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 ) .