A Generalization of the Levine-Tristram Link Invariant

Lawrence Smolinsky
1989 Transactions of the American Mathematical Society  
Invariants to w-component links are defined and are shown to be link cobordism invariants under certain conditions. Examples are given. In knot and link theory there are certain signature invariants which determine up to torsion a knots cobordism class. These invariants were defined by Tristram [11], Levine [7] and Milnor [8] and are known as the Levine-Tristram or p-signatures. This paper investigates a generalization of the Levine-Tristram signature. The invariant can be viewed as a function
more » ... ewed as a function which assigns «i-roots of unity, one for each link component, to an integer. This invariant is a cobordism invariant in the same sense the Levine-Tristram signature is. However, unlike the Levine-Tristram signature, it is not a weak cobordism invariant. Our approach is geometric along the lines of O. Ja. Viro's geometric interpretation of the Levine-Tristram signature [12] . We discuss the relation of this invariant to the invariant of boundary links defined by Cappell and Shaneson in [2] . Applications of this invariant to the computation of Casson-Gordon invariants has been given in [9 and 10]. The author is grateful to J. Levine for many helpful discussions and to the referee for pointing out two errors. Definitions. An m component link of dimension « or an w-link is an ordered collection of m disjoint smooth oriented submanifolds of S"+ , each of which is homeomorphic to S" . We assume « > 2 and our links will always be ordered. We will denote a link by (S"+2 ; L, , ... , Lm) when we wish to emphasize this point. We also write L = Lx U L2 U • • • U Lm to denote the link. Every link bounds an oriented manifold called a Seifert surface. If an mlink has an m component Seifert surface so that each component is bounded by exactly one link component then the link is a boundary link. A link is sliced if it bounds a disjoint union of disks in the (« + 3)-ball. A link is weakly sliced if it bounds a disk with punctures. An analysis of the obstructions for a link to be weakly sliced can be carried out for high dimensional links following the usual analysis for knots.
doi:10.2307/2001380 fatcat:ayif32xeajavtd7eoyvbnwr73m