How essential is a spanning surface?

TL;DR

This paper introduces new invariants to measure how far spanning surfaces are from being compressible, extends existing theorems about plumbing, and explores their behavior in various 3-manifolds.

Contribution

It defines algebraic and geometric essence invariants for spanning surfaces, extends Ozawa's theorem to these invariants, and introduces twisted plumbing to analyze their properties.

Findings

Plumbing respects the algebraic essence invariant.

The new invariants distinguish between essential and non-essential surfaces.

Extended results apply to arbitrary 3-manifolds.

Abstract

Gabai proved that any plumbing, or Murasugi sum, of $\pi_1$-essential Seifert surfaces is also $\pi_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show that the analogous statement about geometrically essential surfaces is untrue. We then introduce new numerical invariants, the algebraic and geometric essence of a spanning surface $F\subset S^3$, which measure how far $F$ is from being compressible, and we extend Ozawa's theorem by showing that plumbing respects the algebraic version of this new invariant. We also introduce a ``twisted'' generalization of plumbing and use it to compute essence for many examples, including checkerboard surfaces from reduced alternating diagrams. Finally, we extend all of these results to plumbings and twisted plumbings of spanning surfaces in arbitrary 3-manifolds.