett. 1, No. 6, , 809-822 (1994; Zbl 0853.57019); 2, No. 1, 9-13 (1995; Zbl 0854.57019); No. 2, 221-238 (1995; Zbl 0854.57020); J. Am. Math. Soc. 9, No. 3, 845-918 (1996; Zbl 0867.53025)] concerning Seiberg-Witten and Gromov invariants of symplectic manifolds to give a similar relationship between Gromov invariants in the symplectic case as well as to show that if \(\Delta_K(t) \) is not monic, then \(X_K\) does not possess a symplectic structure. The adjunction inequality is used to show that the existence of certain surfaces in \(X\) (which are readily found in many cases) will imply that \(X_K\) with the opposite orientation also has no symplectic structure. One case where these constructions apply is when \(X\) is the \(K3\) surface with Seiberg-Witten invariant 1. A corollary is that any \(A\)-polynomial \(P(t)\) can occur as the Seiberg-Witten invariant of an irreducible homotopy \(K3\) surface; if \(P(t)\) is not monic, then the homotopy \(K3\) surface does not admit a symplectic structure with either orientation; any monic \(A\)-polynomial can occur as the Seiberg-Witten invariant of a symplectic homotopy \(K3\) surface. The authors also show that their new examples cannot occur through a series of log transforms because of differing Seiberg-Witten invariants. The authors generalize their construction to the situation of \(n\)-component links where they form a fibered sum along the link components \(\times S^1\) with \(n\) 4-manifolds \(X_i.\) They derive a formula for the Seiberg-Witten of the result and show it is the product of the Seiberg-Witten invariants of the fiber sums \(E(1)\#_{F=T_j}X_j\) and the multivariable Alexander polynomial \(\Delta_L(t_1,\dots,t_n).\) In particular, the Seiberg-Witten invariant when all of the \(X_j\) are the rational elliptic surface \(E(1)\) will just be \(\Delta(t_1,\dots,t_n).\) When \(L\) is a two component link with odd linking number, then the manifold \(E(1)_L\) constructed from two copies of \(E(1)\) will have polynomials which are not products of \(A\)-polynomials and will occur as Seiberg-Witten invariants of homotopy \(K3\) surfaces. In particular, Z. Szabó’s examples \(X(k)\), \((k \in {\mathbb Z}\), \(k \neq 0,1)\) [Invent. Math. 132, No. 3, 457-466 (1998; Zbl 0906.57014)] of nonsymplectic simply connected irreducible smooth 4-manifolds occur as \(E(1)_{W(k)}\), where \(W(k)\) is the 2-component \(k\)-twisted Whitehead link, and the authors’ examples \(Y(k)\) of nonsymplectic \(K3\) surfaces occur as \(K3_{T(k)},\) where \(T(k)\) is the \(k\)-twist knot. One critical tool for the computations in the paper are gluing theorems of J. W. Morgan, T. S. Mrowka and Z. Szabó [Math. Res. Lett. 4, No. 6, 915-929 (1997; Zbl 0892.57021)], together with independent and joint work of these authors with Taubes that compute relative Seiberg-Witten invariants for \(c\)-embedded tori in terms of the absolute Seiberg-Witten invariant, as well as to compute Seiberg-Witten invariants of fiber sums and internal fiber sums. The authors also use gluing formulas for generalized log transforms. These gauge theory tools are then combined with geometric results of J. Hoste [Pac. J. Math. 112, 347-382 (1984; Zbl 0539.57004)] extending earlier work of W. R. Brakes [Lond. Math. Soc. Lect. Note Ser. 48, 27-37 (1982; Zbl 0483.57009)] on sewn-up link exteriors. These results are combined to show that the effect of certain constructions given here on the Seiberg-Witten invariants satisfies the same axiomatic rules as occur in a computation of the Alexander polynomial due to Conway via skein relations, and this allows the authors to prove their main formulas on Seiberg-Witten invariants.


57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
57R55 Differentiable structures in differential topology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R50 Differential topological aspects of diffeomorphisms
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