Knots with arbitrarily high distance bridge decompositions (Q2872188)

A link \(L\) in a closed orientable 3-manifold \(M\) admits a \((g,b)\)-bridge splitting if there is a genus \(g\) Heegaard splitting \((V_1, V_2, S)\) of \(M\) with \(b\) trivial arcs of \(L\) in each handlebody. The distance of a \((g,b)\)-bridge splitting is the length of the shortest path in the curve complex of the surface \(S_0= S-\)int\(N(S\cap L)\) which connects the subcomplexes \(D(V_1)\) and \(D(V_2)\), where \(D(V_i)\) is the subcomplex consisting of all vertices that correspond to simple closed curves bounding disks in \(V_i -\)int\((V_i\cap L)\), for \(i=1,2\).NEWLINENEWLINEThe aim of this paper is to prove that for any 3-manifold \(M\) with a genus \(g\) Heegaard splitting \((V_1, V_2, S)\), any positive integers \(b\) and \(n\), there is a knot \(K\) in \(M\) which admits a \((g,b)\)-bridge splitting of distance greater than \(n\) with respect to \((V_1, V_2, S)\), except for \((g,b)=(0,1), (0,2)\).NEWLINENEWLINEThe authors consider a trivial link \(L \subset M\) with \(b\) components, such that \(L\) admits a \((g,b)\)-bridge splitting with respect to \((V_1, V_2, S)\). The idea is to replace \(V_2\) in \(M\) to obtain a manifold \(M_m\), homeomorphic to \(M\), and a knot \(K_m\) which satisfies the required conditions. The replacement is done via a power of a pseudo-Anosov homeomorphism \(\Phi _N : S_0 \rightarrow S_0\) that satisfies \(d(D(V_1), \Phi _N ^mD(V_2)) \rightarrow \infty\) as \(m\rightarrow \infty\) and the capped off homeomorphism \(\widehat \Phi_N: S \rightarrow S\) is isotopic to the identity of \(S\). The existence of such homeomorphism is essentially given by \textit{K. Ichihara} and \textit{K. Motegi} [Tokyo J. Math. 28, No. 2, 527--538 (2005; Zbl 1100.37025)].NEWLINENEWLINEAs an application the authors prove that for any integers \(t\geq 1\) and \(m \geq 0\) with \(m \leq t+1\), there is a knot in \(S^3\) of tunnel number \(t\) and of meridional destabilizing number \(m\). The concept of meridional destabilizing number was introduced by the second author in [\textit{T. Saito}, Algebr. Geom. Topol. 11, No. 2, 1205--1242 (2011; Zbl 1221.57011)].