Generating the Free Group of Rank Two with Dynnikov Coordinates

Abstract

It is well known that if the geometric intersection number of two simple closed curves is at least two, then the Dehn twists about these curves generate a free group of rank two. In this paper, we consider a pair of intersecting standard curves in the three-punctured disk D3 and show that the corresponding Dehn twists generate a free group of rank two. This result is proved using a coordinate-based alternative approach formulated entirely in terms of Dynnikov coordinates, which allows the ping–pong dynamics providing a sufficient criterion for freeness to be seen explicitly