Heegaard Splittings and Branched Coverings of B3.

Steven A. Bleiler; Yoav Moriah

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For a knot $K$ with a strong inversion $i$ induced by an unknotting tunnel, we have a double covering projection $Π : S^{3} \to S^{3} / i$ branched over a trivial knot $Π (fix (i))$ , where $fix (i)$ is the axis of $i$ . Then a set $Π (fix (i) \cup K)$ is called a $θ$ -curve. We construct $θ$ -curves and the $Z_{2} \oplus Z_{2}$ cyclic branched coverings over $θ$ -curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.