Generating the Free Group of Rank Two with Dynnikov Coordinates
CUMHURIYET SCIENCE JOURNAL, vol.47, no.2, pp.356-360, 2026 (TRDizin)
- Publication Type: Article / Article
- Volume: 47 Issue: 2
- Publication Date: 2026
- Doi Number: 10.17776/csj.1860028
- Journal Name: CUMHURIYET SCIENCE JOURNAL
- Journal Indexes: TR DİZİN (ULAKBİM)
- Page Numbers: pp.356-360
- TED University Affiliated: Yes
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.