A new bridge index for links with trivial knot components

Let L = K-1 boolean OR K-2 be a 2-component link in the 3-sphere such that K-1 is a trivial knot. In this paper, we introduce a new bridge index, denoted by b(K1=1)([L]), for L. Roughly speaking, b(K1=1)([L]) is the minimum of the bridge numbers of the links ambient isotopic to L under the constraint that all of the bridge numbers of the components corresponding to K-1 are 1. We provide a lower bound estimate of b(K1-1)([L]) in the case when L is a non-split satellite link. By using this result, we show that for each integer n(>= 2), there exists a link L-n = K-1n boolean OR K-2n with K-1n a trivial knot such that b(K1n=1)([L-n]) - b([L-n]) = n - 1, where b([L-n]) is the bridge index of L-n.