Laver’s results and low-dimensional topology

Abstract

Abstract. In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.Richard Laver established two remarkable results that might lead to significant applications in low-dimensional topology, namely the existence of a series of finite structures satisfying the left-selfdistributive law, now known as the Laver tables, and the well-foundedness of the standard ordering of Artin's positive braids. In this text, we shall explain the precise meaning of these results and discuss their (past or future) applications in topology. In one word, the current situation is that, although the depth of Laver's results is not questionable, few topological applications have been found. However, the example of braid groups orderability shows that, once initial obstructions are solved, topological applications of algebraic results involving selfdistributivity can be found; the situation with Laver tables is presumably similar, and the only reason explaining why so few applications are known is that no serious attempt has been made so far, mainly because the results themselves remain widely unknown in the topology community.Therefore this text is more a program than a report on existing results. Our aim is to provide a self-contained and accessible introduction to the subject, hopefully helping the algebraic and topological communities to better communicate. Most of the results mentioned below have already appeared in literature, a number of them even belonging to the folklore of their domain (whereas ignored outside of it). However, at least the observations about cocycles for Laver tables mentioned in Subsection 1.3 (and established in another paper) are new.Very naturally, the text comprises two sections, one devoted to Laver tables, and one devoted to the well-foundedness of the braid ordering. It should be noted that the above two topics (Laver tables, well-foundedness of the braid ordering) do not exhaust Laver's contributions to selfdistributive algebra and, from there, to potential topological applications: in particular, Laver constructed powerful tools for investigating free LD-structures, leading to applications of their own [68,69,70]. However, connections with topology are less obvious in these cases and we shall not develop them here (see the other articles in this volume).

Cohomology of finite monogenic self-distributive structures

“…However, the example of free shelves (which are conjecturally approximated by Laver tables) confirms that general shelf colorings may be adapted to arbitrary braids, yielding extremely strong invariants [Deh94,Deh00]. This led Patrick Dehornoy to launch a challenging project of developing braid-theoretic applications of Laver tables [Deh14]. As a first step, Dehornoy and the author [DL14] explicitly described Z k (A n , Z), B k (A n , Z), and H k (A n , Z) for k 3, revealing in particular rich combinatorics behind the 2-cocycles of the A n .…”

Applications of self-distributivity to Yang-Baxter operators and their cohomology

“…Observe that for the Burau representation, modding out the diagonal part (mentioned in Table 1) yields reduced Burau. For a discussion of potential braid-theoretic applications of Laver tables, see [DL14,Deh16]. The last row is the author's work in progress.…”

Classification of Connected Shelves with Applications to Positive Knots

“…Laver tables were introduced in 1995 by Richard Laver while investigating self-embedding in set theory. Recently they have been investigated from the topology point of view, see for example [12], which discusses the use of Laver tables in low-dimensional topology, and [11], which classifies 2-and 3-cocycles on Laver's tables.…”