Geodesic
θ-curves inducing two different knots with the same 2-fold branched covering spaces
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 199-209.

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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:S3→S3/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 Z2⊕Z2 cyclic branched coverings over θ-curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.
Per un nodo K con un'inversione forte i indotta da un tunnel di scioglimento abbiamo una proiezione Π:S3→S3/i che è un ricoprimento doppio ramificato sopra un nodo banale Π(fix(i)), dove fix(i) è l'asse i. Allora un insieme Π(fix(i)∪K) è chiamato θ-curva. Costruiamo θ-curve e i ricoprimenti Z2⊕Z2 ciclici ramificati sopra θ-curve, che hanno due decomposizioni di Heegaard non isotopiche che sono uno stabilmente equivalenti. 
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Kim, Soo Hwan; Kim, Yangkok. $\theta$-curves inducing two different knots with the same $2$-fold branched covering spaces. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 199-209. https://geodesic-test.mathdoc.fr/item/BUMI_2003_8_6B_1_a11/

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