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THE COCHRAN SEQUENCES OF SEMI-BOUNDARY LINKS

For a 1-dimensional semi-boundary link, Cochran constructed a sequence of Sato-Levine invariants of successively derived links. This is a linear recurrence sequence and conversely any linear recurrence sequence can be constructed in this way. An upper bound for the growth of this sequence is obtained. 1. Introduction. An ordered pair L = (M, K) of oriented circles which are disjointly and smoothly embedded in S 3 is called a semiboundary link if the linking number lk(M, K) is equal to zero [Sa]. If L = (M9 K) is a semi-boundary link, then there exist Seifert surfaces V of M and W of K which intersect only in the interiors and transversely. We call (V, W) a Seifert pair for L. The orientations on V and W 9 which induce the orientation of L on the boundaries, determine a normal 2-frame field on V n W. The Sato-Levine invariant β(L) of the link L is the homotopy class in π-$S2 = Z which is represented by the Thom-Pontryagin construction on the 2-frame field. It is proven in [Sa] that this integer is independent of the choice of the Seifert surfaces and that it is a linkconcordance invariant. The following well-known proposition is included for completeness. PROPOSITION 1.1. If L = (M, K) is a semi-boundary link, then (a) β(L) = 0 if L is a boundary link.