Chord diagrams and the Mapping class group

The combinatorial description of surfaces in terms of triangulations, or equivalently the dual notion of marked fatgraphs, has been used to derive results concerning the mapping class groups of these surfaces. In particular, the Ptolemy groupoid, which is defined in terms of elementary moves on trivalent marked fatgraphs, gives a simple combinatorial infinite presentation of the mapping class group. In this talk, I will discuss a linear variation on this theme for surfaces with one boundary component, where attention is restricted to fatgraphs of the form of linear chord diagrams, and the elementary moves, called chord slides, generate an analogous groupoid. I will also discuss one advantage of this viewpoint, which is an algorithm called fatgraph Nielsen reduction, used to represent a given mapping class in terms of chord slides.