Systematic Singular Triangulations of All Orientable Seifert Manifolds

In this paper, we construct singular triangulations [1] of all orientable Seifert manifolds [2]. Especially, we consider singular triangulations with only one vertex, called one-vertex triangulation. Our construction is useful to calculate the state sum type invariant, for example, Turaev-Viro invariant, Turaev-Viro-Ocneanu invariant or Dijkgraaf-Witten invariant; this subject will be seen in forthcoming paper [3]. Also our work is made use of the introduction of a new complexity invariant of closed 3-manifold, see [4]. Let M be a Seifert manifold and P be a special spine [5] of M. Considering a dual complex for M relative to P , we obtain a one-vertex triangulation of M. Now, how to construct a special spine P of M? Our construction is based on the fact that any orientable Seifert manifold is obtained by gluing Mn, J and Vp,q , which are homeomorphic to (S2 − ∐n i=1 D2 i ) × S1, (S1 × S1 − D2) × S1 and (p, q)-type fibered solid torus respectively. The first step is to make special spines PMn , PJ , PVp,q of three compact manifolds Mn, J and Vp,q satisfying the following conditions: each connected component of ∂ Mn ∩ PMn , ∂ J ∩ PJ and ∂ Vp,q ∩ PVp,q is the theta-curve shown in Figure 1 and the loop γ ᾱ is a fiber, where ᾱ means the reverse direction of the edge labeled α. As an example, the solid torus V1,1