Trivial double-torus knot
Chuichiro Hayashi
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A knot $K$ in a closed connected orientable $3$-manifold $M$ is called a double-torus knot, if it is in a genus two Heegaard splitting surface $H$ of $M$. We give a necessary and sufficient condition for a double-torus knot to be the trivial knot in words of meridian disks of genus two handlebodies obtained by splitting $M$ along $H$.
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