Meridional destabilizing number of knots
In this paper, the author introduces the meridional destabilizing number of a knot, \(md(K)\), and studies its behavior under connected sum of tunnel number one knots. Let \(K\) be a knot in a connected closed orientable 3-manifold \(M\). A Heegaard splitting \((V_1,V_2)\) of \(M\) is called a Heegaard splitting of \((M,K)\) if \(K\subset V_i\), say \(i=1\), and the exterior of \(K\) in \(V_1\) is a compression body. We denote by \(hg(K)\) the Heegaard genus of \((M,K)\). Note that \(t(K)=hg(K)-1\), where \(t(K)\) is the tunnel number of \(K\). The meridional destabilizing number of \(K\), denoted by \(md(K)\), is the maximal number \(m\) such that \((M,K)\) admits an \((hg(K)-m,m)\)-bridge decomposition. Suppose \(K\) is a knot in the 3-sphere. We describe \(K\in\mathcal K^{m}_{g}\) if \(hg(K)=g\) and \(md(K)=m\). Then, \(K\) is the trivial knot if and only if \(K\in\mathcal K^{1}_{1}\). We can divide tunnel number one knots into three families: \(\mathcal K^{2}_{2},\,\mathcal K^{1}_{2}\) and \(\mathcal K^{0}_{2}\). Knots in \(\mathcal K^{2}_{2}\) are non-trivial 2-bridge knots, those in \(\mathcal K^{1}_{2}\) are \((1,1)\)-knots which are not 2-bridge knots, and those in \(\mathcal K^{0}_{2}\) are the other tunnel number one knots. The following is the main theorem of this paper: Theorem. (1) If \(K_i\in\mathcal K^{1}_{2}\,(i=1,2,3)\), then \(K_1\sharp K_2\in \mathcal K^{1}_{3}\) and \(K_1\sharp K_2\sharp K_3\in\mathcal K^{1}_{4}\). (2) If \(K_j\in\mathcal K^{0}_{2}\,(j=1,2)\), then \(K_1\sharp K_2\in \mathcal K^{0}_{4}\) or \(\mathcal K^{1}_{4}\).
scientific article; zbMATH DE number 5899707
scientific article