Tunnel number one genus one non-simple knots
A knot in the 3-sphere is said to be tunnel number one if there exists an arc attached to the knot at its endpoints so that the complement of a regular neighborhood of the resulting complex is a genus two handlebody. For a complex \(C\) in \(S^3\), \(N(C)\) means the regular neighborhood of \(C\) in \(S^3\), and \(E(C)\) means the exterior \(S^3- \operatorname{int} N(C)\). Using the work of \textit{K. Morimoto} and \textit{M. Sakuma} [Math. Ann. 289, No. 1, 143-167 (1991; Zbl 0697.57002)] on tunnel number one non-simple knots, the authors classify tunnel number one genus one non-simple knots as follows: Let \(K_0\) be a non-trivial knot \(T(p,q)\) of type \((p,q)\) in \(S^3\) and \(L=K_1 \cup L_2\) a 2-bridge link \(S(\alpha, \beta)\) of type \((\alpha, \beta)\) in \(S^3\) with \(\alpha \geq 4\). Then there is an orientation preserving homeomorphism \(f: E(K_2) \rightarrow N(K_0)\) which takes a meridian \(m_2 \in \partial E(K_2)\) of \(K_2\) to a regular fiber \(h \in \partial N(K_0) = \partial N(K_0)\) of the Seifert fibration of \(E(K_0)\). Denote the knot \(f(K_1) \in N(K_0) \in S^3\) by \(K(\alpha, \beta; p,q)\). Then a tunnel number one genus one non-simple knot \(K\) in \(S^3\) is \(K=K(8m,4m+1;p,q)\) for some \(m \neq 0\).
scientific article; zbMATH DE number 1340490