A sequence of new bridge indices for links each of which has a trivial knot component

摘要

Let L = K_1∪ K_2 be a 2-component link in S~3 such that K_1 is a trivial knot. In this paper, we introduce for each na new bridge index b_K1=n([L]) of L called a constrained bridge index with respect to n-bridge K_1. Roughly speaking, b_K1=n([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 n. We give exact values bK1=n([L]) (n = 1, 2,...) for particular $L = K-1 cup K-mst $ constructed as follows: There exist unknotted solid tori V_1V_2? ? V_m in S~3, where the core of each V_j(j = 1, 2,..., m - 1) is a (1, _j)-torus knot (_j) in V_(j+1), such that K_1 is the core of V_1, and $K-mst $ is the core of the exterior of V_m.