The complete splitting number of a lassoed link

In this paper we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s−1, and less than or equal to r+split(K). In particular, we obtain from a knot by r-iterated component-lassoings an algebraically completely splittable link L with split(L)=r. Moreover, we construct a link L whose unlinking number is greater than split(L).