Greg Friedman

Abstract:   The cobordism theory of smooth high-dimensional sphere knots was completely solved (or at least turned into an algebraic problem) in the 1960s through the work of Kervaire, who showed that all even-dimensional knots are cobordant, and J. Levine, who showed that the cobordism type of an odd-dimensional knot is determined by its Seifert matrix. In fact, two odd-dimensional knots are cobordant if and only if their Seifert matrices satisfy a certain algebraic cobordism equivalence.

We study the cobordism theory of disk knots, i.e. PL locally-flat proper embeddings D^n-2 into Dⁿ. Disk knots are very closely related to sphere knots with point singularities, and our study is motivated by an attempt to understand singular knots in general. We study both cobordism of disk knots and cobordism rel boundary, holding the boundary sphere knot fixed. It is fairly easy to classify cobordisms except in the odd-dimensional rel boundary case. In this case, we introduce a disk knot analogue of the Seifert matrix and show that it also provides a classification. Further, we study the realization of matrix invariants given a fixed boundary knot. The question is settled in all cases except those for which the middle-dimensional Alexander module of the boundary knot has 2-torsion. The solution is provided by defining a Blanchfield pairing for disk knots and determining close ties with Farber-Levine torsion pairing of the boundary knot (in fact the former determines the latter under certain connectivity assumptions).

Here are some directions to Northeastern University. Lake Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street.