We define quasi-Garside groups and prove a theorem about them parallel to Garside’s result on the word problem for the usual braid groups. We introduce a group which we call the braid group B of Zn, and which bears some resemblance to mapping class groups. We prove that B is a quasi-Garside group. We give a ‘small presentation’ for B.

Let g be a semisimple simply-laced Lie algebra of finite type. C an abelian categorical representation the quantum group Uq(g) categorifying integrable V. The Artin braid B acts on Db(C) by Rickard complexes, providing triangulated equivalenceΘw0:Db(Cμ)→Db(Cw0(μ)) where μ is weight V, and Θw0 positive lift longest element Weyl group. We prove that this equivalence t-exact up to shift when V iso...

Using the band representation of the 3-strand braid group, it is shown that the genus of 3-braid links can be read off their skein polynomial. Some applications are given, in particular a simple proof of Morton’s conjectured inequality and a condition to decide that some polynomials, like the one of 949, are not admitted by 3-braid links. Finally, alternating links of braid index 3 are classified.