Smoothly embedding Seifert fibered spaces in

Using an obstruction based on Donaldson’s theorem, we derive strong restrictions on when a Seifert fibered space Y = F ( e ; p 1 q 1 , … , p k q k ) Y = F(e; \frac {p_1}{q_1}, \ldots , \frac {p_k}{q_k}) over an orientable base surface F F can smoothly embed in S 4 S^4 . This allows us to classify precisely when Y Y smoothly embeds provided e > k / 2 e > k/2 , where e e is the normalized central weight and k k is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant μ ¯ \overline {\mu } , we make some conjectures concerning Seifert fibered spaces which embed in S 4 S^4 . Finally, we also provide some applications to doubly slice Montesinos links, including a classification of the smoothly doubly slice odd pretzel knots up to mutation.