Title : Distance between toroidal surgeries on hyperbolic knots in the $3$-sphere
Speaker : Masakazu Teragaito (Hiroshima University)
Abstract :

For a hyperbolic knot in the $3$-sphere $S^3$, at most finitely many Dehn surgeries yield non-hyperbolic $3$-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed $3$-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. There are many examples of integral toroidal surgery, and Eudave-Mu\~{n}oz constructed an infinite family of hyperbolic knots $k(\ell,m,n,p)$ admitting half-integral toroidal surgeries. Recently, Gordon and Luecke proved that the Eudave-Mu\~{n}oz knots are the only hyperbolic knots with half-integral toroidal surgeries. We consider the distance (the minimal geometric intersection number) between toroidal slopes on a hyperbolic knot in $S^3$. The figure-eight knot admits exactly three toroidal slopes $0, 4$ and $-4$. Note that $\Delta(-4,4)=8$. If a hyperbolic knot is not the figure-eight knot, then the distance between two toroidal slopes is at most $5$ by Gordon's general result. (There are exactly four hyperbolic $3$-manifolds which admit two toroidal slopes with distance at least $6$. They all are obtained from the Whitehead link by some Dehn surgery on one component. Among those, only the figure-eight knot exterior can be embedded in $S^3$.) This upper bound $5$ is sharp. For example, the $(-2,3,7)$-pretzel knot has toroidal slopes $16$ and $37/2$ with $\Delta(16,37/2)=5$. The purpose of this talk is to show that we can reduce the upper bound when both of toroidal slopes are integral. \begin{theorem}\label{main1} Let $K$ be a hyperbolic knot in $S^3$, which is not the figure-eight knot. If $\alpha$ and $\beta$ are two integral toroidal slopes for $K$, then $\Delta(\alpha,\beta)\le 4$. \end{theorem} This is sharp. For example, the twist knot $C[2n,2]$ in Conway's notation with $n\ge 1$ admits two integral toroidal slopes $0$ and $4$. \vspace{-1pt} \begin{corollary}\label{main2} If a hyperbolic knot $K$ in $S^3$ admits two toroidal slopes $\alpha$ and $\beta$ with $\Delta(\alpha,\beta)=5$, then $K$ is the Eudave-Mu\~{n}oz knot $k(2,-1,n,0)$ for some integer $n\ne 1$, and $\{\alpha,\beta\}=\{25n-\frac{37}{2},25n-16\}$. \end{corollary} \vspace{-1pt} It is conjectured that a hyperbolic knot in $S^3$ admits at most three toroidal surgeries. This holds for Eudave-Mu\~{n}oz knots. By Gordon's result stated above, a hyperbolic knot admits at most $6$ toroidal surgeries. Our main theorem also gives an improvement of this upper bound. \vspace{-1pt} \begin{corollary} A hyperbolic knot in $S^3$ admits at most $5$ toroidal surgeries. \end{corollary}

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