Jointly with the opening talk of Knots in Washington XLIII
(Introduction by Ali Eskandarian, Dean & Professor of Physics)
Title: Knot colorings by quandles and their animations
Speaker: Masahico Saito, University of South Florida
Abstract: A Fox coloring of a knot diagram is defined by assigning integers modulo n to arcs of the diagram with a certain condition at every crossing. The number of colorings is independent of the choice of a diagram, and is a knot-invariant. This idea leads to a concept of algebraic systems called quandles, that have self-distributive binary operations with few other conditions. Knot colorings are defined with quandles and yield knot invariants. This was further generalized to knot invariants called quandle cocycle invariants, incorporating ideas from quantum knot invariants and group cohomology. After a review of these concepts, we consider quandle cocycle invariants with matrix groups. A continuous family of knot colorings is represented by animations of polygons moving on the sphere. These animations will be presented.
Short bio: Masahico Saito received PhD from the University of Texas at Austin in 1990. After a few post-doc positions, he has been at University of South Florida in Tampa since1995. His research interests include knot theory, related algebraic structures, and applications to DNA recombination.