Exotically knotted closed surfaces from Donaldson's diagonalization for families

Abstract

We introduce a method to detect exotic surfaces without explicitly using a smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the construction. Our main tools are two versions of families (Seiberg-Witten) generalizations of Donaldson's diagonalization theorem, including a real and families version of the diagonalization. This leads to an example of a pair of exotically knotted $\mathbb{R}P^2$'s embedded in a closed 4-manifold whose complements are diffeomorphic, making it the first example of a non-orientable surface with this property. In particular, any invariant of a 4-manifold-surface pair (including invariants from real Seiberg-Witten theory such as Miyazawa's invariant) fails to detect such an exotic $\mathbb{R} P^2$. One consequence of our construction reveals that non-effective embeddings of corks can still be useful in pursuit of exotica. Precisely, starting with an embedding of a cork $C$ in certain a 4-manifold $X$ where the cork-twist does not change the diffeomorphism type of $X$, we give a construction that provides examples of exotically knotted spheres and $\mathbb{R}P^2$'s with diffeomorphic complements in $ C \# S^2 \times S^2 \subset X \# S^2 \times S^2$ or $C \# \mathbb{C}P^2 \subset X \# \mathbb{C}P^2 $. In another direction, we provide infinitely many exotically knotted embeddings of orientable surfaces, closed surface links, and 3-spheres with diffeomorphic complements in once stabilized corks, and show some of these surfaces survive arbitrarily many internal stabilizations. By combining similar methods with Gabai's 4D light-bulb theorem, we also exhibit arbitrarily large difference between algebraic and geometric intersections of certain family of 2-spheres, embedded in a 4-manifold.