Generating the Free Group of Rank Two with Dynnikov Coordinates
CUMHURIYET SCIENCE JOURNAL, cilt.47, sa.2, ss.356-360, 2026 (TRDizin)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 47 Sayı: 2
- Basım Tarihi: 2026
- Doi Numarası: 10.17776/csj.1860028
- Dergi Adı: CUMHURIYET SCIENCE JOURNAL
- Derginin Tarandığı İndeksler: TR DİZİN (ULAKBİM)
- Sayfa Sayıları: ss.356-360
- TED Üniversitesi Adresli: Evet
Özet
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.