A note on fundamental groups of symplectic torus complements in 4-manifolds

Abstract

Previously, we constructed an infinite family of knotted symplectic tori representing a fixed homology class in the symplectic four-manifold E(n) K, which is obtained by Fintushel-Stern knot surgery using a nontrivial fibered knot K in S-3, and distinguished the (smooth) isotopy classes of these tori by indirectly computing the Seiberg-Witten invariants of their complements. In this note, we compute the fundamental groups of the complements of these knotted tori and show that for each nontrivial fibered knot K these groups constitute an infinite collection of nonisomorphic groups. We also review some other constructions of symplectic tori in 4-manifolds and show that the fundamental groups of the complements do not distinguish homologous tori in those cases.