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Spun normal surfaces in $3$–manifolds, I: $1$–efficient triangulations

Ensil Kang and Joachim Hyam Rubinstein

Algebraic & Geometric Topology 22 (2022) 3533–3576

Spun normal surfaces are a useful way of representing proper essential surfaces using ideal triangulations for 3–manifolds with tori and Klein bottle boundaries. Walsh (2011) established that proper essential surfaces which are not virtual fibers can be put into spun normal form, so long as the triangulations have no edges isotopic into the boundary. We use a different approach to study spun normal surfaces in 1–efficient ideal triangulations. 1–efficiency is a stronger condition than Walsh used, but we are able to deal with semifiber surfaces. Moreover, fiber surfaces are shown to spin normalize, so long as there is a closed normal surface in the infinite cyclic covering dual to the fibering. In two followup papers, spinning essential surfaces in general ideal triangulations will be considered and in the case of a single boundary component of the 3–manifold, all the boundary slopes of proper essential surfaces will be shown to occur at vertices of the projective solution space, answering a question of Dunfield and Garoufalidis.

3–manifolds, normal surfaces, ideal triangulation, spun normal surfaces
Mathematical Subject Classification 2010
Primary: 57M99
Secondary: 57M10
Received: 1 September 2014
Revised: 19 January 2021
Accepted: 6 September 2021
Published: 14 March 2023
Ensil Kang
Department of Mathematics
Chosun University
South Korea
Joachim Hyam Rubinstein
Department of Mathematics and Statistics
The University of Melbourne
Melbourne VIC