Fibers as normal and spun-normal surfaces in link manifolds

Abstract

The technique of inflating ideal triangulations developed by Jaco and Rubinstein gives a procedure starting with an ideal triangulation 𝒯* of the interior of a compact 3-manifold 𝑀, that constructs a triangulation 𝒯⌄𝛬 of 𝑀 with real boundary components. This construction is carried out in such a way that combinatorial information from 𝒯* persists in 𝒯⌄𝛬. Viewed as an inverse operation, crushing 𝒯⌄𝛬 along the boundary of 𝑀 recovers exactly 𝒯*. We present results from joint work with Jaco and Rubinstein showing, for 𝒯* and 𝒯⌄𝛬, there is a bijection between the closed normal surfaces of 𝒯* and the closed normal surfaces of 𝒯⌄𝛬. Further corresponding surfaces are homeomorphic. Given the previous relationship for closed normal surfaces, it is natural to inquire about surfaces with boundary. That is, if a surface is normal in 𝒯⌄𝛬, is there a corresponding spun-normal surface in 𝒯*? In general the answer is no. However, we show that an affirmative answer can be given if the normal surface in 𝒯⌄𝛬 is in `C-position.' Cooper Tillmann and Worden pose the question: For a fibered knot complement or fibered once-cusped 3-manifold 𝑀, is there always some ideal triangulation of 𝑀 such that the fiber is realized as an embedded spun-normal surface. We present an algorithm that will construct an inflated triangulation 𝒯⌄𝛬 in which the fiber is a normal surface in C-position, thus in the underlying ideal triangulation the fiber is realized as a spun-normal surface; answering the question in the affirmative. Further the algorithm will find the spun-normal representation of the fiber.