Alex Bene, USC

TITLE: Factoring mapping classes of once bordered surfaces into elementary moves

ABSTRACT: It has long been known that mapping classes of a surface can be decomposed into elementary diagonal flips on triangulations of the surface.  Dually, one can explore the mapping class group by elementary Whitehead moves on fatgraphs embedded in the surface, where the collection of all such moves comprises the so-called Ptolemy groupoid, an object of interest in various fields of topology and geometry.  While several algorithms have been proposed to determine a sequence of Whitehead moves representing a given mapping class, all are "topological" in that they rely on resolving intersections of curves and arcs in the surface.  In this talk, I will describe an "algebraic" algorithm which explicitly determines a sequence of Whitehead moves representing any mapping class for a once bordered surface, when that mapping class is given purely by its action on the fundamental group of the surface.  The proof relies on the combinatorics of a certain kind of fatgraphs called linear chord diagrams and the elementary chord slide moves on them.  In particular, we show that there exists a certain "energy function" on the set of all embedded linear chord diagrams, and the above mentioned algorithm is given by an energy reducing path.