An algorithm for finding parameters of tunnels
A knot \(K\) in \(S^3\) has tunnel number 1 if a 1-handle can be added to a regular neighbourhood of \(K\) in a way that results in an unknotted handlebody. Such a 1-handle is called a tunnel. \textit{S. Cho} and \textit{D. McCullough} [Geom. Topol. 13, No. 2, 769--815 (2009; Zbl 1191.57005)] associated a parameter to a tunnel number \(1\) knot. This parameter arises from the fact that every tunnel can be obtained from the unique tunnel of a trivial knot through the application of a unique sequence of cabling constructions, each of which is determined by a rational number. The parameters for all tunnels of 2-bridge knots and torus knots were given by \textit{S. Cho} and \textit{D. McCullough} in [loc. cit.] and [Algebr. Geom. Topol. 9, No. 1, 1--20 (2009; Zbl 1170.57005)], respectively. In the paper under review, an algorithm is given for computing the parameter of a given tunnel of a tunnel number 1 knot by using its Heegaard diagram. The author then provides an interesting application of his algorithm by showing that the twisted torus knot \(T(7,7,2)\) contains a non-(1,1)-tunnel. This provides an example of a knot with both a (1,1)-tunnel and a non-(1,1)-tunnel.
scientific article; zbMATH DE number 5934837
scientific article