Common stabilizations of Heegaard splittings of link exteriors
A \textit{compression body} \(H\) is a 3-manifold obtained from a compact closed surface \(S\) by attaching 2-handles to \(S\times [0,1]\) on \(S\times \{1\}\) and capping off any resulting 2-sphere boundary component with 3-handles. The surface \(S\times \{0\}\) is denoted by \(\partial_{+}H\) and \(\partial H-\partial_{+}H\) is denoted by \(\partial_{-}H\). If a compact 3-manifold \(M\) is the union of two compression bodies \(H_1\) and \(H_2\) along their boundary \(S=\partial_{+}H_1=\partial_{+}H_2\), we call the decomposition \(M=H_1\cup_{S}H_2\) a \textit{Heegaard splitting } of \(M\). The genus of \(S\) is called the genus of the Heegaard splitting of \(H_1\cup_{S}H_2\). Two Heegaard splittings \(H_1\cup_S H_2\) and \(H'_1\cup_{S'}H'_2\) of \(M\) are \textit{isotopic} if there is an ambient isotopy \(h_t\,\,\,(0\leq t \leq 1)\) of \(M\) such that \(h_1(H_1)=H'_1\). Let \(V_1\cup_{T}V_2\) be the genus one Heegaard splitting of the 3-sphere \(S^3\). Note that genus \(g\) Heegaard splittings of \(S^3\) are mutually isotopic for each \(g\) by \textit{F. Waldhausen} [Topology 7, 195--203 (1968; Zbl 0157.54501)]. We say \(H'_{1}\cup_{S'}H'_{2}\) is obtained from \(H_1\cup_{S}H_2\) by a \textit{stabilization} if \(H'_{1}\cup_{S'}H'_{2}\) is a connected sum of \(H_1\cup_{S}H_2\) and \(V_1\cup_{T}V_2\). \textit{M. Reidemeister} [Abhandlungen Hamburg 10, 109--118 (1934; JFM 60.0514.01)] and \textit{J. Singer} [Trans. Am. Math. Soc. 35, 88--111 (1933; Zbl 0006.18501)] showed that any two Heegaard splittings of \(M\) become isotopic by a sequence of stabilizations, and then the next conjecture was presented by \textit{M. Scharlemann} in [Handbook of geometric topology, Amsterdam: Elsevier, 921--953 (2002; Zbl 0985.57005)]: Conjecture. Suppose \(H_1\cup_{S}H_2\) and \(H'_1\cup_{S'}H'_2\) are Heegaard splittings of the same 3-manifold: genus \(g\leq g'\), respectively. Then the splittings obtained by one stabilization of \(H'_1\cup_{S'}H'_2\) and \(g'-g+1\) stabilizations of \(H_1\cup_{S}H_2\) are isotopic. In this paper, the authors give a sufficient condition for a non-trivial tunnel number one link exterior to have a common stabilization when it has two disjoint unknotting tunnels. Moreover, they confirm that any 2-bridge link satisfies this condition and the above conjecture. The details are as follows. A tunnel of a knot or a link \(K\) in \(S^3\) is a collection of disjoint embedded arcs \(t_1,t_2,\ldots, t_n\) in \(S^3\) with \(K\cap\cup_{i=1}^{n}t_i=\cup_{i=1}^{n}\partial t_i\) such that \(H=\text{ cl}(S^3-N(K\cup\cup_{i=1}^{n}t_i))\) is a genus \(n+1\) handlebody. The tunnels gives rise to a Heegaard splitting of the exterior of \(K\) \[ \text{ cl}(S^3-N(K))=H\cup_{\partial H}N(K\cup\cup_{i=1}^{n}t_n-N(K)). \] The minimum number of such number \(n\) is called the \textit{tunnel number} of \(K\). If the tunnel number of \(K\) is one, the tunnel is called \textit{unknotting tunnel} of \(K\). The main theorem of this paper is: \noindent Theorem. Let \(L\) be a non-trivial tunnel number one link in \(S^3\) and \(t_1\) and \(t_2\) two disjoint unknotting tunnels of \(L\) such that a meridian disk of the genus two handlebody \(H_1=\text{ cl}(S^3-N(L\cup t_1))\) does not intersect \(t_2\). Then \(\text{ cl}(S^3-N(L\cup t_1\cup t_2))\) is a genus three handlebody.
scientific article; zbMATH DE number 2173257