Abstract
We study a family of 2-bridge knots with 2-tangles in the 3-sphere admitting a genus two 1-bridge splitting. We also observe a geometric relation between (g i 1;1)-splitting and (g;0)- splitting for g = 2;3. Moreover we construct a family of closed orientable 3-manifolds which are n-fold cyclic coverings of the 3- sphere branched over those 2-bridge knots. 1. Preliminaries and deflnitions By a handlebody, we mean a bounded connected oriented 3-manifold V which admits mutually disjoint proper disks such that they split V into solid tori. Let M be a closed connected oriented 3-manifold. A Heegaard handlebody of M is a handlebody V in M such that V 0 = Cl(M i V ), the closure of M i V , is also a handlebody. The surface H = V V 0 and V (H V 0 are called a Heegaard surface(or Heegaard diagram) and a Heegaard splitting of M, respectively. Every closed connected oriented 3-manifold M admits a Heegaard splitting.
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