The disk complex of a surface in a 3–manifold is used to define its topological index. Surfaces with well-defined topological index are shown to generalize well known classes, such as incompressible, strongly irreducible and critical surfaces. The main result is that one may always isotope a surface with topological index to meet an incompressible surface so that the sum of the indices of the components of is at most . This theorem and its corollaries generalize many known results about surfaces in 3–manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel’s distance to surfaces with topological index .
"Topological Index Theory for surfaces in 3–manifolds." Geom. Topol. 14 (1) 585 - 609, 2010. https://doi.org/10.2140/gt.2010.14.585