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θ -curves inducing two different knots with the same 2 -fold branched covering spaces

Soo Hwan KimYangkok Kim — 2003

Bollettino dell'Unione Matematica Italiana

For a knot K with a strong inversion i induced by an unknotting tunnel, we have a double covering projection Π : S 3 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 K is called a θ -curve. We construct θ -curves and the Z 2 Z 2 cyclic branched coverings over θ -curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.

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