Triple linking numbers, Hopf invariants and integral formulas for 3-component links

Geometry Topology Seminar
Monday, October 26, 2009 - 14:00
1 hour (actually 50 minutes)
Skiles 269
Columbia University
To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link is the degree of its associated Gauss map from the 2-torus to the 2-sphere.In the case where the pairwise linking numbers are all zero, I will present an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.