Colóquio 

03/10/2000, 17:00 — 18:00 — Anfiteatro Va5, Pavilhão de Eng. Civil
Louis B. Kauffman, University of Illinois, Chicago

Virtual Knot Theory

Virtual Knot theory is to classical knot theory as all graphs are to planar graphs. In virtual knot theory one studies Gauss codes representing "knots" that have an abstract existence, but require virtual crossings when drawn in the plane. Many new phenomena appear in this generalization of classical knot theory. The Jones polynomial, quantum link invariants and Vassiliev invariants generalize to virtuals as does the fundamental group, rack and quandle. There are non-trivial virtual knots with trivial Jones polynomial. There are non-trivial virtual knots with non-trivial Jones polynomial, but with the infinite cyclic fundamental group. A combinatorial theory of flat virtuals is interesting in its own right. Virtual braids can be analyzed and, surprisingly, virtuals have applications not only to knots and links in thickened surfaces but also to the homotopy theory of infinite loop spaces and to the embeddings of surfaces in four dimensional space.