New obstructions for doubly slicing knots

Taehee Kim

KNOTS in WASHINGTON XV (2nd Japan-USA Workshop in Knot Theory)
January 10-15, 2003
George Washington University and Johns Hopkins University
Washington, DC and Baltimore, MD, USA

Organizers
Kazuaki Kobayashi, Jozef H. Przytycki, Yongwu Rong, Shin-ichi Suzuki, Kouki Taniyama, Tatsuya Tsukamoto, Akira Yasuhara

Conference Homepage

New obstructions for doubly slicing knots
by
Taehee Kim
Indiana University, Bloomington

A knot is doubly slice if it is the intersection of a three sphere with a trivially embedded two sphere in a four sphere. The resulting knot splits the two sphere into two distinct slicing disks for the knot. Thus, the term "doubly slice". The definition goes back to Fox and Sumners in the 60's, who developed an initial obstruction theory. Although extensive efforts have been made to better understand this natural four manifold relation on knots, only elementary obstructions have been discovered.

In recent years, Cochran, Orr, and Teichner have gained a deeper understanding of classical topological knot concordance using von Neumann signatures and new Blanchfield duality pairings on knots. Similarly, we seek insights into doubly slice knots.

We develop a bi-sequence of new obstructions for a knot being doubly slice containing the classical obstructions as initial cases. Examples are constructed to illustrate the non-triviality of these obstructions at all levels. Here analytic signatures play a key role. Also we induce a bi-filtration of the double concordance group.

Date received: October 13, 2002