-knots via the mapping class group of the twice punctured torus.
We describe a combinatorial algorithm for constructing all orientable 3-manifolds with a given standard bidimensional spine by making use of the idea of bijoin (Bandieri and Gagliardi (1982), Graselli (1985)) over a suitable pseudosimplicial triangulation of the spine.
For a knot with a strong inversion induced by an unknotting tunnel, we have a double covering projection branched over a trivial knot , where is the axis of . Then a set is called a -curve. We construct -curves and the cyclic branched coverings over -curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.