How natural of a geometric operation is Murasugi sum?

Abstract

Gabai proved that any Murasugi sum of $π_1$-essential Seifert surfaces is also $π_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show, however, that the analogous statement about geometrically essential surfaces is untrue. (A spanning surface is geometrically essential if it cannot be compressed or boundary compressed to another spanning surface.) We also ask when plumbing an unknotted annulus (with any number of twists) onto a compressible spanning surface yields a compressible surface. We provide positive and negative examples, and we establish a simple sufficient condition. As an application, we obtain a new constructive proof of a result of Hatcher and Thurston about essential spanning surfaces for 2-bridge knots and links.