Twisted fiber sums of Fintushel-Stern’s knot surgery 4-manifolds

Abstract

Abstract. In the article, we study Fintushel-Stern's knot surgery four-manifold E(n) K and its monodromy factorization. For fibered knots we provide a smooth classification of knot surgery 4-manifolds up to twisted fiber sums. We then show that other constructions of 4-manifolds with the same SeibergWitten invariants are in fact diffeomorphic.

Lefschetz fibration structures on knot surgery 4-manifolds

“…Then, for any knot K ⊂ S 3 , one can construct a new 4-manifold, called a knot surgery 4-manifold, X K = X♯ T =Tm (S 1 × M K ) by taking a fiber sum along a torus T in X and T m = S 1 × m in S 1 × M K , where M K is the 3-manifold obtained by doing 0-framed surgery along K and m is the meridian of K. Then Fintushel and Stern proved that, under a mild condition on X and T , the knot surgery 4-manifold X K is homeomorphic, but not diffeomorphic, to a given X [3]. Furthermore, if X is a simply connected elliptic surface E(2), T is the elliptic fiber, and K is a fibred knot, then it is also known that the knot surgery 4-manifold E(2) K admits not only a symplectic structure but also a genus 2g(K)+1 Lefschetz fibration structure [5,22]. Note that there are only two inequivalent genus one fibred knots, but there are infinitely many inequivalent genus g fibred knots for g ≥ 2.…”

“…Furthermore, Fintushel and Stern [5] also questioned whether any two in the following 4-manifolds {Y (2; K 1 , K 2 ) := E(2) K1 ♯ id:Σ2g+1→Σ2g+1 E(2) K2 | K 1 , K 2 are genus g fibred knots} are mutually diffeomorphic or not. The second author obtained a partial result related to this question under the constraint that one of K i (i = 1, 2) is fixed [22].…”

Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds

“…Theorem 3 [7,19] Let K ⊂ S 3 be a fibered knot of genus g. Then the knot surgery 4-manifold E(n) K , as a genus (2g + n − 1) Lefschetz fibration, has a monodromy factorization…”

“…Lemma 5 [2,19] Let W i = w i,n i · · · w i,2 · w i,1 (i = 1, 2) be a sequence of right handed Dehn twists along a simple closed curves on Σ g such that W )) ∼ = G(W ) because we take a conjugate to each letter in W .…”

“…Then Fintushel and Stern proved that, under a mild condition on X and T , the knot surgery 4-manifold X K is homeomorphic, but not diffeomorphic, to a given X [6]. Furthermore, if X is a simply connected elliptic surface E(n), T is the elliptic fiber, and K is a fibered knot, then it is also known that the knot surgery 4-manifold E(n) K admits not only a symplectic structure but also a genus 2g(K ) + n − 1 Lefschetz fibration structure [7,19].…”