Link Types under Twisting Solid Tori 103

The link types do not always vary under sorne coniposition of twist rnaps. For exaníple, the knot type 1< iii Figure 1 is invariant under fil> ~ fV í) • Then, when the link types vaHes? In the case of knots, the following theorení is given by M. Kouno, 1<. Motegi an(l 1. Shibuya ([KMS 1], [KMS2]). Theorem 1.1. Let 1< be u knot in S3, l~1 and i”2 salid tan witittite bonudanies being;nutuafly disjoint and essential in tite exterior of It’, ffl> un n-twist alang .Suppose K G 1/1 G V 2, wv,(K) =2 and wv2(careVi) =2. If(ni,n2) # (0,0), then fI~’~(K) ~ fh>(K). Here coreV denotes a core of V. Tbey also got sorne results about twisting of knots along a standard solid torus. Moreover, the author ([12]) showed Ube following theorern. Theorem 1.2. Let 1< be a knot iii S 3,V 1,...,Vk be salid tan witi