Spun normal surfaces in 3-manifolds III: Boundary slopes

Abstract

Spun normal surfaces are a useful way of representing proper essential surfaces, using ideal triangulations for 3-manifolds with tori and Klein bottle boundaries. In this paper, we consider spinning essential surfaces in an irreducible P 2 P^2 -irreducible, anannular, atoroidal 3-manifold with tori and Klein bottle boundary components. We can assume that such a 3-manifold is equipped with an ideal 1-efficient triangulation. In particular, we prove that for a given choice of a set of boundary slopes for a proper essential surface, there is a set of essential vertex solutions for the projective solution space at these boundary slopes, answering a question of Dunfield and Garoufalidis [Trans. Amer. Math. Soc. 364 (2012), pp. 6109–6137], so long as the slopes are not of a fiber of a bundle structure.