J. Aust. Math. Soc.  78 (2005), 305-321
Normal surfaces in non-compact 3-manifolds

Ensil Kang
  Department of Mathematics
  College of Natural Sciences
  Chosun University
  Gwangju 501-759

We extend the normal surface $Q$-theory to non-compact 3-manifolds with respect to ideal triangulations. An ideal triangulation of a 3-manifold often has a small number of tetrahedra resulting in a system of $Q$-matching equations with a small number of variables. A unique feature of our approach is that a compact surface $F$ with boundary properly embedded in a non-compact 3-manifold $M$ with an ideal triangulation with torus cusps can be represented by a normal surface in $M$ as follows. A half-open annulus made up of an infinite number of triangular disks is attached to each boundary component of $F$. The resulting surface $\hat{F}$, when normalized, will contain only a finite number of $Q$-disks and thus correspond to an admissible solution to the system of $Q$-matching equations. The correspondence is bijective.
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