Paper detail

Cobordism of disk knots

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, 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 Levine on cobordism of sphere knots, we define 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 find necessary and sufficient conditions to realize a Seifert matrix cobordism class among the disk knots corresponding to a fixed boundary knot, assuming the boundary knot has no middle-dimensional 2-torsion. This classification is performed by relating the Seifert matrix of a disk knot to its Blanchfield pairing and by establishing a close connection between this Blanchfield pairing and the Farber-Levine torsion pairing of the boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfield pairing will determine the boundary Farber-Levine pairing).