Tunnel one, fibered links (Q692402)

An unknotting tunnel \(\tau\) for a link \(L\) in the \(3\)--sphere \(S^3\) is a properly embedded arc in \(E(L) = S^3 - \mathrm{int}N(L)\) such that \(S^3 - \mathrm{int}(N(L) \cup N(\tau))\) is a genus two handlebody. If a link \(L\) has an unknotting tunnel, then \(L\) has at most two components, and if it has exactly two components \(K_1\) and \(K_2\), then \(\tau\) connects \(N(K_1)\) and \(N(K_2)\). Suppose that \(L\) has an unknotting tunnel \(\tau\) and the exterior \(E(L)\) admits a fibration over the circle with a fiber surface \(F\). In the paper under review, the author proves that \(\tau\) can be isotoped to lie in the fiber surface \(F\). In the proof, analogous to the proof of a result by \textit{M. Scharlemann} and \textit{A. Thompson} [Proc. Lond. Math. Soc., III. Ser. 87, No. 2, 523--544 (2003; Zbl 1047.57008)] the author begins by showing that \(\tau\) may be slid and isotoped until it is disjoint from \(F\). Then cut \(E(L)\) along \(F\) to obtain a product \(3\)--manifold \(M \cong F \times [0, 1]\), which contains the properly embedded arc \(\tau\). Let \(\widehat{M} \cong \widehat{F} \times [0, 1]\) be the \(3\)--manifold obtained from two copies of \(M\) by identifying their vertical boundary \(\partial F \times [0, 1]\), where \(\widehat{F}\) is the double of \(F\). Note that the product \(3\)--manifold \(\widehat{M}\) contains the double \(\widehat{\tau}\) of \(\tau\). The author observes that \(\widehat{\tau}\) admits a nontrivial surgery such that \(\widehat{F} \times \{ 0 \}\) compresses in the resulting \(3\)--manifold, and then applies a result by \textit{M. Scharlemann} and \textit{A. Thompson} [Algebr. Geom. Topol. 9, No. 3, 1825--1835 (2009; Zbl 1197.57011)] to conclude that \(\tau\) can be pushed into the fiber surface \(F\).