Dual Garside structure and reducibility of braids

Details Dual Garside structure and reducibility of braids Dual Garside structure and reducibility of braids

2011-05-18

arxiv.org/abs/1105.3675v2

Journal of Algebra 356 (2012) pp. 355-373

Mathematics

Group Theory

Scientific paper

10.1016/j.jalgebra.2012.01.022

Benardete, Gutierrez and Nitecki showed an important result which relates the geometrical properties of a braid, as a homeomorphism of the punctured disk, to its algebraic Garside-theoretical properties. Namely, they showed that if a braid sends a curve to another curve, then the image of this curve after each factor of the left normal form of the braid (with the classical Garside structure) is also standard. We provide a new simple, geometric proof of the result by Benardete-Gutierrez-Nitecki, which can be easily adapted to the case of the dual Garside structure of braid groups, with the appropriate definition of standard curves in the dual setting. This yields a new algorithm for determining the Nielsen-Thurston type of braids.

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