Geodesic
$\theta$-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

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For a knot $K$ with a strong inversion $i$ induced by an unknotting tunnel, we have a double covering projection $\Pi \colon S^{3}\rightarrow S^{3}/i$ branched over a trivial knot $\Pi(\text{fix}(i))$, where $\text{fix}(i)$ is the axis of $i$. Then a set $\Pi(\text{fix}(i)\cup K)$ is called a $\theta$-curve. We construct $\theta$-curves and the $\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$ cyclic branched coverings over $\theta$-curves, having two non-isotopic Heegaard decompositions which are one stable equivalent. 
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     author = {Kim, Soo Hwan and Kim, Yangkok},
     title = {$\theta$-curves inducing two different knots with the same $2$-fold branched covering spaces},
     journal = {Bollettino della Unione matematica italiana},
     pages = {199--209},
     publisher = {mathdoc},
     volume = {Ser. 8, 6B},
     number = {1},
     year = {2003},
     zbl = {1150.57002},
     mrnumber = {MR1955705},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_1_a11/}
}
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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. http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_1_a11/