A specious unlinking strategy

We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components. The unlinking number u(L) of a link L in S is the minimal number of crossing changes required to turn a diagram of L into a diagram of the unlink. Here we take the minimum over all diagrams of L. Similarly, the splitting number sp(L) is the minimal number of crossing changes required to turn a diagram of L into a diagram of a split link. Once again the minimum is taken over all diagrams of L. Here recall that an m-component link L is split if there are m disjoint balls in S, each of which contains a component of L. The detailed study of unlinking numbers and splitting numbers of links was initiated by Kohn [Ko91, Ko93] in the early 1990s, and was continued by several other researchers, see e.g. [Ad96, Ka96, Sh12, BS13, Ka13, CFP13]. See also [BW84, Tr88] for some early work. Somewhat to our surprise, these are still relatively unstudied topics, and many basic questions remain unanswered. In our investigations we wondered whether the computation of unlinking numbers can be separated into the two problems of splitting links and unknotting of knots. More precisely, the following question arose. Question 1. Is the following always the most efficient strategy for unlinking? First split the link with the minimal number of crossing changes, and then unknot the resulting knots. In this short note we will give a negative answer to the above question. More precisely, we have the following theorem. Theorem 2. The link L in Figure 1 has the property that any sequence of crossing changes for which the initial sp(L) crossing changes turn L into a split link S, and where the remaining crossing changes unknot the components of S, has length greater than u(L). The remainder of this note comprises the demonstration of this theorem. First we 2010 Mathematics Subject Classification. 57M25, 57M27.