  We give an explicit geometric argument that Artin's braid group $B_n$ is
right-orderable. The construction is elementary, natural, and leads to a new,
effectively computable, canonical form for braids which we call left-consistent
canonical form. The left-consistent form of a braid which is positive
(respectively negative) in our order has consistently positive (respectively
negative) exponent in the smallest braid generator which occurs. It follows
that our ordering is identical to that of Dehornoy, constructed by very
different means, and we recover Dehornoy's main theorem that any braid can be
put into such a form using either positive or negative exponent in the smallest
generator but not both.
  Our definition of order is strongly connected with Mosher's normal form and
this leads to an algorithm to decide whether a given braid is positive,
trivial, or negative which is quadratic in the length of the braid word.

Geometric Topology