Generalized \(\#\)-unknotting operations (Q1389857)

In this note certain crossing change operations on knot diagrams are considered. Take the usual crossing change operation but replace each of the two strands by a number of parallel strands. If the two collections each number, algebraically, a multiple of \(p\), it is called an \(\#_p\)-move. For \(p=1, 2\), this has been considered earlier. The main results of this paper are: 1. Any knot can be unknotted by a succession of \(\#_p\)-moves. Then, if the \(\#_p\)-unknotting number of a knot is defined in the obvious way: 2. For any \(p\), there are knots with arbitrarily large \(\#_p\)-unknotting number. 3. There is a knot whose \(\#_p\)-unknotting number is \(>1\) for every \(p\). Finally it is shown that a \(2\)-component link in the boundary of \(P\), a punctured \(S^2\times S^2\), does not necessarily bound disjoint imbedded disks in \(P\), as contrasted with the case of a knot in \(\partial P\) which always bounds an imbedded disk in \(P\).