$\mathbb{Z}$-Torus Exteriors and Small Knot-Surgery Four-Manifolds

Abstract

We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of $M_K\times S^1$ from $Σ_g\times T^2$, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block $Y_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to $S^2\times S^2$. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of $Y_K\setminusνΣ_2$, rather than the involutive gluing defining the original $X_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and $σ=0$. An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to $\#_3(S^2\times S^2)$. The Case II small-perturbation Seiberg--Witten invariant vanishes. The Seiberg--Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.