On critical Heegaard splittings of tunnel number two composite knot exteriors (Q2872785)

The definitions of topologically minimal surface and topological index for a surface contained in an orientable compact 3-manifold where introduced by \textit{D. Bachman} [Geom. Topol. 14, No. 1, 585--609 (2010; Zbl 1206.57020)]. These concepts are a way to regard incompressible, strongly incompressible and critical surfaces in a unified viewpoint. For instance a surface \(F\) is incompressible if and only if \(F\) has topological index 0; it is strongly incompressible if and only if it has topological index 1, and it is critical if and only if it has topological index 2.NEWLINENEWLINEA way to determine the nature of a Heegaard splitting \((\mathcal{V}, \mathcal{W}; F)\) of a 3-manifold \(M\), is to examine compressing disks for the Heegaard surface \(F\). A pair of compressing disks \((V,W)\) for \(F\) with \(V \subset \mathcal{V}\) and \(W \subset{W}\) is a weak reducing pair if \(V\cap W = \emptyset\).NEWLINENEWLINEA Heegaard splitting is called critical if the Heegaard surface is a critical surface.NEWLINENEWLINEIn the paper under review, the author gives a condition for a genus three unstabilized Heegaard splitting \((\mathcal{V}, \mathcal{W}; F)\) to be a critical. The condition is the existence of two weak reducing pairs \((V_0, W_0)\) and \((V_1, W_1)\) where \(V_0\) and \(V_1\) are non-isotopic, non-separating disks in \(V\). The author applies this result to the exterior of a tunnel number two knot. He shows that the Heegaard splitting induced by a tunnel system is critical if there are two weak reducing pairs such that each weak reducing pair contains the cocore disk of each tunnel. Finally, the analogous result is proved for the connected sum of two 2-bridge knots and the connected sum of two \((1,1)\)-knots.