Cobordism of disk knots

Abstract

We study cobordisms and cobordisms rel boundary of PL locally-∞at disk knots D ni2 ,! D n . Any two disk knots are cobordant if the cobordisms are not required to flx the boundary sphere knots , and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of J. Levine on the cobordism of sphere knots, we deflne disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cobordant rel boundary if and only if their disk knot Seifert matrices are algebraically cobordant. We also ask which algebraic cobordism classes can be realized given a flxed boundary knot and provide a complete classiflcation when the boundary knot has no 2-torsion in its middle-dimensional Alexander module. In the course of this classiflcation, we establish a close connection between the Blanchfleld pairing of a disk knot and the Farber-Levine torsion pairing of its boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfleld pairing will determine the boundary Farber-Levine pairing). In addition, we study the dependence of disk knot Seifert matrices on choices of Seifert surface, demonstrating that all such Seifert matrices are rationally S-equivalent, but not necessarily integrally S-equivalent. 2000 Mathematics Subject Classiflcation: Primary 57Q45; Secondary 57Q60, 11E39, 11E81