Toroidal Dehn surgery on hyperbolic knots and hitting number
Let \(K\) be a hyperbolic knot in the \(3\)-sphere \(S^3\). Suppose that \(r\)-Dehn surgery on \(K\) yields a \(3\)-manifold \(K(r)\) which contains an incompressible torus. Such a Dehn surgery is called a toroidal Dehn surgery. Since \(K\) is hyperbolic, the core \(K^*\) of the filled solid torus in \(K(r)\) must intersect incompressible tori nontrivially, and the \textit{hitting number} of the toroidal Dehn surgery \((K, r)\) defined as min\(\{|K^* \cap T| : T\;\text{is an incompressible torus in}\;K(r) \}\) is positive. It is known that if the hitting number is odd, then \(r = 0\) and \(K\) is of genus one [\textit{D. Gabai}, J. Differ. Geom. 26, 479--536 (1987; Zbl 0639.57008)], and hence the hitting number is forced to be one. Thus it is interesting to consider the realization of even positive integers as hitting numbers of toroidal Dehn surgeries. \textit{M. Eudave-Muñoz} [J. Knot Theory Ramifications 6, No.~5, 659--676 (1997; Zbl 0887.57014)] gave examples of hyperbolic knots having toroidal Dehn surgery with hitting number four. Furthermore, \textit{J. K. Osoinach jun.} [Manifolds obtained by Dehn surgery on infinitely many distinct knots in \(S^3\), dissertation, the University of Texas at Austin, (1998), cf. Topology 45, No.~4, 725--733 (2006; Zbl 1092.57018)] shows that there is no bound on hitting numbers by giving an infinite family of toroidal Dehn surgeries \((K_n, 0)\). In the paper under review, the author studies Osoinach's examples in detail and proves that the toroidal Dehn surgery \((K_n, 0)\) has the hitting number \(2n\), and thus, each positive even integer is realized as the hitting number of a toroidal Dehn surgery on a hyperbolic knot. This gives a complete answer to the realization problem of even integers as hitting numbers.
scientific article; zbMATH DE number 5641769
scientific article