Braid Order, Sets, and Knots

We survey two of the many aspects of the standard braid order, namely its set theoretical roots, and the known connections with knot theory, including results by Netsvetaev, Malyutin, and Ito, and very recent work in progress by Fromentin and Gebhardt. It has been known since 1992 [7, 8] that Artin’s braid groups Bn are leftorderable, by an ordering that has several remarkable properties. In particular, it was proved in [23] that its restriction to the braid monoid B n is a well-ordering. Many subsequent results were established using different approaches, and the subject has developed so as to become the whole content of the monograph [11]. The first main point is that many different approaches, some of them algebraic or combinatorial, others geometrical of topological, developed by a number of researchers, in particular Burckel, Dynnikov, Fenn, Fromentin, Funk, Greene, Larue, Rolfsen, Rourke, Short, Wiest, and the author, lead to one and the same distinguished ordering on braid groups, the so-called Dehornoy ordering, hereafter referred as the D-ordering. The second main point is that the family of all left-invariant braid orderings turns out to be an interesting space in which the D-ordering plays a significant role, as shown in works by Clay, Ito, Navas, Rolfsen, Short, Wiest. The purpose of the current survey paper is not to repeat the material that is developed in [11], nor even to give a comprehensive introduction to that text, but rather to point out some aspects that are not, or not fully, described there, namely the set-theoretical roots of the D-ordering, as well as its known connections with knot theory. These aspects are alluded to in Sections III.2 and IV.5 of [11], but the latter is a quite short introduction, whereas the former is already obsolete due to several recent developments.