A note on tunnel number of composite knots
For a knot \(K\) in the \(3\)-sphere, its tunnel number \(t(K)\) is essentially the Heegaard genus of the knot exterior. There are various results about the behavior of tunnel number under the connected sum operation between knots. It is easy to see that \(t(K_1\sharp K_2)\leq t(K_1)+t(K_2)+1\) for two knots \(K_1\) and \(K_2\). In fact, \textit{T. Kobayashi} [J. Knot Theory Ramifications 3, No.~2, 179--186 (1994; Zbl 0818.57006)] proved that the gap between \(t(K_1\sharp K_2)\) and \(t(K_1)+t(K_2)\) can be arbitrarily large. On the other hand, the equality \(t(K_1\sharp K_2)=t(K_1)+t(K_2)+1\) holds for various cases. For example, \textit{F. Li, G. Yang} and \textit{F. Lei} [Topology Appl. 157, No.~7, 1188--1194 (2010; Zbl 1189.57020)] showed that if the distance \(d(K_i)\) is at least \(2t(K_i)+5\) for \(i=1,2\), then the above equality holds. Here, the distance \(d(K)\) is defined to be the minimal distance among all the minimal Heegaard splittings of the knot exterior of a knot \(K\). The main result of the paper under review is to give a new sufficient condition for the equality. More precisely, if \(d(K_1)\geq 3\) and \(d(K_2)\geq 2(t(K_1)+t(K_2)+2)\), then \(t(K_1\sharp K_2)=t(K_1)+t(K_2)+1\). Editor's remark: the erratum does not concern the contents of the present article but only the first and second author's affiliations.
scientific article; zbMATH DE number 5956074
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