Knot adjacency and satellites. (Q1426489)

The \(n\)-triviality is a multiple unknotting crossing change operation as below, and relates to finite type knot invariants. The research in the paper under review is motivated by the following question: if a nontrivial satellite knot \(K\) is \(n\)-trivial is there a companion torus of \(K\) that is disjoint from all the crossing changes that exhibit \(K\) as \(n\)-trivial? It is shown that the answer is yes if the condition \(\lq\lq n\)-trivial'' is replaced by the stronger one \(\lq\lq n\)-adjacent to the unknot''. A generalized crossing of order \(q \in {\mathbb Z}\) of a knot diagram is a set \(C\) of \(q\) twist crossings on two strings of opposite orientations. A generalized crossing change substitutes \(-q\) twist crossings for \(C\). This can also be achieved by \(1/q\)-Dehn surgery on a crossing circle which is unknotted and linking the two strands. A knot \(K\) is called \(n\)-adjacent to the unknot if some diagram of \(K\) contains \(n\) disjoint generalized crossings such that changing of any \(0 < m \leq n\) of them yields the unknot. It is \(n\)-trivial if the condition \(\lq\lq n\) disjoint generalized crossings'' is replaced by \(\lq\lq n\) disjoint sets of crossings''. Precisely, the main theorem states that the crossing circles (\(n\)-trivializer) can be taken in the companion solid torus \(V\) and the Dehn surgeries unknot \(K\) in \(V\). It is also shown that \(2\)-bridge knots of the form \([2q_1, 2q_2]\) are the only genus one knots which are \(2\)-adjacent to the unknot.