The Dehornoy Order On Braid Group B_n And An Algorithm Of Handle Reduction

This artical mainly reviews the (left-)orderablility of braid group Bn, showing three methods to proof that Bn is a (left-) order able group.The (left-)orderablility of braid group Bn was got by Dehornoy at first, this artical minutely shows the way Dehornoy defines the left-invarant total order of Bn,namely Dehornoy order.Then presents the other two approachs to construct the order of braid groups,one is put forward by Wiest and four others, based on mapping class groups, and the other one is given by Thurston according to hyperbolic geometry.The order got through mapping class groups is the same as Dehornoy order. By using hyperbolic geometry we can get a family of orders which include Dehornoy order.During proofing that Dehornoy order is left-invarant total order,Dehornoy puts forward the theory of handle reduction and two algorithms "FullHRed" and" GreedyHRed" to compare braid words and give another way to solve the word problem on braid group.At last this artical gives a new algorithm to compare braid words,namely "BIHRed",based on Dehornoy’s theory of handle reduction, and compares these three algorithms briefly.