KNOTS, LINKS, AND 4-MANIFOLDS

RONALD FINTUSHEL AND RONALD J. STERN

Abstract. In this paper we investigate the relationship between isotopy classes of knots and links in S^{3} and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 surface.

1. Introduction

In this paper we investigate the relationship between isotopy classes of knots and links in S^{3} and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism 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 b_{2}^{+}(X) > 1 is an integer valued function which is defined on the set of spin^{c} structures over X, (cf. [W], [KM],[Ko1],[T1]). In case H_{1}(X, Z) has no 2-torsion (which will be the situation in this paper) there is a natural identification of the spin^{c} structures of X with the characteristic elements of H^{2}(X, Z). In this case we view the Seiberg-Witten invariant as

SW_{X} : {k ∈ H^{2}(X, Z)|k ≡ w_{2}(T X) (mod 2))} → Z.

The Seiberg-Witten invariant SW_{X} is a diffeomorphism invariant whose sign depends on an orientation of H^{0}(X, R) ⊗ det H_{+}^{2}(X, R) ⊗ det H^{1}(X, R). If SW_{X}(β) ≠ 0, then we call β a basic class of X. It is a fundamental fact that the set of basic classes is finite. Furthermore, if β is a basic class, then so is −β with

SW_{X}(−β) = (−1)^{(e+sign)(X)/4} SW_{X}(β)

where e(X) is the Euler number and sign(X) is the signature of X.

The first author was partially supported NSF Grant DMS9401032 and the second author by NSF Grant DMS9626330.

Now let {±β_{1}, …, ±β_{n}} be the set of nonzero basic classes for X. For the purposes of this paper we define the Seiberg-Witten invariant of X to be the formal series

SW_{X} = b_{0} + ∑_{j=1}^{n} b_{j}(exp(β_{j}) + (−1)^{(e+sign)(X)/4} exp(−β_{j}))

where b_{0} = SW_{X}(0) and b_{j} = SW_{X}(β_{j}). Letting t_{j} = exp(β_{j}), we have that the Seiberg-Witten invariant is the ‘symmetric’ Laurent polynomial

SW_{X} = b_{0} + ∑_{j=1}^{n} b_{j}(t_{j} + (−1)^{(e+sign)(X)/4} t_{j}^{−1})

in the (formal) variables t_{1}, …, t_{n}.

We refer to any symmetric Laurent polynomial P (t) = a_{0} + ∑_{j=1}^{n} a_{j}(t^{j} + t^{−j}) of one

variable with coefficient sum a_{0} + 2 ∑_{j=1}^{n} a_{j} = ±1 as an A-polynomial. If, in addition,

a_{n} = ±1 we refer to P (t) as a monic A-polynomial.

Let X be any simply connected smooth 4-manifold with b^{+} > 1. We define a cusp in X to be a PL embedded 2-sphere of self-intersection 0 with a single nonlocally flat point whose neighborhood is the cone on the right-hand trefoil knot. (This agrees with the notion of a cusp fiber in an elliptic surface.) The regular neighborhood N of a cusp in a 4-manifold is a cusp neighborhood; it is the manifold obtained by performing 0-framed surgery on a trefoil knot in the boundary of the 4-ball. Since the trefoil knot is a fibered knot with a genus 1 fiber, N is fibered by smooth tori with one singular fiber, the cusp. If T is a smoothly embedded torus representing a nontrivial homology class [T], we say that T is c-embedded if T is a smooth fiber in a cusp neighborhood N; equivalently, T has two vanishing cycles. Note that a c-embedded torus has self-intersection 0. We can now state our first theorem.

Theorem 1.1. Let X be any simply connected smooth 4-manifold with b^{+} > 1. Suppose that X contains a smoothly c-embedded torus T with π_{1}(X \ T) = 1. Then for any A-polynomial P (t), there is a smooth 4-manifold X_{P} which is homeomorphic to X and has Seiberg-Witten invariant

SW_{X_{P}} = SW_{X} · P (t)

where t = exp(2[T]).

The basic classes of X were defined above to be elements of H^{2}(X). To make sense of the statement of the theorem, we need to replace [T] by its Poincaré dual. Throughout this paper we allow ourselves to pass freely between H^{2}(X) and H_{2}(X) without further comment.

As a corollary to the construction of X_{P} we shall show:

Corollary 1.2. Suppose further that X is symplectic and that T is symplectically embedded. If P (t) is a monic A-polynomial, then X_{P} can be constructed as a symplectic manifold.

Using the work of Taubes [T1–T5] concerning the nature of the Seiberg-Witten invariants of symplectic manifolds, we shall also deduce:

Corollary 1.3. If P (t) is not monic, then X_{P} does not admit a symplectic structure. Furthermore, if X contains a surface Σ_{g} of genus g disjoint from T with 0 ≠ [Σ_{g}] ∈ H_{2}(X; Z) and with [Σ_{g}]^{2} < 2 − 2g if g > 0, or [Σ_{g}]^{2} ≤ 0 if g = 0, then X_{P} with the opposite orientation does not admit a symplectic structure.

As a corollary, we have many interesting new homotopy K3 surfaces (i.e. manifolds homeomorphic to the K3 surface). In particular, since SW_{K3} = 1 we have:

Corollary 1.4. 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. Furthermore, any monic A-polynomial can occur as the Seiberg-Witten invariant of a symplectic homotopy K3 surface.

In fact, there are three disjoint c-embedded tori T_{1}, T_{2}, T_{3} in the K3 surface representing distinct homology classes [T_{j}], j = 1, 2, 3 (cf. [GM]). Theorem 1.1 then implies that the product of any three A-polynomials P_{j}(t_{j}), j = 1, 2, 3, can occur as the Seiberg-Witten invariant of a homotopy K3 surface K3_{P_{1}P_{2}P_{3}} with t_{j} = exp(2[T_{j}]). Furthermore, if all three of the polynomials are monic, the resulting homotopy K3 surface can be constructed as a symplectic manifold. If any one of the A-polynomials P_{j}(t_{j}) is not monic, then the resulting homotopy K3 surface admits no symplectic structure.

A common method for constructing exotic manifolds is to perform log transforms on c-embedded tori. One might ask whether these new homotopy K3 surfaces K3_{P_{1}P_{2}P_{3}} can be constructed in this fashion. However, it is shown in [FS1] that if X′ is the result of performing log transforms of multiplicities p_{1}, …, p_{n} on c-embedded tori in the K3 surface, then SW_{X′}(1, …, 1) = ±p_{1} ⋯ p_{n}. However, SW_{K3_{P_{1}P_{2}P_{3}}}(1, 1, 1) = P_{1}(1)P_{2}(1)P_{3}(1) = ±1; so K3_{P_{1}P_{2}P_{3}} cannot be built using log transforms in this way. (Note that the ‘1’ above is exp(0) corresponding to the zero class of H_{2}.)

The A-polynomials appearing in Theorem 1.1 are familiar to knot theorists. It is known that any A-polynomial occurs as the Alexander polynomial Δ_{K}(t) of some knot K in S^{3}. Conversely, the Alexander polynomial of a knot K is an A-polynomial. Furthermore, if the A-polynomial is monic then the knot can be constructed as a fibered knot, and if K is fibered, then Δ_{K}(t) is a monic A-polynomial. Indeed, it is a knot K in S^{3} which we use to construct X_{P}.

Consider a knot K in S^{3}, and let m denote a meridional circle to K. Let M_{K} be the 3-manifold obtained by performing 0-framed surgery on K. Then m can also be viewed as a circle in M_{K}. In M_{K} × S^{1} we have the smooth torus T_{m} = m × S^{1} of self-intersection 0. Since a neighborhood of m has a canonical framing in M_{K}, a neighborhood of the torus T_{m} in M_{K} ×S^{1} has a canonical identification with T_{m}×D^{2}. Let X_{K} denote the fiber sum

X_{K} = X#_{T =T_{m}}(M_{K} × S^{1}) = [X \ (T × D^{2})] ∪ [(M_{K} × S^{1}) \ (T_{m} × D^{2})]

where T × D^{2} is a tubular neighborhood of the torus T in the manifold X (with π_{1}(X) = π_{1}(X \ T) = 0). The two pieces are glued together so as to preserve the homology class [pt × ∂D^{2}]. This latter condition does not, in general, completely determine the diffeomorphism type of X_{K} (cf. [G]). We take X_{K} to be any manifold constructed in this fashion. Because M_{K} has the homology of S^{2}×S^{1} with the class of m generating H_{1}, the complement (M_{K} ×S^{1})\(T ×D^{2}) has the homology of T^{2} ×D^{2}. Thus X_{K} has the same homology (and intersection pairing) as X. Furthermore, the class of m normally generates π_{1}(M_{K}); so π_{1}(M_{K} × S^{1}) is normally generated by the image of π_{1}(T). Since π_{1}(X \ F) = 1, it follows from Van Kampen’s Theorem that X_{K} is simply connected. Thus X_{K} is homotopy equivalent to X. It is conceptually helpful to note that X_{K} is obtained from X by removing a neighborhood of a torus and replacing it with the complement of the knot K in S^{3} crossed with S^{1}. Also, in order to define Seiberg-Witten invariants, the oriented 4-manifold X must also be equipped with an orientation of H_{+}^{2}(X; R). The manifold X_{K} inherits an orientation as well as an orientation of H_{+}^{2}(X_{K}; R) from X.

Let [T] be the class in H_{2}(X_{K}; Z) induced by the torus T in X, and let t = exp(2[T]). Our first main theorem, from which Theorem 1.1 is an immediate corollary, is:

Theorem 1.5. If the torus T is c-embedded, then the Seiberg-Witten invariant of X_{K} is

SW_{X_{K}} = SW_{X} · Δ_{K}(t).

Let E(1) be the rational elliptic surface with elliptic fiber F. If T is a smoothly embedded self-intersection 0 torus in X, then E(1)#_{F =T} X_{K} = (E(1)#_{F =T} X)#_{T =T_{m}}(M_{K}× S^{1}), and in E(1)#_{F =T} X, the torus T = F is c-embedded. Thus we have a slightly more general result:

Corollary 1.6. Let X be any simply connected smooth 4-manifold with b^{+} > 1. Suppose that X contains a smoothly embedded torus T of self-intersection 0 with π_{1}(X \ T) = 1 and representing a nontrivial homology class [T]. Then

SW_{E(1)#_{F =T} X_{K}} = SW_{E(1)#_{F =T} X} · Δ_{K}(t).

More can be said if K is a fibered knot. Consider the normalized Alexander polynomial A_{K}(t) = t^{d}Δ_{K}(t), where d is the degree of Δ_{K}(t). If K is a fibered knot in S^{3} with a punctured genus g surface as fiber, then M_{K} fibers over the circle with a closed genus g surface Σ_{g} as fiber. Thus M_{K} × S^{1} fibers over S^{1} × S^{1} with Σ_{g} as fiber and with T_{m} = m × S^{1} as section. It is a theorem of Thurston [Th] that such a 4-manifold has a symplectic structure with symplectic section T_{m}. Thus, if X is a symplectic 4−manifold with a symplectically embedded torus with self-intersection 0, then X_{K} = X#_{T =T_{m}}(M_{K} × S^{1}) is symplectic since it can be constructed as a symplectic fiber sum [G]. As a corollary to Theorem 1.5 and the theorems of Taubes relating the Seiberg-Witten and Gromov invariants of a symplectic 4−manifold [T3, T5] we have:

Corollary 1.7. Let X be a symplectic 4-manifold with b^{+} > 1 containing a symplectic c-embedded torus T. If K is a fibered knot, then X_{K} is a symplectic 4-manifold whose Gromov invariant is

Gr_{X_{K}} = Gr_{X} · A_{K}(τ)

where τ = exp([T]).

Proof. The homology H_{2}(M_{K}×S^{1}) is generated by the classes of the symplectic curves T_{m} and Σ_{g}; so the canonical class of M_{K} × S^{1} has the form κ_{M_{K}×S^{1}} = a[T_{m}] + b[Σ_{g}]. Applying the adjunction formula (and using [T_{m}]^{2} = [Σ_{g}]^{2} = 0 and [T_{m}] · [Σ_{g}] = 1)

gives b = 0 and a = 2g−2. But note that the degree of Δ_{K}(τ) = a_{0}+ ∑_{n=1}^{d} a_{n}(τ^{n}+τ^{−n})

is d = g. Hence κ_{M_{K}×S^{1}} = (2d − 2)[T_{m}]. This means that the canonical class of the symplectic structure on X_{K} is κ_{X_{K}} = κ_{X} + κ_{M_{K}×S^{1}} + 2[T] = κ_{X} + 2d[T]. Taubes’ theorem now implies that for any α ∈ H_{2}(X_{K}), the coefficient of exp(α) in Gr_{X_{K}} is:

Gr_{X_{K}}(α) = SW_{X_{K}}(2α − κ_{X_{K}}) = SW_{X_{K}}(2α − κ_{X} − 2d[T])

= ∑_{n=−d}^{d} a_{n} SW_{X}(2α − κ_{X} − 2(d + n)[T]) = ∑_{n=−d}^{d} a_{n}Gr_{X}(α − (d + n)[T]),

and this is the coefficient of exp(α) in Gr_{X} · A_{K}(τ).

Of course, if X is simply connected and π_{1}(X\T) = 1, then X_{K} is homeomorphic to X. This implies Corollary 1.2. As corollary of the initial work of Taubes [T1, T2] on the Seiberg-Witten invariants of symplectic manifolds and the adjunction inequality, we have the following corollary which also implies Corollary 1.3.

Corollary 1.8. If Δ_{K}(t) is not monic, then X_{K} does not admit a symplectic structure. Furthermore, if X contains a surface Σ_{g} of genus g disjoint from T with 0 ≠ [Σ_{g}] ∈ H_{2}(X; Z) and with [Σ_{g}]^{2} < 2 − 2g if g > 0 or [Σ_{g}]^{2} < 0 if g = 0, then X_{K} with the opposite orientation does not admit a symplectic structure.

Proof. Suppose that X_{K} admits a symplectic structure with symplectic form ω and canonical class κ. Taubes has shown that SW_{X_{K}}(κ) = ±1 and that if k is any other basic class then |k · ω| < κ · ω.

Since SW_{X_{K}} = SW_{X} ·Δ_{K}(t) and since Δ_{K}(t) = a_{0}+ ∑_{n=1}^{d} a_{n}(t^{n}+t^{−n}) is a polynomial

in one variable t = exp(2[T]), any nontrivial basic class, and in particular, the canonical class, is of the form κ = α + 2n[T] where SW_{X}(α) ≠ 0 and |n| ≤ d. Let m be the maximum integer satisfying SW_{X}(α+m[T]) ≠ 0. Note that m ≥ 0. Set β = α+m[T]. Because of the maximality of m, we have SW_{X_{K}}(β + 2d[T]) = a_{d} · SW_{X}(β) ≠ 0.

Replacing [T] with −[T] allows us to assume that [T] · ω ≥ 0. First assume that [T] · ω > 0. Then

(β + 2d[T]) · ω = κ · ω + (m + 2(d − n))[T] · ω ≥ κ · ω

because m ≥ 0 and d − n ≥ 0, and equality occurs only if m = 0 and n = d. But a strict inequality contradicts Taubes’ theorem, thus α = β and n = d; so κ = β+2d[T]. This means that ±1 = SW_{X_{K}}(κ) = a_{d} · SW_{X}(β); so a_{d} = ±1, i.e. Δ_{K} is monic.

If [T] · ω = 0,

(β + 2d[T]) · ω = (α + 2n[T]) · ω = κ · ω,

which means that κ = β + 2d[T], and again we see that a_{d} = ±1.

Finally, if any manifold Y contains a homologically nontrivial surface Σ_{g} of genus g with [Σ_{g}]^{2} > 2g − 2, then, if g > 0, it follows from the adjunction inequality [KM, MST] that the Seiberg-Witten invariants of Y vanish, and hence Y does not admit a symplectic structure [T1]. If g = 0 and [Σ_{g}]^{2} ≥ 0 one can also show that the Seiberg-Witten invariants of Y must vanish [FS2, Ko2].

The authors are unaware of any simply connected smooth oriented 4−manifold X with b^{+} > 1 and SW_{X} ≠ 0 which does not contain either a sphere with selfintersection −2 or a torus with self-intersection −1.

The techniques used in proving Theorem 1.5 generalize to the more general setting of links. Let L = {K_{1}, …, K_{n}} be an (n ≥ 2)-component ordered link in S^{3} and suppose that X_{j}, j = 1, …, n, are simply connected smooth 4-manifolds with b^{+} ≥ 1 and each containing a smoothly embedded torus T_{j} of self-intersection 0 with π_{1}(X_{j} \ T_{j}) = 1 and representing a nontrivial homology class [T_{j}]. If (ℓ_{j}, m_{j}) denotes the standard longitude-meridian pair for the knot K_{j}, let

α_{L} : π_{1}(S^{3} \ L) → Z

denote the homomorphism characterized by the property that α_{L}(m_{j}) = 1 for each j = 1, …, n. Now, mimicking the knot case above, let M_{L} be the 3-manifold obtained by performing α_{L}(ℓ_{j}) surgery on each component K_{j} of L. (The surgery curves form the boundary of a Seifert surface for the link.) Then, in M_{L} × S^{1} we have smooth tori T_{m_{j}} = m_{j} × S^{1} of self-intersection 0 and we can construct the n−fold fiber sum

X(X_{1}, … X_{n}; L) = (M_{L} × S^{1})#_{T_{j}=T_{m_{j}}} ∐_{j=1}^{n} X_{j}

Here, the fiber sum is performed using the natural framings T_{m_{j}} × D^{2} of the neighborhoods of T_{m_{j}} = m_{j} ×S^{1} in M_{L}×S^{1} and the neighborhoods T_{j} ×D^{2} in each X_{j} and glued together so as to preserve the homology classes [pt×∂D^{2}]. As in the knot case, it follows from Van Kampen’s Theorem that X(X_{1}, … X_{n}; L) is simply connected. Furthermore, the signature and Euler characteristic (i.e. the rational homotopy type) of X(X_{1}, … X_{n}; L) depend only on the rational homotopy type of the X_{i} and the number of components in the link L. In the special case that all the X_{j} are the same manifold X we denote the resulting construction by X_{L}. It is conceptually helpful to note that X(X_{1}, … X_{n}; L) is obtained from the disjoint union of the X_{j} by removing a neighborhood of the tori T_{j} and replacing them with the complement of the link L in S^{3} crossed with S^{1}.

If Δ_{L}(t_{1}, …, t_{n}) denotes the symmetric multivariable Alexander polynomial of the n ≥ 2 component link L, and E(1) denotes the rational elliptic surface, our second theorem is:

Theorem 1.9. The Seiberg-Witten invariant of E(1)_{L} is

SW_{E(1)_{L}} = Δ_{L}(t_{1}, …, t_{n})

where t_{j} = exp(2[T_{j}]).

As a corollary we shall show:

Corollary 1.10. The Seiberg-Witten invariant of X(X_{1}, … X_{n}; L) is

SW_{X(X_{1},...X_{n};L)} = Δ_{L}(t_{1}, …, t_{n}) · ∏_{j=1}^{n} SW_{E(1)#_{F =T_{j}} X_{j}}

where t_{j} = exp(2[T_{j}]).

As before, the work of Taubes [T3, T5] implies that if L is a fibered link, and each (X_{j}, T_{j}) is a symplectic pair, then X(X_{1}, … X_{n}; L) is a symplectic 4-manifold whose Gromov invariant is

Gr_{X(X_{1},...X_{n};L)} = A_{K}(t_{1}, …, t_{n}) · ∏_{j=1}^{n} Gr(E(1)#_{F =T_{j}}X_{j}).

Two words of caution are in order here. First, there may be relations in X(X_{1}, … X_{n}; L) among the homology classes [T_{j}] represented by the tori T_{j} = T_{m_{j}}. These relations are determined by the linking matrix of the link L. In particular, if all the linking numbers are zero, then the [T_{j}] are linearly independent. At the other extreme, if the n-component link is obtained from the Hopf link by pushing off one component (n − 2) times, then all the [T_{j}] are equal. Second, the ordering of the components of the link can affect these relations.

If L is a two component link with odd linking number, then E(1)_{L} is a homotopy K3-surface and there are many interesting new polynomials which are not products of A-polynomials (cf. [Hi]) that can occur in this way as the Seiberg-Witten invariants of a homotopy K3-surface.

The first examples of nonsymplectic simply connected irreducible smooth 4-manifolds were constructed by Z. Szabo [S1]. These manifolds X(k) (k ∈ Z, k ≠ 0, ±1) can be shown to be diffeomorphic to E(1)_{W (k)}, where W (k) is the 2-component k-twisted Whitehead link (see Figure 1) with Alexander polynomial k(t_{1}^{1/2} −t_{1}^{−1/2})(t_{2}^{1/2} −t_{2}^{−1/2}). By Theorem 1.9 (and by the computation first given in [S1])

SW_{X(k)} = k(t_{1}^{1/2} − t_{1}^{−1/2})(t_{2}^{1/2} − t_{2}^{−1/2}).

Thus by Taubes’ Theorem [T1], X(k) does not admit a symplectic structure with either orientation (since X(k) contains spheres with self-intersection −2). Note also that for k = ±1 the k−twisted Whitehead link is fibered; so X(±1) is, in fact, symplectic.

The first examples of nonsymplectic homotopy K3-surfaces were constructed by the authors. These manifolds Y (k) can be shown to be diffeomorphic to K3_{T (k)} where T (k) is the k-twist knot (see Figure 1) with Alexander polynomial kt−(2k+1)+kt^{−1}. By Theorem 1.5

SW_{Y (k)} = kt − (2k + 1) + kt^{−1}

and, by Corollary 1.8, if k ≠ 0, ±1, Y (k) does not admit a symplectic structure with either orientation. Again, for k = ±1 the k−twist knot is fibered; so Y (±1) is symplectic.

[figure: k RH-; twists; W (k) = k-twisted Whitehead link; k RH-; twists; T (k) = k-twist knot; Figure 1]

Our next task is to prove Theorem 1.5 and Theorem 1.9. The proof of Theorem 1.5 is constructive and gives an algorithm which relates the Seiberg-Witten invariants of X_{K} with those of X by performing a series of topological log transforms on nullhomologous tori in X_{K} which reduce it to X. This turns out to be the same algorithm used to compute the Alexander polynomial Δ_{K}(t). This proof relies upon important analytical work of Morgan, Mrowka, and Szabo [MMS] (cf. [S1, S2]) and Taubes [T6] regarding the effect on the Seiberg-Witten invariants of removing neighborhoods of tori and sewing in manifolds with nonnegative scalar curvature. These we present in Section 2 in the form of gluing theorems. The proof of Theorem 1.9 can take one of two routes. The first is to utilize the algorithm provided by Conway [C] to compute the Alexander polynomial (more precisely the potential function) of a link. The proof then proceeds in a (tedious) manner, similar to the proof of Theorem 1.5. However, we choose to present a more direct proof by showing that the Seiberg-Witten invariants for the manifolds E(1)_{L} satisfy the axioms for the Alexander polynomial of a link as provided by Turaev [T]. The advantage of this proof, aside from its brevity, is that it isolates the required gluing properties of the Seiberg-Witten invariants and perhaps lays the foundation for determining the axioms for an appropriate gauge theory which may expose the other, more sophisticated, knot and link invariants. In the final section we discuss examples with b^{+} = 1 which are given by our construction.

It was Meng and Taubes [MT] who first discovered the relationship between Seiberg-Witten type invariants and the Alexander polynomial. In [MT] they defined a 3manifold invariant by dimension-reducing the Seiberg-Witten invariants, and they showed that this 3-manifold invariant was related to the Milnor torsion. We fell upon Theorems 1.5 and 1.9 by attempting to understand our above mentioned constructions of nonsymplectic homotopy K3-surfaces.

We end this introduction with three items. First, we conjecture that if K and K′ are two distinct knots (or n-component links) then X_{K} is diffeomorphic to X_{K′} if and only if K is isotopic to K′. Second, we wish to thank Jim Bryan, Bob Gompf, Elly Ionel, Dieter Kotschick, Wladek Lorek, Dusa McDuff, Terry Lawson, Tom Parker, and Cliff Taubes for useful conversations. Finally, we wish to make it clear that the contributions of the present paper are of a purely topological nature. The gauge theoretic input to our theorems is due to Morgan, Mrowka, and Szabo and to Taubes.

2. Background for the proofs of Theorems 1.5 and 1.9

In this section we shall survey the recent gluing theorems of Morgan, Mrowka, and Szabo [MMS] (cf. [S1, S2]) and Taubes [T6] (cf. [MT]) which are used in the proof of Theorem 1.5. Also we shall review some of the work of W. Brakes and J. Hoste on ‘sewn-up link exteriors’ which will be used in our constructions.

The context for the first of the gluing results is as follows. We are given a smooth 4manifold X with b_{X}^{+} > 1 and with an embedded torus T which represents a nontrivial homology class [T] of self-intersection 0 in H_{2}(X; Z). Any Seiberg-Witten basic class α ∈ H_{2}(X; Z) (i.e. any α with SW_{X}(α) ≠ 0) must be orthogonal to the homology class [T] since the adjunction inequality states that 0 ≥ [T]^{2} + |α · [T]|. The relative Seiberg-Witten invariant SW_{(X;T)} is formally defined to be

SW_{(X;T)} = SW_{X#_{T =F} E(1)}

where E(1) is the rational elliptic surface with smooth elliptic fiber F.

It is an interesting consequence of the gluing theorems of Morgan-Mrowka-Szabo [MMS] and Taubes [T6] that

Theorem 2.1. Suppose that b_{X}^{+} > 1 and that the torus T is c-embedded. Then

SW_{(X;T)} = SW_{X} · (t^{1/2} − t^{−1/2})

where t = exp(2[T]).

Note that E(1) has a metric of positive scalar curvature. A much more general (and difficult to prove) gluing theorem is:

Theorem 2.2 (Morgan, Mrowka, and Szabo [MMS]). In the situation above

SW_{X_{1}#_{T_{1}=T_{2}}X_{2}} = SW_{(X_{1};T_{1})} · SW_{(X_{2};T_{2})}.

The fact that Corollary 1.10 follows from Theorem 1.9 is now an easy consequence of the gluing theorems Theorem 2.1 and Theorem 2.2; for note that X(X_{1}, …, X_{n}; L) = X(X_{2}, …, X_{n}; L)#_{T_{m_{1}}=T_{1}}X_{1}. Thus Theorem 2.1 implies that

SW_{X(X_{1},...,X_{n};L)} = SW_{X(E(1),X_{2},...,X_{n};L)} · SW_{X_{1}#_{T_{1}=F} E(1)},

and continuing inductively completes the argument. This is the only time we shall need to use the general gluing theorem (2.2).

For our proof of Theorem 1.5 we will form an ‘internal fiber sum’. For this construction suppose that we have a pair of c-embedded tori T_{1}, T_{2}.

In our manifold X with b_{X}^{+} > 1 we formally define the relative Seiberg-Witten invariant SW_{(X;T_{1},T_{2})} to be

SW_{(X;T_{1},T_{2})} = SW_{X} · (τ_{1} − τ_{1}^{−1}) · (τ_{2} − τ_{2}^{−1}).

where τ_{j} = exp([T_{j}]). We construct the internal fiber sum X_{T_{1},T_{2}} by identifying the boundaries of neighborhoods of the T_{i}, again preserving the homology classes [pt × ∂D^{2}].

The first gluing theorem we need for the proof of Theorem 1.5 is:

Theorem 2.3 (Morgan, Mrowka, and Szabo [MMS], Taubes [T6](cf.[MT])). The internal fiber sum X_{T_{1},T_{2}} has Seiberg-Witten invariant

SW_{X_{T_{1},T_{2}}} = SW_{(X;T_{1},T_{2})}|_{τ_{1}=τ_{2}}.

The other gluing result we need concerns generalized log transforms on nullhomologous tori. Let p and q be relatively prime nonzero integers (or (1, 0) or (0, 1)). If T is any embedded self-intersection 0 torus in a 4-manifold Y with tubular neighborhood

N = T × D^{2} = S^{1} × S^{1} × D^{2},

let φ = φ_{p,q} be the diffeomorphism S^{1} × S^{1} × ∂D^{2} → ∂N given by

φ(x, y, z) = (x, y^{s}z^{q}, y^{r}z^{p}), det (\begin{matrix}p & q \\ r & s \end{matrix}) = −1.

The manifold

Y (p/q) = (Y \ N) ∪_{φ} (S^{1} × S^{1} × D^{2})

is called the (generalized) (p/q)-log transform of Y along T. Our notation and terminology are incomplete since the splitting T = S^{1} × S^{1} is necessary information. Throughout this paper, whenever a log transform is performed on a torus T, there will be a natural identification T = S^{1} × S^{1} and we always perform the transform with φ_{p,q} in the coordinates S^{1} × S^{1} × D^{2} as above.

We shall need to study the situation where a log transform is performed on a nullhomologous torus in Y. If T is such a torus, then in Y (0/1) there appears a new 2-dimensional homology class which is represented by the torus

T_{0} = S^{1} × S^{1} × pt ⊂ S^{1} × S^{1} × D^{2} ⊂ (X \ N) ∪_{φ} (S^{1} × S^{1} × D^{2}) = Y (0/1).

(The old torus pushed to ∂(Y \ N) is now φ(S^{1} × pt × ∂D^{2}).) Notice that each homology class α ∈ H_{2}(Y; Z) may be viewed as a homology class in each Y (p/q).

Theorem 2.4 (Morgan, Mrowka, and Szabo [MMS], Taubes [T6] (cf. [MT])). Let Y be a smooth 4-manifold with b^{+} ≥ 3, and suppose that Y contains a nullhomologous torus T with tubular neighborhood N = T ×D^{2} = S^{1}×S^{1}×D^{2}. Let τ be the homology class of T_{0} in Y (0/1). Then for each characteristic homology class α ∈ H_{2}(Y; Z),

SW_{Y (p/q)}(α) = pSW_{Y} (α) + q ∑_{i=−∞}^{∞} SW_{Y (0/1)}(α + 2iτ).

The sum in the above formula reduces to a single term in all the situations which are encountered in the proofs of Theorem 1.5 and Theorem 1.9. Specifically,

2.5. In Theorem 2.4 suppose that there is a torus in Y (0/1) which is disjoint from T and which represents a homology class σ of self-intersection 0 whose intersection number with τ in Y (0/1) is 1. Suppose furthermore that σ·α = 0 for all α ∈ H_{2}(Y) ⊂ H_{2}(Y (0/1)). Then

SW_{Y (p/q)} = pSW_{Y} + qSW_{Y (0/1)}.

Proof. Note that H_{2}(Y (0/1)) = H_{2}(Y) ⊕ H(σ, τ) where H(σ, τ) is a hyperbolic pair. If α ∈ H_{2}(Y; Z) satisfies SW_{Y (0/1)}(α + 2iτ) ≠ 0, the adjunction inequality implies that

0 ≥ σ^{2} + |(α + 2iτ) · σ| = |2i|.

Thus i = 0, and SW_{Y (p/q)}(α) = pSW_{Y} (α) + qSW_{Y (0/1)}(α). Since for p ≠ 0, each α ∈ H_{2}(Y (p/q)) arises from a class in H_{2}(Y), the lemma follows.

We next wish to describe a method for constructing 3-manifolds which was first studied by W. Brakes [B] and extended by J. Hoste [Ho]. Let L be a link in S^{3} with two oriented components C_{1} and C_{2}. Fix tubular neighborhoods N_{i} ≅ S^{1} × D^{2} of C_{i} with S^{1} × (pt on ∂D^{2}) a longitude of C_{i}, i.e. nullhomologous in S^{3} \ C_{i}. For any A ∈ GL(2; Z) with det A = −1, we the get a 3-manifold

s(L; A) = (S^{3} \ int(N_{1} ∪ N_{2}))/A

called a sewn-up link exterior by identifying ∂N_{1} with ∂N_{2} via a diffeomorphism inducing A in homology. For n ∈ Z, let A_{n} = ( \begin{matrix}−1 & 0 \\ −n & 1 \end{matrix} ). A simple calculation shows that H_{1}(s(L; A_{n}); Z) = Z ⊕ Z_{n−2ℓ} where ℓ is the linking number in S^{3} of the two components C_{1}, C_{2}, of L. (See [B].) The second summand is generated by the meridian to either component.

J. Hoste [Ho] has given a recipe for producing Kirby calculus diagrams for s(L; A_{n}). First we review the notion of a ‘band sum’ on an oriented knot or link L. Consider a portion of L consisting of a pair of strands, oriented in opposite directions, and separated by a band B. We identify B with an embedding of I^{2} = [0, 1] × [0, 1] in S^{3} such that B ∩ L = ({0} × I) ∪ −({1} × I). Let K be the (oriented) knot or link obtained by trading the segments B ∩ L = {0, 1} × I of ∂B for the complementary oriented segments I × {0} and −(I × {1}). The process of exchanging L for K in this fashion is called a band sum. Associated with the band move are two unknots: U_{ν}, an unknotted circle which bounds a disk whose interior meets B = I^{2} in the arc \{\frac{1}{2}\} × I and is disjoint from L, and U_{o}, which spans a disk whose interior meets B in I × \{\frac{1}{2}\} and is disjoint from K. (See Figure 2.)

[figure: U_{o}; L; C_{1}; C_{2}; B; ⟶; U_{ν}; K; Figure 2]

Proposition 2.6 (Hoste [Ho]). Let L = C_{1} ∪ C_{2} be an oriented link in S^{3}. Consider a portion of L consisting of a pair of strands, one from each component, oriented in opposite directions, and separated by a band B. The band sum of C_{1} and C_{2} is a knot K, and U_{ν} links K twice geometrically and 0 times algebraically. The sewn-up link exterior s(L; A_{n}) is obtained from surgery on the the two component link K ∪ U_{ν} in S^{3} with surgery coefficient 0 on U_{ν} and n − 2ℓ on K, where ℓ is the linking number of C_{1} and C_{2}.

Next consider a related situation. Let Z = s(L; A_{n}) where n = 2ℓ; so Z has H_{1}(Z; Z) = Z ⊕ Z. Suppose that B is a band in S^{3} meeting L as in Proposition 2.6, with the circle U_{o}, which links L twice geometrically and 0 times algebraically. Then U_{o} gives rise to a loop, Ū_{o} in Z. To get a Kirby calculus picture of this situation, apply Hoste’s formula. We obtain a two component framed link K ∪ U_{ν} with 0-framing on each. In S^{3}, U_{o} bounds a disk which is disjoint from K and meets a disk spanning U_{ν} in two points with opposite orientations. Thus U_{o} bounds a punctured torus in S^{3} \ (K ∪ U_{ν}); and so Ū_{o} is nullhomologous in Z. This means that Ū_{o} has a naturally defined longitude in Z; so p/q- Dehn surgery on Ū_{o} is well-defined. Let Z_{0} denote the result of 0-surgery on Ū_{o} in Z.

In Z, let m denote the meridian circle to the sewn-up torus. Then in Z × S^{1} we have the torus T_{m} = m × S^{1} of self-intersection 0. Form the fiber sum X#_{T} (Z × S^{1}) of X with Z × S^{1} by identifying T_{m} with the torus T of X. (In the name of brevity, we shall sometimes make, as in this case, a mild change in our notation for fiber sum.) Similarly we can form the fiber sum X#_{T} (Z_{0} × S^{1}), which is seen to be the result of performing a (0/1)-log transform on the torus Ū_{o} × S^{1} in X#_{T} (Z × S^{1}). We wish to compute the Seiberg-Witten invariant of this 4-manifold.

By cutting open s(L; A_{n}) along one of the two components of L, we obtain the exterior of the link L in S^{3}, and if this is done after performing 0-surgery on U_{o}, we obtain a link exterior in S^{2} × S^{1}. Now perform the corresponding task in X#_{T} (Z_{0} × S^{1}), removing a torus C_{i} ×S^{1}. We obtain X#_{T} (S^{2} ×S^{1} ×S^{1}) with a pair of tubular neighborhoods of self-intersection 0 tori removed. Call this manifold Q(L). To reiterate — Q(L) is obtained by performing 0-surgery on U_{o} in S^{3} \ L, crossing with S^{1}, and then fiber-summing with X along T_{m}. The boundary components of Q(L) have a natural framing λ, μ, σ, coming from the longitude and meridian of the link components in S^{3} and the S^{1} in the last coordinate. We can re-obtain X#_{T} (Z_{0} × S^{1}) by sewing up the boundary 3-dimensional tori of Q(L) using the matrix A_{n}⊕(1). Instead, let us fill in each of the boundary components of Q(L) with a copy of S^{1}×D^{2}×S^{1}. This can be done in many ways. We wish to do it so that, using the framings obtained from our copies of S^{1} × D^{2} × S^{1}, we obtain X#_{T} (Z_{0} × S^{1}) by sewing up the boundary of the resultant manifold with a neighborhood of the (new) link (S^{1}×pt×S^{1})∪(S^{1}×pt×S^{1}) removed using the matrix ±A_{0} ⊕ (1). Using the obvious framing for S^{1} × D^{2} × S^{1}, we claim that this is done by gluing each S^{1} × D^{2} × S^{1} to a component of ∂Q(L)

by a diffeomorphism with matrix B_{ℓ} ⊕ (1) where B_{ℓ} = (\begin{matrix}0 & 1 \\ 1 & −ℓ \end{matrix}). We shall denote

by W (L) the manifold formed using B_{ℓ} ⊕ (1) to sew in the neighborhoods of the tori C′_{i} × S^{1} = S^{1} × pt × S^{1}.

Let V (L) be the result of filling in the exterior of L in S^{3} via the diffeomorphism B_{ℓ} on each component. This is just the result of (−ℓ)-framed surgery on each component of L. Now if we sew up the link complement V (L) \ L via −A_{0}, we get the result of sewing up S^{3} \ L using the diffeomorphism B_{ℓ}(−A_{0})B_{ℓ}^{−1} = A_{2ℓ}. Thus s(L; A_{n}) = s(L′; −A_{0}) where L′ is the link C′_{1}, C′_{2}. Denote by V_{0}(L) the result of performing 0surgery on U_{o} in V (L) \ L′ = S^{3} \ L. Then W (L) = X#_{T} (V_{0}(L) × S^{1}) and X#_{T} (Z_{0} × S^{1}) = W (L)_{T_{1}=T_{2}} where T_{i} = C′_{i} × S^{1}.

Proposition 2.7. Suppose that T is a c-embedded torus in X. Then with the above notation,

SW_{X#_{T} (Z_{0}×S^{1})} = (t^{1/2} − t^{−1/2})^{2} · SW_{X_{K}}

as Laurent polynomials, where K is the band sum of C_{1} and C_{2} using the band B, and t = exp(2[T]).

Proof. Since the matrix −A_{0} ⊕ (1) identifies the tori C′_{i} × S^{1} in the boundary components of Q(L), Theorem 2.3 tells us that SW_{X#_{T} (Z_{0}×S^{1})} = SW_{W (L)_{T_{1}=T_{2}}} is obtained from the relative invariant SW_{(W (L);C′_{1},C′_{2})} by identifying the homology classes in Q(L) represented by the tori T_{i} = C′_{i} × S^{1}.

The gluing diffeomorphism B_{ℓ} identifies the homology class of a longitude of C′_{i} with the meridian of C_{i} in S^{3} \ L. Thus Theorem 2.3 implies that

SW_{X#_{T} (Z_{0}×S^{1})} = SW_{W (L)} · (τ − τ^{−1})^{2},

where τ = exp([m × S^{1}]).

It remains to identify the manifold W (L) as X_{K}. By construction, W (L) is obtained from the 3-component link C_{1} ∪ C_{2} ∪ U_{o} in S^{3} by performing (−ℓ)-framed surgery on C_{1} and C_{2} and 0-framed surgery on U_{o}, crossing with S^{1}, and fiber summing to X along T_{m} and T. The result of framed surgery on the 3-component link is, by sliding C_{1} over C_{2}, seen to be the same as 0-framed surgery on K. Thus W (L) = X_{K}, and the handle slide carries the meridian m to a meridian of K. Letting t = exp(2[T]) = τ^{2}, we get the calculation as claimed.

3. The proof of Theorem 1.5

We first recall a standard technique for calculating the (symmetrized) Alexander polynomial of a knot. This uses the skein relation

Δ_{K_{+}}(t) = Δ_{K_{−}}(t) + (t^{1/2} − t^{−1/2}) · Δ_{K_{0}}(t) (1)

where K_{+} is an oriented knot or link, K_{−} is the result of changing a single oriented positive (right-handed) crossing in K_{+} to a negative (left-handed) crossing, and K_{0} is the result of resolving the crossing as shown in Figure 3.

Note that if K_{+} is a knot, the so is K_{−}, and K_{0} is a 2-component link. If K_{+} is a 2-component link, then so is K_{−}, and K_{0} is a knot.

[figure: K_{+}; K_{−}; K_{0}; Figure 3]

The point of using (1) to calculate Δ_{K} is that K can be simplified to an unknot via a sequence of crossing changes of a projection of the oriented knot or link to the plane. To describe this well-known technique, consider such a projection and choose a basepoint on each component. In the case of a link, order the components. Say that such a projection is descending, if starting at the basepoint of the first component and traveling along the component, then from the basepoint of the second component and traveling along it, etc., the first time that each crossing is met, it is met by an overcrossing. Clearly a link with a descending projection is an unlinked collection of unknots. Our goal is to start with a knot K and perform skein moves so as to build a tree starting from K and at each stage adding the bifurcation of Figure 4, where each K_{+}, K_{−}, K_{0} is a knot or 2-component link, and so that at the bottom of the tree, we obtain only unknots, and split links. Then, because for an unknot U we have Δ_{U}(t) = 1, and for a split link S (of more than one component) we have Δ_{S}(t) = 0, we can work backwards using (1) to calculate Δ_{K}(t).

[figure: K_{+}; K_{−}; K_{0}; Figure 4]

The recipe for constructing the tree is, in the case of a knot, to change the crossing of the first ‘bad’ crossing encountered on the traverse described above. In this case, the result of changing the crossing is still a knot, and the result of resolving the crossing is a 2-component link. In the case of a 2-component link, one changes the first ‘bad’ crossing between the two components which is encountered on the traverse. The result of changing the crossing is still a 2-component link, and the result of resolving the crossing is a knot. In this way we obtain a tree whose top vertex is the given knot K and which branches downward as in the figure above. We shall call this tree a resolution tree for the knot K. We claim that the tree can be extended until each bottom vertex represents an unknot or a split link.

For the projection of an oriented, based knot K, let c(K) be the number of crossings and b(K) be the number of bad crossings encountered on a traverse starting at the basepoint. The complexity of the projection is defined to be the ordered pair (c(K), b(K)). For the projection of an oriented, based 2-component link L, let c(L) be the total number of crossings and let b(L) be the number of bad crossings between the two components. Again the complexity is defined to be (c(L), b(L)). Consider a vertex which represents a knot or 2-component link A. Note that c(A_{−}) = c(A_{+}), b(A_{−}) < b(A_{+}), and c(A_{0}) < c(A_{+}). Thus in the lexicographic ordering, (c(A_{−}), b(A_{−})) < (c(A_{+}), b(A_{+})) and (c(A_{0}), b(A_{0})) < (c(A_{+}), b(A_{+})). Now a knot K_{1} with c(K_{1}) = 0 or with b(K_{1}) = 0 is the unknot, and a link L with c(L) = 0 is the unlink and with b(L) = 0, it is at least a split link. This completes the proof that we can construct the resolution tree as described. (We remark that for the sake of simplicity we have considered only the case where we have changed a positive to a negative crossing in the skein move. Of course, we may as well have to change a negative to a positive crossing in order to lower b at various steps, but this does not change the proof.)

Consider an oriented, based knot K in S^{3} and a knot projection of K. We shall use the resolution tree for this projection as a guide for simplifying X_{K} in a way which leads to a calculation of X_{K}. Let us consider the first step, say K = K_{+} → {K_{−}, K_{0}}. At the crossing of K_{−} that is in question, there is an unknotted circle U linking K algebraically 0 times, so that the result of +1 surgery on U turns K_{−} into K_{+} = K. (See Figure 5.) In X_{K_{−}} we have the nullhomologous torus S^{1} × U. (It bounds the product of S^{1} with a punctured torus in S^{3} \ K_{−}.) The fact that +1 surgery on U turns K_{−} into K_{+} means that X_{K_{+}} is the result of a (1/1)-log transform on X_{K_{−}} along S^{1} × U. Let X_{K_{−}}(0/1) denote the result of performing a (0/1)-log transform on S^{1} × U in X_{K_{−}}. We now use (2.4) and (2.5) to compute SW(X_{K}). Two tori are central to this calculation. Letting m_{U} be a meridional circle to U, we get the torus S^{1} × m_{U} which represents the homology class τ of (2.4). Also, in X_{K_{−}}(0/1), the boundary of the punctured torus described above is spanned by a disk, and we obtain a torus of self-intersection 0 representing a class σ such that σ · τ = 1. Note that H_{2}(X_{K_{−}}(0/1)) = H_{2}(X_{K_{−}}) ⊕ H(σ, τ); so (2.5) applies. Hence SW_{X_{K}} = SW_{X_{K_{−}}} + SW_{X_{K_{−}}(0/1)}.

[figure: +1; K_{−}; Figure 5]

Recall that X_{K_{−}}(0/1) is obtained by performing 0-framed surgery on both components of the link K_{−} ∪U, crossing with S^{1} and fiber-summing with X, using the torus T obtained from a meridian of K_{−} crossed with S^{1}. Hoste’s recipe, (2.6), allows us to interpret the result of 0-framed surgery on both components of K_{−} ∪ U in S^{3} as s(K_{0}; A_{2ℓ}) where K_{0} is the 2-component link obtained by resolving the crossing under consideration (see Figure 6), and ℓ is the linking number of the two components of K_{0}.

[figure: 0; K_{−}; 0; ⟶; s (; ); ; A_{2ℓ}; K_{0}; Figure 6]

Let X(s(K_{0}; A_{2ℓ})) = (s(K_{0}; A_{2ℓ}) × S^{1})#_{T} X where T is the product of S^{1} with a meridian to either component of K_{0}. Because A_{2ℓ} sends meridians to meridians, this definition does not depend on the choice of component. Then X_{K_{−}}(0/1) ≅ X(s(K_{0}; A_{2ℓ})); so

SW_{X_{K_{+}}} = SW_{X_{K_{−}}} + SW_{X(s(K_{0};A_{2ℓ}))}, (2)

mimicking the skein move which gives the second tier of the resolution tree.

Now consider the next stage of the resolution tree and the skein move L → {L_{−}, L_{0}} where K_{0} = L = L_{+}. This move corresponds to changing a bad crossing involving both components of L. If the bad crossing under consideration is, say, right-handed, then L = L_{+} can be obtained from L_{−} = C_{1} ∪ C_{2} by +1-surgery on an unknotted circle U_{o} as in Figure 7. This means that X(s(L; A_{2ℓ})) is the result of a (1/1)-log transform on the torus S^{1} × U_{o} in X(s(L_{−}; A_{2ℓ_{−}})) where ℓ_{−} is the linking number of the components C_{1}, C_{2} of L_{−} and is determined by the fact that H_{1}(s(L_{−}; A_{2ℓ_{−}}); Z) = H_{1}(s(L; A_{2ℓ}); Z) = Z ⊕ Z.

[figure: +1; L_{−}; C_{1}; C_{2}; Figure 7]

In X(s(L_{−}; A_{2ℓ_{−}})), the torus S^{1} × U_{o} is nullhomologous. This is precisely the situation of (2.7) where Z = s(L_{−}; A_{2ℓ_{−}}). We wish to apply (2.5) to this situation. Let X(s(L_{−}; A_{2ℓ_{−}}))(0/1) denote the 4-manifold obtained by performing a (0/1)-log transform on S^{1} ×U_{o} in X(s(L_{−}; A_{2ℓ_{−}})). This is the manifold X#_{T} (Z_{0} ×S^{1}) of (2.7), and the 3-manifold Z_{0} is the result of 0-surgery on U_{o} in s(L_{−}; A_{2ℓ_{−}}). Let m_{U_{o}} be a meridional circle to U_{o} ⊂ s(L_{−}; A_{2ℓ_{−}}). The torus m_{U_{o}} × S^{1} in X(s(L_{−}; A_{2ℓ_{−}}))(0/1) is the T_{0} mentioned in the statement of (2.4). As we argued in the proof of (2.7), in Z, the loop U_{o} bounds a punctured torus, and this gets completed to a torus of self-intersection 0 in Z_{0}. Let σ be its homology class in X(s(L_{−}; A_{2ℓ_{−}}))(0/1) = X#_{T} (Z_{0} × S^{1}). Since the class σ satisfies the hypothesis of (2.5), we have

SW_{X(s(L_{+};A_{2ℓ}))} = SW_{X(s(L_{−};A_{2ℓ_{−}}))} + SW_{X#_{T} (Z_{0}×S^{1})}.

Applying (2.7), this becomes:

SW_{X(s(L_{+};A_{2ℓ}))} = SW_{X(s(L_{−};A_{2ℓ_{−}}))} + (t^{1/2} − t^{−1/2})^{2} · SW_{X_{L_{0}}} (3)

where L_{0} is the result of resolving the crossing of L which is under consideration, and t = exp(2[T]).

In order to see that this process calculates Δ_{K}(t), for fixed X, we define a formal Laurent series Θ, which is an invariant of knots and 2-component links. For a knot K, define Θ_{K} to be the quotient, Θ_{K} = SW_{X_{K}}/SW_{X}, and for a 2-component link with linking number ℓ between its components, Θ_{L} = (t^{1/2} − t^{−1/2})^{−1} · SW_{X(s(L;A_{2ℓ}))}/SW_{X}, where as usual t = exp(2[T]). It follows from (2) and (3) that for knots or 2component links, Θ satisfies the skein relation

Θ_{K_{+}} = Θ_{K_{−}} + (t^{1/2} − t^{−1/2}) · Θ_{K_{0}}.

For a split 2-component link, L, the 3-manifold s(L; A_{2ℓ}) contains an essential 2sphere (coming from the 2-sphere in S^{3} which splits the link). This means that X(s(L; A_{2ℓ})) contains an essential 2-sphere of self-intersection 0, and this implies that SW_{X(s(L;A_{2ℓ}))} = 0 (see [FS2]). Thus for a split link, Θ_{L} = 0. For the unknot U, the manifold X_{U} is just X#_{T} (S^{2} × T^{2}) = X, and so Θ_{U} = 1. Subject to these initial values, the resolution tree and the skein relation (1) determine Δ_{K}(t) for any knot K. It follows that Θ_{K} is a Laurent polynomial in a single

variable t, and Θ_{K}(t) = Δ_{K}(t), completing the proof of Theorem 1.4.

4. The proof of Theorem 1.9

We first review the axioms which determine the Alexander polynomial of a link. The reference for this material is [T]. The fact, proved in [T], that we shall use here is that there is but one map ∇ which assigns to each n−component ordered link L in S^{3} an element of the field Q(t_{1}, …, t_{n}) with the following properties:

1. ∇(L) is unchanged under ambient isotopy of the link L.

2. If L is the unknot, then ∇(L) = 1/(t − t^{−1}).

3. If n ≥ 2, then ∇(L) ∈ Z[t_{1}, t_{1}^{−1}, …, t_{n}, t_{n}^{−1}].

4. The one-variable function ˜∇(L)(t) = ∇(L)(t, t, …, t) is unchanged by a renumbering of the components of L.

5. (Conway Axiom). If L_{+}, L_{−}, and L_{0} are links coinciding (except possible for the numbering of the components) outside a ball, and inside this ball have the form depicted in Figure 8, then

˜∇(L_{+}) = ˜∇(L_{−}) + (t − t^{−1}) · ˜∇(L_{0}).

6. (Doubling Axiom). If the link L′ is obtained from the link L = {K_{1}, …, K_{n}} by replacing the K_{j} by its (2, 1)−cable, then

∇(L′)(t_{1}, …, t_{n}) = (T + T^{−1}) · ∇(L)(t_{1}, …, t_{j−1}, t_{j}^{2}, t_{j+1}, …, t_{n})

where T = t_{j} ∏_{i≠j} t_{i}^{ℓk(K_{j},K_{i})}.

[figure: L_{+}; L_{−}; L_{0}; Figure 8]

The Alexander polynomial Δ_{L} in Theorem 1.9 is just

Δ_{L}(t_{1}^{2}, …, t_{n}^{2}) ≐ ∇(L)(t_{1}, …, t_{n})

where the symbol ≐ denotes equality up to multiplication by −1 and powers of the variables.

Recall the construction of E(1)_{L}. If L = {K_{1}, …, K_{n}} is an (n ≥ 2)-component ordered link in S^{3} and (ℓ_{j}, m_{j}) denotes the standard longitude-meridian pair for the knot K_{j}, we let

α_{L} : π_{1}(S^{3} \ L) → Z

denote the homomorphism characterized by the property that α_{L}(m_{j}) = 1 for each j = 1, …, n. Define M_{L} to be the 3-manifold obtained by performing α_{L}(ℓ_{j}) surgery on each component K_{j} of L. Then, in M_{L} × S^{1} we use the smooth tori T_{m_{j}} = m_{j} × S^{1} to construct the n−fold fiber sum

E(1)_{L} = (M_{L} × S^{1})#_{T_{m_{j}} =F} ∐_{j=1}^{n} E(1),

the fiber sum being performed using the natural framings T_{m_{j}} × D^{2} of the neighborhoods of T_{m_{j}} = m_{j} × S^{1} in M_{j} and the neighborhoods F × D^{2} of an elliptic fiber in each copy of E(1) and glued together so as to preserve the homology classes [pt × ∂D^{2}]. It is amusing to note that this later condition is unnecessary in this special situation. For, since E(1) \ F has a big diffeomorphism group, we can fiber sum in the E(1) with any gluing map and end up with diffeomorphic 4-manifolds.

Now let ∇ be that function which associates to every ordered n-component link L, the polynomial ∇(L)(t_{1}, …, t_{n}) = SW_{E(1)_{L}}(t_{1}^{2}, …, t_{n}^{2}), with t_{j} = exp(2[T_{m_{j}}]). We show that ∇ satisfies the above stated axioms. However, there is a small obstruction to doing this in a straightforward manner: SW_{X} is only defined when b^{+} > 1 and when L has but one component E(1)_{L} has b^{+} = 1. Although our Theorem 1.9 is stated only for n ≥ 2, the axioms insist that we consider 1-component links. We overcome this problem as follows. Given an ordered n-component link L we always fiber sum in the K3-surface (rather than E(1)) to M_{L} × S^{1} along T_{m_{1}}. In the case that n ≥ 2, the resulting manifold, which we temporarily denote by E(2, 1)_{L}, has

SW_{E(2,1)_{L}} = (t_{1}^{1/2} − t_{1}^{−1/2}) · SW_{E(1)_{L}}

We shall complete the proof of Theorem 1.9 by showing that

∇(L)(t_{1}, …, t_{n}) = \frac{SW_{E(2,1)_{L}}(t_{1}^{2}, …, t_{n}^{2})}{t_{1} − t_{1}^{−1}}

satisfies all the axioms.

Axiom 1 is clear.

For Axiom 2 note that if L is the unknot, then E(2, 1)_{L} is the K3-surface, so that ∇(L)(t) = SW_{E(2,1)_{L}}(t^{2})/(t − t^{−1}) = 1/(t − t^{−1}).

To verify Axiom 3, consider an (n ≥ 2)-component link L with components K_{1}, …, K_{n}. We need to see that the only possible basic classes of E(2, 1)_{L} are the classes T_{m_{i}} (which are identified in E(2, 1)_{L} with the fiber classes F_{i}). A Mayer-Vietoris sequence argument shows that H_{2}(E(2, 1)_{L}) ≅ im(φ) ⊕ G where

H_{2}(E(2) \ F) ⊕ ∑_{1}^{n−1} H_{2}(E(1) \ F) ⊕ H_{2}((S^{3} \ L) × S^{1}) ⟶^{φ} H_{2}(E(2, 1)_{L}) ⟶^{δ}G

and G is isomorphic to the kernel of ∑_{1}^{n} H_{1}(T^{3}) → H_{1}((S^{3} \L)×S^{1}). Now H_{2}(E(2)\

F) ≅ 2E_{8} ⊕ 2H ⊕ 3(0) and H_{2}(E(1) \ F) ≅ E_{8} ⊕ 3(0) where H denotes a hyperbolic pair. Each copy of E_{8} is represented by eight −2-spheres in the usual configuration, say W, with ∂W = Σ(2, 3, 5), the Poincaré homology sphere. Since W embeds in E(2), whose only basic class is 0, and since Σ(2, 3, 5) has positive scalar curvature, a rudimentary gluing formula implies that each basic class of E(2, 1)_{L} is orthogonal to the image of the E_{8} summands. Each of the two hyperbolic pairs H is the homology of a nucleus in E(2) \ F and is generated by a torus τ of self-intersection 0 and a sphere σ of self-intersection −2 which intersect at one point. For any basic class κ of E(2, 1)_{L}, the adjunction formula implies 0 ≥ τ^{2} +|κ·τ|; so κ is orthogonal to τ. Also, τ + σ is represented by another torus of self-intersection 0; so k is in fact orthogonal to H. Furthermore, each of the summands (0) in H_{2}(E(1) \ F) and H_{2}(E(2) \ F) is represented by a torus in the boundary of the tubular neighborhood of F, and each of these tori is glued to a torus in (S^{3} \ L) × S^{1} in E(2, 1)_{L}. It follows that the only possible basic classes lying in the image of φ in fact lie in the image of H_{2}((S^{3}\L)×S^{1}). This image is spanned by the classes of the tori T_{m_{i}}, i = 1, …, n and the tori V_{j}, j = 1, …, n − 1 where V_{i} is the boundary of a tubular neighborhood of K_{i} in S^{3}. The nonzero elements of G determine classes in E(2, 1)_{L} with nonzero Mayer-Vietoris boundary. These are generated by classes γ_{i}, i = 1, …, n and Σ_{j}, j = 1, …, n − 1. A representative of γ_{i} is formed as follows. Let S_{i} denote intersection of a Seifert surface for the knot K_{i} with the link exterior. The intersection of S_{i} with V_{j} (j ≠ i) consists of ℓk(K_{i}, K_{j}) copies of m_{j}. Each of these is glued to a circle on the fiber F of the corresponding E(1) or E(2), and this circle bounds in the elliptic surface. The same is true for the longitude of the knot K_{i}. The result represents γ_{i}. Note that γ_{i}·F_{j} = δ_{ij} and γ_{i}·V_{j} = 0 for each j = 1, …, n. The generators Σ_{j} are constructed by starting with an arc A_{j} in S^{3} \ L from a point on V_{n} to a point on V_{j}. The boundary of A_{j} × S^{1} consists of two circles, and each is identified with a circle in H_{2}(E(1) \ F) or H_{2}(E(2) \ F). In the elliptic surfaces, these circles bound vanishing cycles, disks of self-intersection −1. Thus Σ_{i} is represented by a −2-sphere. We have Σ_{j} · F_{i} = 0 for all i, and Σ_{j} · V_{i} = δ_{ij}. Suppose we have a basic class

κ = ∑_{i=1}^{n} a_{i}F_{i} + ∑_{j=1}^{n−1} b_{j}V_{j} + ∑_{k=1}^{n} c_{k}γ_{k} + ∑_{ℓ=1}^{n−1} d_{ℓ}Σ_{ℓ}

Since F_{i} is a torus of self-intersection 0, the adjunction inequality implies that κ·F_{i} = 0, i.e. that c_{i} = 0. Similarly, V_{j} is a torus of self-intersection 0; so 0 = κ·V_{j} = d_{j}. Also Σ_{ℓ}+V_{ℓ} is represented by a torus of self-intersection 0; thus 0 = κ·(Σ_{ℓ}+V_{ℓ}) = κ·Σ_{ℓ} = b_{ℓ}.

Axiom 4 is clear.

Axiom 5 is verified in the spirit of Theorem 1.5. However, we must first construct an auxiliary manifold Ē(1)_{L} as follows. In E(2, 1)_{L}, L an n-component link, let F_{1}, …, F_{n} be tori with the torus F_{1} the elliptic fiber in the K3-surface and, for j ≥ 1, F_{j} the elliptic fiber in the (j − 1)^{st} copy of E(1). (Note that F_{i} = T_{m_{i}} in E(2, 1)_{L}.) Now perform (n − 1) internal fiber sums, identifying F_{1} with F_{2}, a parallel copy of F_{2} with F_{3}, and so on. The homology classes represented by the F_{j} in Ē(1)_{L} are all equal, and we denote this homology class by [F]. Let t = exp(2[F]). It follows from the gluing formula Theorem 2.3 that

SW_{Ē(1)_{L}}(t) = SW_{E(2,1)_{L}}(t, …, t) · (t^{1/2} − t^{−1/2})^{(2n−2)},

or by defining S̃W_{Ē(1)_{L}}(t) = SW_{E(2,1)_{L}}(t, …, t),

S̃W_{Ē(1)_{L}}(t) = \frac{SW_{Ē(1)_{L}}(t)}{(t^{1/2} − t^{−1/2})^{(2n−2)}} .

Suppose now that L_{+}, L_{−}, and L_{0} are links which coincide (except possibly for the numbering of the components) outside a ball, and inside this ball have the form depicted in Figure 8. Furthermore, assume L_{±} has n components. Then, as in the proof of Theorem 1.5, there is a nullhomologous torus T in Ē(1)_{L_{−}} so that Ē(1)_{L_{+}} is the result of a (1/1)-log transform on T. By the log transform formula (Theorems 2.4 and 2.5),

SW_{Ē(1)_{L_{+}}} = SW_{Ē(1)_{L_{−}}} + SW_{Ē(1)_{L_{−}}(0/1)}.

There are two cases. For the first case, the two strands of L_{+} are from distinct components K_{i_{1}}, K_{i_{2}} of L_{+}. The manifold Ē(1)_{L_{−}}(0/1) is obtained as follows: Perform surgeries on the link components {K_{i}} of L_{−} with surgery coefficient α_{L_{−}}(ℓ_{i}) on K_{i} and perform 0-surgery on the unknotted component U which links K_{i_{1}} and K_{i_{2}} as shown in Figure 9.

[figure: 0; α(ℓ_{i_{1}}); α(ℓ_{i_{2}}); m_{i_{1}}; m_{i_{2}}; K_{i_{1}}; K_{i_{2}}; ⟶; m_{0}; m_{0}; α_{L_{0}}(ℓ_{0}); K_{0}; Figure 9]

Let M^{3} be the resulting 3-manifold. Next form

E(2, 1)_{L_{−}}(0/1) = (M^{3} × S^{1})#_{T_{m_{1}}=F_{1}}E(2)#_{T_{m_{2}}=F_{2}}E(1)# ⋯ #_{T_{mn}=F_{n}}E(1). (4)

Finally, Ē(1)_{L_{−}}(0/1) is obtained from E(2, 1)_{L_{−}}(0/1) by performing n − 1 internal fiber sums, as described above. If we slide the handle corresponding to K_{i_{2}} over the handle corresponding to K_{i_{1}} then we obtain a new Kirby calculus description of M^{3}: it is obtained from surgery on the link L_{0} with surgery coefficients again given by the α_{L_{0}}(ℓ_{i}), as in Figure 9. (Note that if the new component is called K_{0} then for j ≠ i_{1}, i_{2}, the linking number ℓk_{L_{0}}(K_{0}, K_{j}) = ℓk_{L_{−}}(K_{i_{1}}, K_{j}) + ℓk_{L_{−}}(K_{i_{2}}, K_{j}); so the total linking number for the longitude ℓ_{0} is α_{L_{0}}(ℓ_{0}) = α_{L_{−}}(ℓ_{i_{1}})+α_{L_{−}}(ℓ_{i_{2}})−2ℓk_{L_{−}}(K_{i_{1}}, K_{i_{2}}) which is exactly the framing which is given to K_{0} by the handle slide.)

Now the link L_{0} has n − 1 components. We see that the difference between the constructions for Ē(1)_{L_{−}}(0/1) and Ē(1)_{L_{0}} is that an extra copy of E(1) needs to be fiber-summed into Ē(1)_{L_{0}} and then an extra internal fiber sum needs to be performed on the result, in order to get Ē(1)_{L_{−}}(0/1). Theorem 2.1 and Theorem 2.3 then imply that SW_{Ē(1)_{L_{−}}(0/1)}(t) = (t^{1/2} − t^{−1/2})^{3}SW_{Ē(1)_{L_{0}}}(t), where one factor (t^{1/2} − t^{−1/2}) comes from the extra fiber sum with E(1), and (t^{1/2} − t^{−1/2})^{2} comes from the extra internal fiber sum. We have:

S̃W_{Ē(1)_{L_{+}}}(t) = \frac{SW_{Ē(1)_{L_{+}}}(t)}{(t^{1/2} − t^{−1/2})^{(2n−2)}}

= \frac{SW_{Ē(1)_{L_{−}}}(t) + SW_{Ē(1)_{L_{−}}(0/1)}(t)}{(t^{1/2} − t^{−1/2})^{(2n−2)}}

= S̃W_{Ē(1)_{L_{−}}}(t) + \frac{(t^{1/2} − t^{−1/2})^{3} · SW_{Ē(1)_{L_{0}}}(t)}{(t^{1/2} − t^{−1/2})^{(2n−2)}}

= S̃W_{Ē(1)_{L_{−}}}(t) + \frac{(t^{1/2} − t^{−1/2}) · SW_{Ē(1)_{L_{0}}}(t)}{(t^{1/2} − t^{−1/2})^{(2n−4)}}
= S̃W_{Ē(1)_{L_{−}}}(t) + (t^{1/2} − t^{−1/2}) · S̃W_{Ē(1)_{L_{0}}}(t),

as desired.

For the second case, the two strands of L_{+} are from the same component (say the j^{th}) of L_{+}; so L_{0} has (n + 1) components. Then Ē(1)_{L_{−}}(0/1) is obtained by first performing surgeries on the components {K_{i}} of L_{−} with surgery coefficient α_{L_{−}}(ℓ_{i}) on K_{i} and then performing 0-surgery on an unknotted circle which links K_{j} twice geometrically and 0-times algebraically as in Figure 10.

[figure: m_{j}; 0; K_{j}; α(ℓ_{j}); ⟶; s (; ); ; A_{n}; K′′; K′; m′; Figure 10]

This gives a 3-manifold, M^{3}. Then we form E(2, 1)_{L_{−}}(0/1) as given by (4), and Ē(1)_{L_{−}}(0/1) is obtained from this by performing (n − 1) internal fiber sums.

Denote the components of L_{0} by K_{1}, …, K_{j−1}, K′, K′′, K_{j+1}, …, K_{n}. Let L_{j} denote the 2-component link L_{j} = {K′, K′′}. It follows from Hoste’s theorem (2.6) that M^{3} may be obtained from the sewn-up link exterior s(L_{j}; A_{n}) by further surgering the K_{i}, i ≠ j with framing α_{L_{−}}(ℓ_{i}) (where n = α_{L_{−}}(ℓ_{j}) + 2ℓk(K′, K′′)). Again see Figure 10. Because α_{L_{0}}(ℓ′) + α_{L_{0}}(ℓ′′) = ∑_{i≠j} ℓk(K_{i}, K_{j}) + 2ℓk(K′, K′′) = n, and also

α_{L_{−}}(ℓ_{i}) = α_{L_{0}}(ℓ_{i}), i ≠ j, the discussion preceding Proposition 2.7 relates the Seiberg-Witten invariant of E(2, 1)_{L_{−}}(0/1) with the Seiberg-Witten invariant of E(2, 1)_{L_{0}}. We need to keep in mind, however, that because L_{0} has n + 1 components, there is an extra fiber sum with E(1) in the construction for E(2, 1)_{L_{0}}. Thus,

SW_{E(2,1)_{L_{−}}(0/1)} = (t_{j}^{1/2} − t_{j}^{−1/2}) · SW_{E(2,1)_{L_{0}}}|_{t′=t′′=t_{j}}

Furthermore, Ē(1)_{L_{0}} has one more internal fiber sum than does Ē(1)_{L_{−}}(0/1); so

SW_{Ē(1)_{L_{−}}(0/1)}(t) = \frac{1}{(t^{1/2} − t^{−1/2})} · SW_{Ē(1)_{L_{0}}}(t).

Thus

S̃W_{Ē(1)_{L_{+}}}(t) = \frac{SW_{Ē(1)_{L_{+}}}(t)}{(t^{1/2} − t^{−1/2})^{(2n−2)}}

= \frac{SW_{Ē(1)_{L_{−}}}(t) + SW_{Ē(1)_{L_{−}}(0/1)}(t)}{(t^{1/2} − t^{−1/2})^{(2n−2)}}

= S̃W_{Ē(1)_{L_{−}}}(t) + \frac{\frac{1}{(t^{1/2}−t^{−1/2})} · SW_{Ē(1)_{L_{0}}}(t)}{(t^{1/2} − t^{−1/2})^{(2n−2)}}

= S̃W_{Ē(1)_{L_{−}}}(t) + \frac{(t^{1/2} − t^{−1/2}) · SW_{Ē(1)_{L_{0}}}(t)}{(t^{1/2} − t^{−1/2})^{2n}}
= S̃W_{Ē(1)_{L_{−}}}(t) + (t^{1/2} − t^{−1/2}) · S̃W_{Ē(1)_{L_{0}}}(t),

completing the proof of Axiom 5.

Finally, to verify Axiom 6 we first note that E(2, 1)_{L′} is obtained from E(2, 1)_{L} by performing an order 2 log transform on the torus T̄_{j} = K̄_{j} × S^{1} where K̄_{j} is the core of the surgered K_{j} in M_{L}. The homology class of the resulting meridian m′_{j} is twice that of m_{j} so that [T_{m′_{j}}] = 2[T_{m_{j}}]. The log transform formulas of [FS1], then state that

SW_{E(2,1)_{L′}}(t_{1}, …, t_{j−1}, t′_{j}, t_{j+1}, …, t_{n})

= (t̄_{j}^{1/2} + t̄_{j}^{−1/2}) · SW_{E(2,1)_{L}}(t_{1}, …, t_{j−1}, t_{j}^{2}, t_{j+1}, …, t_{n}).

The result now follows since

[T̄_{j}] = [T_{j}] + ∑_{i≠j} ℓk(K_{j}, K_{i})[T_{i}].

5. Examples with b^{+} = 1

In this section we shall discuss examples which have b^{+} = 1. For such manifolds, the Seiberg-Witten invariant depends on a choice of metric and self-dual 2-form as follows. Let X be a simply connected oriented 4-manifold with b_{X}^{+} = 1 with a given orientation of H_{+}^{2}(X; R) and a given metric g. Since b_{X}^{+} = 1, there is a unique gself-dual harmonic 2-form ω_{g} ∈ H_{+}^{2}(X; R) with ω_{g}^{2} = 1 and corresponding to the positive orientation. Fix a characteristic cohomology class k ∈ H^{2}(X; Z). Given a pair (A, ψ), where A is a connection in the complex line bundle corresponding to k and ψ a section of the bundle W^{+} of self-dual spinors for the associated spin^{c} structure, the perturbed Seiberg-Witten equations are:

D_{A}ψ = 0 (5)

F_{A}^{+} = q(ψ) + iη^{+}

where F_{A}^{+} is the self-dual part of the curvature of A, D_{A} is the twisted Dirac operator, η^{+}, is a self-dual 2-form on X, and q is a quadratic function. Write SW_{X,g,η^{+}}(k) for the corresponding invariant evaluated on the class k (= \frac{i}{2π}[F_{A}]). As the pair (g, η^{+}) varies, SW_{X,g,η^{+}}(k) can change only at those pairs (g, η^{+}) for which there are solutions of (5) with ψ = 0. These solutions occur for pairs (g, η^{+}) satisfying (2πk+η^{+})·ω_{g} = 0. This last equation defines a codimension 1 subspace (‘wall’) in H^{2}(X; R). The point ω_{g} lives in the double cone C_{X} = {α ∈ H^{2}(X; R)|α·α > 0}, and, if (2πk+η^{+})·ω_{g} ≠ 0 for a generic η^{+}, SW_{X,g,η^{+}}(k) is well-defined, and its value depends only on the sign of (2πk+η^{+})·ω_{g}. A useful lemma, which follows from the Cauchy-Schwarz inequality (see [LL]) is:

Lemma 5.1. Suppose that α and β are nonzero elements of H^{2}(X; R) which lie in the closure of the same component of C_{X}. Then α · β ≥ 0 with equality if and only if α = λβ for some λ > 0.

It follows from this lemma that that SW_{X,g,η^{+}}(k) depends only on the component of C_{X} which contains the g-self-dual projection of 2πk + η^{+}. Furthermore, if the g_{i}-self-dual projections of 2πk +η_{i}^{+} lie in different components of C_{X} and are oriented so that (2πk + η_{1}^{+}) · ω_{g_{1}} > 0 > (2πk + η_{0}^{+}) · ω_{g_{0}}, then

SW_{X,g_{1},η_{1}^{+}}(k) − SW_{X,g_{0},η_{0}^{+}}(k) = (−1)^{\frac{1}{2} δ(k)+1} (6)

where δ(k) = \frac{1}{4}(k^{2} − (3sign + 2e)(X)) is the formal dimension of the moduli spaces. Thus, as the pair (g, η^{+}) is varied, there are exactly two values of SW_{X,g,η^{+}}(k). Furthermore, in case b^{−} ≤ 9, for any fixed a ∈ H_{2}(X; Z) with δ(a) = a^{2} + b^{−} − 9 ≥ 0, the self-dual projections of 2πa all lie in the same component of C_{X}; so, if a = k is characteristic, then for small enough perturbations, the Seiberg-Witten invariants agree, independent of metric [KM, S2].

Suppose that X contains a smooth essential torus T of self-intersection 0. By Lemma 5.1, the class [T] orients C_{X} by declaring C_{X}^{+} to be the component of C_{X} which contains classes α with α · [T] > 0. Denote the other component by C_{X}^{−} and the corresponding Seiberg-Witten invariants by SW_{X}^{±}; i.e. SW_{X}^{+}(k) = SW_{X,g,η^{+}}(k) where the g-self-dual projection of 2πk+η^{+} has positive intersection with [T], and we define SW_{X}^{−}(k) similarly. The [T]^{⊥}-restricted Seiberg-Witten invariants are defined to be

SW_{X,T}^{±} = ∑_{k·[T]=0} SW_{X}^{±}(k) exp(k).

When k^{2} ≥ 0 and k · [T] = 0, Lemma 5.1 implies that k = λ[T]. In particular, if b_{X}^{−} ≤ 9 and δ(k) ≥ 0, then k = λ[T] if k · [T] = 0.

Lemma 5.2. Let X be a simply connected smooth 4-manifold with b_{X}^{+} = 1, and suppose that X contains a smooth homologically nontrivial torus T of self-intersection 0. Let t = exp(2[T]). Then

(t^{1/2} − t^{−1/2}) · SW_{X,T}^{+} = (t^{1/2} − t^{−1/2}) · SW_{X,T}^{−} .

Proof. The coefficient of exp(k) in (t^{1/2} − t^{−1/2}) · SW_{X,T}^{+} is

c^{+}(k) = SW_{X}^{+}(k − [T]) − SW_{X}^{+}(k + [T]).

Since k · [T] = 0, we have (k − [T])^{2} = (k + [T])^{2}; so the wall-crossing formula (6) implies that c^{+}(k) = SW_{X}^{−}(k − [T]) − SW_{X}^{−}(k + [T]), the coefficient of exp(k) in (t^{1/2} − t^{−1/2}) · SW_{X,T}^{−}.

For example, consider the case of the rational elliptic surface E(1) and its fiber class [F]. For a Kahler metric g on E(1), the Kahler form ω is self-dual, and since F is a complex curve, ω · [F] > 0. So the self-dual projection of [F] is a positive multiple of ω, and we see that the small-perturbation component of C_{E(1)} is C_{E(1)}^{+} for n[F], n > 0, and similarly is C_{E(1)}^{−} for n[F], n < 0. Since E(1) carries a metric of positive scalar curvature, this means SW_{E(1)}^{+}(n[F]) = 0 for n > 0, and SW_{E(1)}^{−}(n[F]) = 0 for n < 0. The wall-crossing formula (6) implies that (up to an overall sign) SW_{E(1)}^{−}((2n + 1)[F]) − SW_{E(1)}^{+}((2n + 1)[F]) = 1. Hence (up to that same sign)

SW_{E(1)}^{−}((2n + 1)[F]) = \begin{cases}1, & n ≥ 0 \\ 0, & n < 0. \end{cases}

In other words,

SW_{E(1),F}^{−} = ∑_{n=0}^{∞} t^{(2n+1)/2}

where t = exp(2[F]). Similarly,

SW_{E(1),F}^{+} = − ∑_{n=0}^{∞} t^{−(2n+1)/2}.

Notice that in this case, (t^{1/2} − t^{−1/2}) · SW_{E(1),F}^{+} and (t^{1/2} − t^{−1/2}) · SW_{E(1),F}^{−} both equal −1, as Lemma 5.2 demands.

Let X be a simply connected smooth 4-manifold with b_{X}^{+} = 1, and suppose that X contains a c-embedded oriented torus T. Suppose further that π_{1}(X \ T) = 1. Then for any knot K in S^{3} we can form X_{K}, which is homeomorphic to X. Since [T] = [T_{m}] orients C_{X_{K}}, we have invariants SW_{X}^{±} and SW_{X_{K}}^{±}. Our discussion of § 3 applies in this case as well, once we have the analogues of the log transform and gluing formulas of § 2. These formulas are also due to Taubes and Morgan, Mrowka, and Szabo, cf. [S2].

Theorem (Morgan, Mrowka, and Szabo [MMS], Taubes). Consider the simply connected 4-manifolds X_{1} and X_{2} with b_{X_{1}}^{+} = 1 and b_{X_{2}}^{+} ≥ 1. Suppose that X_{1} and X_{2} contain c-embedded tori T_{1} and T_{2}. Then

SW_{X_{1}#_{T_{1}=T_{2}}X_{2}} = \begin{cases}SW_{X_{1},T_{1}}^{±} · SW_{X_{2},T_{2}}^{±} · (t^{1/2} − t^{−1/2})^{2}, & if b_{X_{2}}^{+} = 1 \\ SW_{X_{1},T_{1}}^{±} · SW_{X_{2}} · (t^{1/2} − t^{−1/2})^{2}, & if b_{X_{2}}^{+} > 1 \end{cases}

where t = exp(2[T]) and [T] = [T_{1}] = [T_{2}] ∈ H_{2}(X_{1}#_{T_{1}=T_{2}}X_{2}).

It follows from Lemma 5.2 that these formulas actually make sense. The restriction to SW_{X_{1},T_{1}}^{±} is accounted for by the fact that b_{X_{1}#_{T_{1}=T_{2}}X_{2}}^{+} ≥ 3, and so any basic class in H_{2}(X_{1}#_{T_{1}=T_{2}}X_{2}) must be orthogonal to [T].

In order to state the internal fiber sum formula, let X be a simply connected 4manifold with b_{X}^{+} = 1. Suppose that X contains a pair of c-embedded tori T_{1}, T_{2}, and let X_{T_{1},T_{2}} denote the internal fiber sum. Denote by SW_{X,T_{1},T_{2}}^{±} the ([T_{1}], [T_{2}])^{⊥}- restricted Seiberg-Witten invariants of X,

SW_{X,T_{1},T_{2}}^{±} = ∑_{\begin{matrix}k·[T_{i}]=0 \\ i=1,2 \end{matrix}} SW_{X}^{±}(k) exp(k).

Theorem (Morgan, Mrowka, and Szabo [MMS], Taubes). Let X be a simply connected 4-manifold with b_{X}^{+} = 1. Suppose that X contains a pair of c-embedded tori T_{1}, T_{2} representing homology classes [T_{1}] and [T_{2}].

SW_{X_{T_{1},T_{2}}} = (SW_{X,T_{1},T_{2}}^{±})|_{τ_{1}=τ_{2}} · (t^{1/2} − t^{−1/2})^{2}

where τ_{i} = exp([T_{i}]) and t = exp(2[T]).

Of course we also need the log transform formula:

Theorem (Morgan, Mrowka, and Szabo [MMS], Taubes). Let Y be a smooth 4-manifold with b^{+} = 1, and suppose that Y contains a nullhomologous torus T. Let τ be the homology class of T_{0} in Y (0/1). For each characteristic homology class α ∈ H_{2}(Y; Z) and p ≠ 0,

SW_{Y (p/q)}^{±}(α) = pSW_{Y}^{±}(α) + q ∑_{i=−∞}^{∞} SW_{Y (0/1)}(α + 2iτ).

Now the arguments of § 3 go through verbatim to give:

Theorem 5.3. Let X be a simply connected smooth 4-manifold with b_{X}^{+} = 1, and suppose that X contains a c-embedded oriented torus T. Suppose further that π_{1}(X \ T) = 1. For t = exp(2[T]) the [T]^{⊥}-restricted Seiberg-Witten invariants of X_{K} are

SW_{X_{K},T}^{±} = SW_{X,T}^{±} · Δ_{K}(t).

As in the b^{+} ≥ 3 case, if X is a symplectic 4-manifold with b^{+} = 1 containing a symplectic embedded torus T of self-intersection 0 and K is a fibered knot, then X_{K} is a symplectic 4-manifold. Conversely, if X is symplectic, choose a metric g for X so that the symplectic form ω is self-dual, and let κ_{X} denote the canonical class. In [T1, T2], Taubes shows that for r << 0, we have SW_{X,g,rω}(−κ_{X}) = ±1 and also that if SW_{X,g,rω}(k) ≠ 0 then −κ_{X} · ω ≤ k · ω, with equality only when k = −κ_{X}. Now let b_{X}^{+} = 1, and let T be an embedded torus of self-intersection 0, such that [T] · ω > 0. For example, this holds if T is symplectically embedded. Then for any k ∈ H_{2}(X; R), we have (2πk^{+} + rω) · [T] < 0 for r << 0. This means that SW_{X,g,rω}(k) = SW_{X}^{−}(k). Hence:

Proposition 5.4 (Taubes). Let X be a symplectic 4-manifold with b^{+} = 1 containing an embedded torus T of self-intersection 0 such that [T] · ω > 0. Then

SW_{X}^{−}(−κ_{X}) = ±1

and if SW_{X}^{−}(k) ≠ 0, then

−κ_{X} · ω ≤ k · ω

with equality only when k = −κ_{X}.

Lemma 5.5. Let X be a simply connected smooth 4-manifold with b_{X}^{+} = 1, and suppose that X contains a c-embedded oriented torus T and that π_{1}(X \ T) = 1. Suppose also that SW_{X#_{T =F} E(1)} ≠ 0. Then there is a Laurent polynomial S_{X} such that

SW_{X,T}^{±} = S_{X} · ∑_{n=0}^{∞} t^{(2n+1)/2} or S_{X} · ∑_{n=0}^{∞} t^{−(2n+1)/2}.

Proof. The gluing formula implies that

SW_{X#_{T =F} E(1)} = SW_{X,T}^{±} · SW_{E(1),F}^{±} · (t^{1/2} − t^{−1/2})^{2} = SW_{X,T}^{±} · (t^{1/2} − t^{−1/2})

by Lemma 5.2 and the above calculation of SW_{E(1),F}^{±}. Because X#_{T =F} E(1) has b^{+} = 3, its Seiberg-Witten invariant is a Laurent polynomial, S_{X} ≠ 0; so the lemma follows.

Lemma 5.6. Suppose that b_{X}^{+} = 1 and α ∈ H^{2}(X), then for any pair (g, η^{+}) with η^{+} · ω_{g} ≠ 0 and |r| >> 0, we have

| SW_{X,g,rη^{+}}(α) ± SW_{X,g,rη^{+}}(−α) | = 1.

Proof. Since FĀ = −F_{A} and q(ψ̄) = −q(ψ), if (A, ψ) is a solution to the Seiberg-Witten equations for α corresponding to (g, rη^{+}), then (Ā, ψ̄) is a solution to the equations for −α corresponding to (g, −rη^{+}). Thus

SW_{X,g,−rη^{+}}(−α) = (−1)^{(sign+e)(X)/4}SW_{X,g,rη^{+}}(α).

For |r| >> 0, the signs of (−2πα − rη^{+}) · ω_{g} and (−2πα + rη^{+}) · ω_{g} are opposite. Thus the wall-crossing formula implies that

| SW_{X,g,−rη^{+}}(−α) ± SW_{X,g,rη^{+}}(−α) | = 1

and the lemma follows.

In the symplectic case, the above lemma is essentially [MS, Prop.2.2].

Corollary 5.7. Let X be a symplectic 4-manifold with b^{+} = 1 containing a symplectic c-embedded torus T. Suppose also that SW_{X#_{T =F} E(1)} ≠ 0. If Δ_{K} is not monic, then X_{K} does not admit a symplectic structure.

Proof. Write Δ_{K}(t) = a_{0} + ∑_{j=1}^{d} a_{j}(t^{j} + t^{−j}) with a_{d} ≠ 0. Suppose that X_{K} admits a

symplectic structure with symplectic form ω and canonical class κ. If [T] · ω = 0, it follows from Lemma 5.1 that [T] = λω for some λ ≠ 0. Such a symplectic form is clearly nongeneric and we may perturb it so that [T] · ω ≠ 0. We may also assume that T is oriented so that [T] · ω > 0. The adjunction inequality of Li and Liu [LL, Thm.E] then implies that κ · [T] ≤ 0.

The hypothesis that SW_{X#_{T =F} E(1)} ≠ 0 and Lemma 5.5 imply that there is a Laurent polynomial S_{X} ≠ 0 such that, for t = exp(2[T]), one of

SW_{X,T}^{−} = S_{X} · ∑_{n=0}^{∞} t^{(2n+1)/2} (7)

SW_{X,T}^{−} = S_{X} · ∑_{n=0}^{∞} t^{−(2n+1)/2} (8)

holds. Let α be any class such that the coefficient of exp(α) in S_{X} is nonzero. There are finitely many such classes; so we may list the integers m_{1} < ⋯ < m_{r} such that the coefficient S_{X}(α+m_{i}[T]) of exp(α+m_{i}[T]) in S_{X} is nonzero. Theorem 5.3 implies that if the case (7) holds, then

SW_{X_{K}}^{−}(α + (m_{1} − 2d + 1)[T]) = a_{d} S_{X}(α + m_{1}[T]) ≠ 0 (9)

and if the case (8) holds then

SW_{X_{K}}^{−}(α + (m_{r} + 2d − 1)[T]) = a_{d} S_{X}(α + m_{r}[T]) ≠ 0. (10)

If κ·ω < 0 then, by a result of Liu and Ohta and Ono, X_{K} is the blowup of a rational or ruled surface [MS, Cor.1.4]. Every such surface has a metric of positive scalar curvature; so it follows from the wall-crossing formula that for each α, SW_{X_{K}}^{±}(α) = ±1 or 0. Equations (9) and (10) imply in this case that a_{d} = ±1, i.e. that Δ_{K} is monic.

Thus we may assume that κ · ω > 0. This means that κ and [T] both lie in the same component of the cone C_{X_{K}}; so Lemma 5.1 implies that κ · [T] ≥ 0. Since we already have the opposite inequality, we must have κ · [T] = 0. By Lemma 5.6,

| SW_{X_{K}}^{−}(κ + 2m[T]) ± SW_{X_{K}}^{−}(−κ − 2m[T]) | = 1

for each m. However [T] · ω > 0; so for m large, we have (−κ − 2m[T]) · ω < −κ · ω. Thus Proposition 5.4 implies that SW_{X_{K}}^{−}(−κ − 2m[T]) = 0 for m >> 0. This means that SW_{X_{K}}^{−}(κ + 2m[T]) = ±1 for m >> 0. But SW_{X_{K},T}^{−} = SW_{X,T}^{−} · Δ_{K}(t) and (κ + 2m[T]) · [T] = 0; so the case (7) must hold.

Again by Proposition 5.4, we have SW_{X_{K}}^{−}(−κ) ≠ 0; so we may write κ = −α+2n[T] where SW_{X}^{−}(α) ≠ 0 and |n| ≤ d, and again we have m_{1} < ⋯ < m_{r} and (9). From (7):

SW_{X}^{−}(α) = ∑_{m_{i} odd<0} S_{X}(α + m_{i}[T]).

But (α+(m_{1}−2d+1)[T])·ω = −κ·ω+(m_{1}+1−2(d−n))[T]·ω ≤ −κ·ω because m_{1} < 0 (since SW_{X}^{−}(α) ≠ 0) and d − n ≥ 0. Thus Proposition 5.4 implies that m_{1} = −1 and n = d; so κ = α + 2d[T]. This means that ±1 = SW_{X_{K}}^{−}(−κ) = a_{d} · SW_{X}^{−}(−α); so a_{d} = ±1, i.e. Δ_{K} is monic.

As in the b^{+} > 1 case, if X contains a surface Σ_{g} of genus g disjoint from T with 0 ≠ [Σ_{g}] ∈ H_{2}(X; Z) and with [Σ_{g}]^{2} < 2 − 2g if g > 0 or [Σ_{g}]^{2} < 0 if g = 0, then X_{K} with the opposite orientation does not admit a symplectic structure. These results along with the blowup formula of [FS2] imply:

Corollary 5.8. For any knot K in S^{3} whose Alexander polynomial Δ_{K}(t) is not monic, the manifolds E(1)_{K} admit no symplectic structure with either orientation, even after an arbitrary number of blowups.

The first examples of this sort were obtained by Szabo [S2], and in fact they are the manifolds E(1)_{K_{k}} where K_{k} is the k-twist knot. For these examples (with T = F)

SW_{E(1)_{K_{k}} ,T}^{−} = (kt − (2k + 1) + kt^{−1}) · ∑_{n=0}^{∞} t^{(2n+1)/2}

SW_{E(1)_{K_{k}} ,T}^{+} = −(kt − (2k + 1) + kt^{−1}) · ∑_{n=0}^{∞} t^{−(2n+1)/2}

Szabo computes the ‘small perturbation’ invariant SW_{E(1)_{K_{k}}}^{0} . As is the case for E(1), for m > 0, this is SW_{E(1)_{K_{k}}}^{0} (m[T]) = SW_{E(1)_{K_{k}}}^{+} (m[T]), and for m < 0, it is SW_{E(1)_{K_{k}}}^{0} (m[T]) = SW_{E(1)_{K_{k}}}^{−} (m[T]). Hence

SW_{E(1)_{K_{k},T}}^{0} = kt^{−1/2} − kt^{1/2},

as Szabo calculates.

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Ronald Fintushel, Department of Mathematics, Michigan State University, East Lansing, MI 48824

  E-mail address: ronfint@math.msu.edu

Ronald J. Stern, Department of Mathematics, University of California, Irvine, CA 92697

  E-mail address: rstern@math.uci.edu